Wittgenstein s Notes on Logic Michael Potter (1)

Wittgenstein’s Notes on Logic

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Wittgenstein’s Notes on Logic

Michael Potter

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One often makes a remark and only sees later how true it is.

Wittgenstein, 10 October 1914

Preface

I wrote the first draft of this book while I was a Senior Research Fellow inthe Department of Philosophy at Stirling University, funded by the AHRC.The friendly welcome I received there, in a department which I came to seeas an oasis of academic seriousness and respect, contributed very markedly tomy well being and hence to the successful completion of the project. Suchconducive research environments are rare; I suspect that university adminis-trators who recognize them when they arise, and cherish them as they should,are even rarer.A happy feature of the AHRC project was a series of workshops and a con-

cluding conference on the Tractatus and its history, at various of which I triedout some of the ideas expressed here. I have pleasant memories of the con-structive atmosphere at these meetings, and am grateful for all the feedback Ireceived, even when it was no more articulate than a raised eyebrow. Some-thing similar goes for a series of seminars on the first draft of the book which Iconducted in Cambridge in the Lent Term, 2006. I am sure the final versionis better as a result.In the archival research for the book I have been assisted by librarians at

the Houghton Library, Harvard University; the Russell Archive, McMasterUniversity; and the Manuscripts Room, Cambridge University Library. I amgrateful to Blackwell Publishing for permission to reprint the Notes on Logic asan appendix. In compiling the version printed here I was grateful for ear-lier editorial work by Michael Biggs. Will Crouch compiled the index forme. Nicholas Griffin, Stephen Read, David Cardwell, and Brian McGuin-ness have responded helpfully to requests for information. I have also bene-fited from detailed comments on the first draft by some unusually generouscolleagues. I am very grateful to Ian Proops, Peter Sullivan, and two anony-mous readers for Oxford University Press, not least because each of them hassaved me from a number of embarrassing displays of my own ignorance andstupidity.Wittgenstein scholars who read this book will, of course, look first for their

own names in the index. Most will, I fear, come away disappointed. If I hadfootnoted everyone I have read, and explained in full what I thought of their

views, the book would have been twice as long and twice as late, but not, Isuspect, twice as useful. My gratitude for what I have gained from their workis no less sincere for remaining unitemized. To Brian McGuinness, however,I owe a particular debt which it would be churlish not to single out: his edito-rial work, most notably on Wittgenstein’s letters, has saved me an enormousamount of time; and his writings on Wittgenstein display a combination ofhistorical accuracy and philosophical acuity which I can only dream of emu-lating.It was Peter Sullivan who arranged the AHRC project in the first place and

hence created the circumstances in which it was possible for me to write thisbook. I must have discussed almost every idea in it with him at some point,and many were no doubt originally his. Since neither of us now knows whichthese are, and some of those which are certainly his he now prefers to deny,this preface is the only feasible place to record that debt.

MDP

Contents

Contents vii

Introduction 1

1 Finding a problem 6

1.1 Early life 61.2 Manchester 71.3 The Principles of Mathematics 101.4 Logicism 131.5 Russell’s paradox 15

2 First steps 18

2.1 Cambridge 182.2 On denoting 202.3 Sense-data 23

3 Matter 26

3.1 The project 263.2 On matter 273.3 Dawes Hicks 303.4 The relation of sense-data to physics 333.5 The atomistic assumption 36

4 Analysis 39

4.1 Inference or construction? 394.2 Wittgenstein’s conception 434.3 Practicalities 45

viii Contents

5 The fundamental thought 49

5.1 Why logic? 495.2 Logical constants as incomplete symbols 515.3 There are no logical constants 525.4 There are no real variables 545.5 Logic as a special science 575.6 Logic as contentless 585.7 The fundamental thought 61

6 The symbolic turn 63

6.1 Propositions 636.2 The rejection of psychologism 646.3 The reliability of language 666.4 Conflicting conceptions 68

7 Simplicity 70

7.1 Realism 707.2 Solipsism 737.3 Idealism 747.4 Reconciliation 76

8 Unity 78

8.1 The copula 788.2 There cannot be different types of things 808.3 The theory of types is superfluous 82

9 Fregean propositions 86

9.1 Frege’s notion of assertion 869.2 Propositions are not names of truth-values 889.3 Whose influence? 909.4 Propositions as articulate 91

10 Assertion 94

10.1 The judgment stroke as force indicator 9410.2 Asserted and unasserted propositions 9510.3 Assertion as psychological 9810.4 Psychology 99

11 Complex and fact 102

11.1 A world of facts, not of things 10211.2 Influences 10511.3 Russell on facts 106

Contents ix

12 Forms 109

12.1 Form as name 10912.2 Form as function 11112.3 The form of a fact 11312.4 The unity of the proposition 11412.5 The symbolic turn again 116

13 Russell’s theory of judgment 118

13.1 The original multiple relation theory 11813.2 A problem for the original theory 12013.3 Russell’s revised theory 12213.4 Wittgenstein’s further objection 12413.5 Acquaintance 12513.6 Another formulation 12613.7 The fate of the multiple relation theory 12813.8 Other accounts 129

14 Meaning 132

14.1 Russell’s lectures on logical atomism 13314.2 Propositions are not names of their meanings 13514.3 Meanings as facts 13714.4 The demise of propositional meaning 140

15 Metaphysics 142

15.1 Disjunctive facts 14215.2 Negative facts 14315.3 Summing facts 14515.4 General facts 14715.5 Logical data 149

16 Sense 151

16.1 Semantic value 15116.2 The semantic value of a form 15316.3 The compass-needle analogy 15516.4 Grain 156

17 Truth-functions 158

17.1 Using primitive signs 15817.2 Truth-tables 16017.3 Truth-diagrams 16017.4 Comparison 163

x Contents

18 Truth-operations 165

18.1 The problem 16518.2 The solution 16618.3 Duality 168

19 Molecular propositions 170

19.1 Terminology 17019.2 Which fact? 17119.3 Poles 17319.4 The inputs 174

20 Generality 177

20.1 Variables as classes of propositions 17720.2 Notation 18020.3 Undecidability 181

21 Resolving the paradoxes 184

21.1 Russell’s theory of types 18421.2 Wittgenstein’s vicious circle principle 18621.3 Types as classes of propositions 18721.4 Types and molecular propositions 18921.5 Types and generality 19021.6 Uniting generality and truth-functions 19121.7 The general form of proposition 19221.8 Unsayability 193

22 Typical ambiguity 196

22.1 Typical ambiguity 19622.2 Independent indefinables 19922.3 Whitehead 200

23 Identity 204

23.1 Russell’s definition 20423.2 Eliminating identity 20623.3 The notational problem 207

24 Sign and symbol 209

24.1 Seeing through to the symbol 20924.2 Same sign, different symbol 21024.3 Same symbol, different sign 21224.4 Symbol in terms of sign 21424.5 The symbol vanishes 216

Contents xi

25 Wittgenstein’s theory of judgment 218

25.1 Russell’s later views 21825.2 The theory of judgment in the Notes 21925.3 Wittgenstein’s later theory of judgment 22025.4 Ramsey 222

26 The picture theory 224

26.1 Coincidence of structure 22426.2 The picturing analogy 22626.3 Truth 22726.4 The identity theory 22926.5 Possibility 231

27 Tractarian objects 232

27.1 Relations as objects 23227.2 Widening the scope 23327.3 Facts in the Tractatus 23627.4 Confusion? 237

28 Philosophy 241

28.1 Metaphysics 24128.2 Psychology 24328.3 Epistemology 24428.4 Value 245

29 Themes 249

29.1 Working methods 24929.2 Characteristics 25029.3 What if 25329.4 Fundamental thoughts 25429.5 Influences on Wittgenstein 25529.6 Influence on Russell 25929.7 Influence on Frege 26029.8 Conclusion 261

A History of the text 263

A.1 Narrative 263A.2 The manuscripts 265A.3 Russell’s labelling 268A.4 The Costello version 271A.5 Wittgenstein’s dissertation 274A.6 Conclusion 274

xii Contents

B The Notes on Logic 276

The Birmingham Notes 276The Cambridge Notes 284Textual notes 290The Costello version 292

Citations 297

Notes on Logic 297Tractatus 297

Index 299

Bibliography 305

Primary sources 305Secondary sources 306

Introduction

In 1911 Wittgenstein arrived in Cambridge to study philosophy with Russell.In 1913, just before he left to spend a year communing with his soul in Nor-way, he produced the Notes on Logic, a sort of summary of the conclusions hehad reached during his time in Cambridge, and his first philosophical work.My intention in what follows is to engage in a study of that period and thiswork.Most of Wittgenstein’s surviving pre-Tractarian writings were published, in

stages, some time ago (in 1957, 1961, and 1979). Since then it has been a fa-miliar method of all serious exegetes of the Tractatus to mine these writings forremarks to support their interpretations. One reason they do this is simply be-cause they can: although Wittgenstein had his prewar Cambridge notebooksdestroyed, and two of his later notebooks are probably missing, what remains(the Notes on Logic already mentioned, the Notes dictated to Moore, three survivingwartime notebooks, and a handful of letters) is a body of evidence of a scalenicely poised to intrigue but not intimidate the diligent scholar. But it is alsobecause they must: some of the remarks in the Tractatus are so obscure thatonly by relying on the earlier notebooks for support do we have any hope ofdivining their meaning.Although I hope in this book to contribute to the same project of Tractarian

exegesis, I aim to do so by a somewhat different method. Instead of studyingthe Tractatus, and drawing on Wittgenstein’s earlier writings only when theycontribute to understanding it, I shall here be focusing on the 1913 Notes onLogic, treating them if not quite as a terminus in Wittgenstein’s work then atleast as worthy of study in their own right.There are several benefits to be derived from this approach. One is that it

gives us a better chance of understanding Wittgenstein’s own reasons for someof the views he held. This is probably a good reason to study the early works ofalmost any major philosopher, but it is especially so with Wittgenstein, whoseown arguments for his views are so often too compressed to be comprehensiblewithout understanding the context in which he formulated them. An obviousexample is that commentators have presented a remarkably large number ofWittgenstein’s logical claims as consequences of his picture theory. But not all

! Introduction

of that theory is present in the Notes. In particular, the analogy with pictureswhich gives the theory its name came to him almost a year after he compiledthem. Understanding the parts of the theory which were already in placeby then puts us in a much better place to understand what further work hethought there was for the picturing analogy to contribute.Another benefit has to do with Wittgenstein’s method of working, by which

remarks were first written down, then compiled and rearranged (almost end-lessly in the case of some of his later work). These rearrangements sometimesgave the remarks, in the resulting juxtapositions, resonances which they didnot originally have. However allusive these resonances may be, however pos-sible it is that Wittgenstein may sometimes have been struck by them and usedhis final arrangement of the text to invite them, the fact remains that it is likelyto be worthwhile to study the remarks in their original setting.But I also hope here to recommend, by exemplification, an approach to

the study of the Tractatus that, if not actually denying the boundary betweenbiography and philosophy, at least regards the interaction between them aspotentially fruitful. I do not apologize for introducing biographical observa-tions into what is primarily a philosophical study; and the reason is that I havefound it often helps, in judging which interpretation of Wittgenstein’s remarksis plausible, to have a sense of how he thought and how he worked. If we areto gain the maximum insight from his work, we need to understand, certainly,what motivated him to address the problem he did in the way that he did.But the exegetical task of understanding him aright can at times seem harderwith Wittgenstein than with some other philosophers. (Russell is an obviousexample.) One thing that makes it easier, though, is the striking unity, if not inWittgenstein’s thought then in his method of thought. Almost all of his ideasare, in a certain sense, simple. Once we have grasped the sort of simplicitythat is in question, it can then become a useful measure by which to assess ourinterpretations in the future. And it is here that an understanding of the manis relevant.Wittgenstein wrote the Tractatus during the First World War, of course, but

it had its birth in the two years he spent working in Cambridge with Russellbetween 1911 and 1913. He compiled the Notes on Logic at the very end of thatperiod, as a summary for Russell (and perhaps, to an extent we shall discusslater, for himself) of the work he had accomplished. The destruction of hisnotebooks, mentioned earlier, makes the Notes almost our only guide to thework he had been doing in Cambridge. By studying them, therefore, we canhope to discover which of his ideasWittgenstein owed to this period and whichto the very different circumstances in which he worked later, first in Norwayand then on active service during the war.We can also hope to lay bare some of the influences which helped to form

his views. The acknowledgment Wittgenstein made in the preface to the Trac-

Introduction #

tatus to ‘the great works of Frege and the writings of my friend Bertrand Rus-sell’ is endlessly quoted. So, too, is a diary entry from 1931 in which he listedFrege and Russell (along with others) as people whose ideas he had used inwhat he called his ‘work of clarification’.1 But there is some gap between ac-knowledging an influence and determining what that influence was; and inany case we are by no means compelled to assume that Wittgenstein was con-scious of, and chose to acknowledge, all the influences that shaped his think-ing. Once again, though, the benefits will not just be biographical: knowingwhere he got his ideas from is often a useful tool for understanding what thoseideas really were.Wittgenstein’s writings have been worked over so thoroughly in the half

century since his death that the lack until now of any book-length study ofthe Notes on Logic is rather remarkable in itself. This is especially so when onepays attention to the significance of their timing, just highlighted. One rea-son for this neglect may be that the Notes are, even by Wittgenstein’s gnomicstandards, hard to understand on a first reading. If they were, to some extentat least, compiled only for Russell’s benefit (or for Wittgenstein’s own), thenthat is of course part of the explanation. But another part of the explana-tion lies in the rather complicated circumstances of their composition, whichhave furnished us with a text (or rather a series of texts) whose convoluted andrepetitive structure has compounded the difficulties in comprehension. Animportant aspect of this book, therefore, will be to disentangle these texts inorder to leave the way to philosophical understanding of Wittgenstein’s inten-tions much clearer.That historical detective work will be the subject of Appendix A, and the

Notes on Logic themselves are reprinted with a critical apparatus as Appendix B.I use Bn and Cn to mean the nth paragraphs of the Birmingham and Cam-bridge versions of the Notes respectively. (Decimal numbers unqualified are, ofcourse, references to the Tractatus.) The bulk of the book itself is taken up withexegesis—not, certainly, of every sentence of the Notes, but at least of what Itake to be their central claims. The aim will be to show that, once the prob-lematic structure of the surviving text is disentangled, the Notes are a muchmore coherent and substantial work than has hitherto been recognized.The reader will no doubt notice, however, that the Notes on Logic are not,

to begin with, mentioned very often. The reason for this lies in the fact thatI have attempted here to describe not only the contents of these Notes, nar-rowly conceived, but the whole of Wittgenstein’s period working with Russellin Cambridge. Just what part of that period Wittgenstein saw the Notes assummarizing is now hard to determine—I shall discuss this question furtherin §11.3—but even if he viewed them as a report on all of his discoveries up tothat point which he thought worth preserving, it would be natural for many

1CV, 19.

$ Introduction

more of those discoveries to have occurred to him in the second year of studythan in the first: that first year was, after all, when he came to Cambridgeas, in effect, a self-taught philosophical novice. In relation to his first year ofstudy, therefore, our evidence concerning what Wittgenstein was thinking ismuch more conjectural: there is little hard evidence apart from a couple ofletters to Russell. The aim of the first half dozen chapters, therefore, will beto make those conjectures that seem possible concerning the work Wittgen-stein did before what is in the Notes on Logic. In practice, this will involve us toa considerable extent at first in examining Russell’s work during this period.Such an examination would be appropriate in any case, since it constitutes thecontext in which Wittgenstein was working. But the closeness and complexityof the working relationship between Wittgenstein and Russell makes the lat-ter’s work during this period of more than usual importance. It is clear that formuch of this period they saw themselves as collaborators in a common project.So Russell’s writings can give us significant clues to what Wittgenstein’s ownviews were.In my discussions of Wittgenstein’s and Russell’s work I shall generally

adopt their logical notation. In particular, the reader needs to be familiarwith the following:

!p not-pp v q p or qp q p and qp ! q If p then q (in Russell’s dubious idiolect: ‘p implies q’)(x) !(x) For all x, !(x) (Russell: ‘!(x) is always true’)(

E

x) !(x) For some x, !(x) (Russell: ‘!(x) is sometimes true’)

Russell usually used dots rather than brackets to indicate scope, but read-ers not at home with this convention can probably let the sense carry themthrough. It is also worth emphasizing at the outset that Wittgenstein did notshow much sign of wanting Quine’s distinction between use and mention.One of the emerging themes of the book will be what Wittgenstein owed toFrege and what to Russell, but one thing which a casual inspection of theNotes on Logic tells us is that he was not inclined while he was compiling themtowards Frege’s pedantry in the use of quotation marks: for Wittgenstein theyare sometimes a naming device, but sometimes no more than a form of paren-thesis. And Wittgenstein’s propositional letters ‘p’, ‘q’, etc. are sometimesschematic, sometimes not. For instance, he plainly intended

‘p’ is true if and only if p

as a schema to stand for

‘Snow is white’ is true if and only if snow is white

Introduction %

and similar propositions. I shall take that as a general licence, in expoundingWittgenstein’s thought, to be no prissier than he was about the use–mentiondistinction, except occasionally when it seems to matter.

Chapter !!!

Finding a problem

What led Wittgenstein to study philosophy with Russell in Cambridge? Thenarrative of Wittgenstein’s life before 1911 is well summarized in the availablebiographies,1 so I shall confine myself in this chapter to picking out a fewpoints that deserve emphasis in the current context.

!.!!.!!.! Early lifeWittgenstein’s father, a steel magnate, was one of the richest men in Austria.He was not an aristocrat, but Wittgenstein evidently acquired in his youthmany of the attitudes of the rich. (One example, perhaps, is the tendency hehad in the early part of his life to dabble in various fields, a tendency whichdisplays a sort of enthusiastic amateurism sometimes to be observed in theindependently wealthy.) Between the ages of 14 and 17 Wittgenstein was ed-ucated at the Oberrealschule in Linz. This was, notoriously, the same school asHitler, but they overlapped for only one year (1903–4), during which Hitlerwas in class IIIA and Wittgenstein in class V, and there is no reason to thinkthat Hitler influenced Wittgenstein’s philosophy any more than that Wittgen-stein influenced Hitler’s anti-Semitism. Wittgenstein’s Jewishness is in anycase not a prominent theme in accounts of his early life: all four of his grand-parents were baptized and, although he was no doubt aware of his Jewishroots, I know of no reason to think that he became at all self-conscious aboutthem until much later. (Some of the remarks his acquaintances made abouthim when he came to Cambridge display casual amusement at the oddity offoreigners, but I have come across none that allude to his Jewishness.)2

In 1906, when he was seventeen, Wittgenstein went to Germany to spendthree semesters studying engineering at the Technische Hochschule in Charlot-tenburg (a suburb of Berlin). Stories from his later life attest to his fascinationwith how things work, his capacity for spatial reasoning, and his ability tomend quite complicated pieces of machinery. Although this may have beento some extent a natural talent, the training at the Technische Hochschule, whichemphasized the practical over the abstract, was specifically designed to foster1McGuinness, Young Ludwig; Monk, The Duty of Genius. 2Cf. McGuinness, ‘Wittgenstein and theidea of Jewishness’.

Manchester &

it. On the other hand, the mathematical component of the training was ratherlimited: higher mathematics (i.e. differential and integral calculus and analyticgeometry) in the first year of the course, and mechanics in both years. In addi-tion there were what the course timetable3 describes as descriptive geometryand graphical statics, but these will have been practical courses, more techni-cal drawing than anything that we would nowadays regard as mathematics.This point bears emphasis, if only because some writers on Wittgenstein’sphilosophy have overestimated his mathematical knowledge—assuming, per-haps, that his training as an engineer included a substantial mathematicaleducation.Evidence that Wittgenstein made some sort of effort to extend his math-

ematical knowledge has survived in his copy of the German translation ofLamb’s Hydrodynamics, which he presumably bought in Berlin around thistime. What is curious about the volume is the markings Wittgenstein made inthe margins—not so much the fact that they occur only in the first four of thetwelve chapters of the book (although that does perhaps suggest that his inter-est was that of a dilettante rather than a serious student), but the nature of thecomments themselves. Apart from a few corrections of obvious misprints inthe text, Wittgenstein’s marginalia are almost all rewordings that he seems tohave regarded as stylistic improvements.4 One might, of course, see this as anearly sign of Wittgenstein’s later deliberate interest in language and precise-ness of expression, but there is something else about them too: the impressionone has is almost that Wittgenstein was not really interested in the mathemat-ics at all; or, if he was, one would like to have been able to explain to him thatthis was not the right way to go about studying a mathematical text.

!."!."!." ManchesterIn 1908, after a short period constructing kites for meteorological research,Wittgenstein became a research student at Manchester. The university hewas joining was one of the leading scientific research institutions in the world.It seems that he originally hoped to work with Rutherford, the professor ofphysics, who had just been awarded the Nobel prize; Chadwick (later towin the Nobel prize for discovering the neutron) was Rutherford’s assistant;Geiger and Marsden were performing their famous experiments on the scat-tering of alpha particles; De Hevesy, who joined the department two yearslater, would receive the Nobel prize for his work on the use of isotopes astracers. The chemistry department, unquestionably the finest in Britain, con-tained organic chemists such as Perkin, Haworth (who first synthesized vita-min C) and Robinson. In the event, though, Wittgenstein ended up working3See Hamilton, ‘Wittgenstein and the mind’s eye’. 4Spielt and McGuinness, ‘Marginalia inWittgenstein’s copy of Lamb’s Hydrodynamics’.

' Finding a problem

not with Rutherford but at the engineering laboratory, whose head, newly ap-pointed in 1908, was Petavel, inventor of a device for measuring variations inpressure caused by exploding gases and later director of the National PhysicalLaboratory.Manchester’s strength was not confined entirely to the experimental sci-

ences. In the mathematics department were Littlewood, one of the best math-ematical analysts in the world, and Lamb. The presence in Manchester ofLamb, whose book Wittgenstein had been so idiosyncratically studying, maywell have been one of the things that attracted Wittgenstein there. When hearrived, Wittgenstein lost no time in approaching Lamb with questions aboutsome equations he had devised. As he related it to his sister, Lamb

will try to solve the equations that I came up with and which I showed him. He saidhe didn’t know for certain whether they are altogether solvable with today’s methodsand so I am eagerly awaiting the outcome of his attempts.5

Perhaps this seemed (or was intended to seem) impressive toWittgenstein’s sis-ter: he had come up with some equations which one of the foremost appliedmathematicians of the day did not know how to solve! But of course what ev-ery applied mathematician knows is that devising equations one cannot solveis easy. The hard part is to model a system in such a way that the resultingequations are soluble.His meeting with Lamb evidently mattered greatly to Wittgenstein at the

time: his letter to his sister makes plain his extreme state of nervous tensionthroughout the day on which he made it. As things turned out, however,Lamb did not play the central role in his life that Wittgenstein seems to havehoped for; and it is hard to avoid the suspicion that what Wittgenstein was re-porting was really a polite brush-off from a professor confronted with a some-what eccentric student and some rather curious equations.In his first year at Manchester Wittgenstein at least began to attend Little-

wood’s lectures on mathematical analysis, but we do not know how long hecarried on. What we do know is that early in his time at Manchester Witt-genstein became interested in the philosophy of mathematics and, after threeyears at Manchester, decided to go to Cambridge to study with Russell. Quitehow this interest in the philosophy of mathematics arose is something of amystery, however. The account offered by McGuinness6 would be hard toimprove on, both as a summary of what is known and as a caution againstspeculation that goes beyond it.Some commentators have presented Wittgenstein’s interest in the philos-

ophy of mathematics as flowing naturally from the mathematics he was en-gaged in, but I am sceptical about this since, as we have just seen, he had not

5To Hermione Wittgenstein, Oct. 1908. 6Young Ludwig, ch. 3.

Manchester (

Wittgenstein’s patented jet rotor

really done very much mathematics by this stage in his life. It has been com-mon, perhaps in an attempt to make this account more plausible, to presentWittgenstein’s research in Manchester as having a strong mathematical com-ponent; and it is true that the reminiscences of Eccles, Wittgenstein’s friendduring his time in Manchester and himself an engineer, refer to a theoreticalaspect to Wittgenstein’s work, but the only evidence we have of his engineer-ing work, a patent he filed in 1910, leaves little trace of it.The patent involved the idea of mounting a jet nozzle on each end of the

rotor blade of a propeller. It is a curious mixture. There are certainly elementsin the design that are original and farsighted: powered flight was still new in1910, and the idea of using any sort of jet engine to power an aeroplane, al-though not itself original, had yet to reach the engineering mainstream. Andthe idea of placing sources of propulsion at the tips of a propellor, althoughagain not original, was eventually used successfully thirty years later by an-other Austrian, Friedrich von Doblhoff, to construct a helicopter with no needof a tail rotor. On the other hand, Wittgenstein’s implementation of the ideadoes not address the practical difficulties involved in turning it into a workableengine. One set of difficulties is created by Wittgenstein’s idea of mountingcombustion chambers on the end of rotating propeller shafts. The propellerblades would have to be very strong in order to withstand the stress generatedby the very high moments of the rotating combustion chambers. This would(at least with the materials then available) require the blades to be heavy, fur-

)* Finding a problem

ther increasing the forces involved. Nor does the patent address the difficultiesinvolved in supplying fuel and oxygen to a combustion chamber on the endof an arm rotating at high speed. A further flaw is that his design has fourseparate combustion chambers: it would be difficult to control them, eitherindependently or together, and any difference in thrust between the engineswould put further strain on the propeller blades and make the assembly unsta-ble. Wittgenstein described his idea in the patent as applying indifferently to‘aeroplanes, helicopters, dirigible balloons, or other forms of aerial machines’,but the practical problems would probably be less serious in a helicopter, sincethe rotation rate of the rotor is typically lower than in an aeroplane propellorand the blades are much longer.The point of labouring these design issues here is not so much to suggest

that Wittgenstein was a poor engineer as to cast doubt on the common repre-sentation of him as a skilled mathematician. One could quite quickly estimatethe moment generated by a combustion chamber rotating at high speed, andyet Wittgenstein’s patent application takes no account of this. Indeed, it isvery hard to believe that he made any calculations before he submitted it. Thedesign shows ingenuity and imagination, it is true, but it would have needed alot more work before it could become a practical engineering project.

!.#!.#!.# The Principles of MathematicsThe plain, if somewhat unsatisfying, fact is that we do not really know whatfirst led Wittgenstein to take an interest in the philosophy of mathematics.What we do know is that Wittgenstein’s interest led him to Russell’s Principlesof Mathematics. How this came about is described in a much later reminiscenceof Rush Rhees.

Wittgenstein himself told me that while he was working in the Engineering Labora-tory, he and two others doing research there began to meet for one evening each weekto discuss questions about mathematics, or ‘the foundations of mathematics’. . . At oneof these meetings Wittgenstein said he wished there were a book devoted to these ques-tions, and one of the others said, ‘Oh there is, a book called The Principles of Mathematics,by Russell: it came out a few years ago.’ Wittgenstein told me that this was the firsthe had heard of Russell: and that this was what led him to write to Russell and to askif he might come and see him. I believe it was from The Principles of Mathematics thatWittgenstein learned of Frege.7

Wittgenstein may, for all we know, have read philosophy before this—he is re-ported8 to have read Schopenhauer’s World as Will and Idea in his youth, andas a consequence to have adopted for a time a version of epistemological ide-alism—but Russell’s Principles is the first philosophical work whose influence

7Recollections of Wittgenstein, 213–14. 8See von Wright, ‘Biographical sketch’, 5.

The Principles of Mathematics ))

on him we can trace directly. (A copy which Wittgenstein bought in October1912 has survived9 but must surely be a duplicate or replacement for one hehad already bought in Manchester.)To modern readers (of whom there are not as many as one might expect,

given its place in the history of the subject) Russell’s Principles comes acrossas a transitional work: it contains extended passages which we can recognizeas analytical philosophy in quite the modern sense, but these are juxtaposedto passages written in a style that strikes us as wholly antiquated, introducingbizarrely elaborate classifications for no apparent reason that develop into anarchitectonic of almost Kantian complexity. Whatever its faults, though, itsinfluence on Wittgenstein is unquestionable. (Tradition has it that he contin-ued to think highly of the book much later in his life.) It will therefore be inplace for us to explain here some of the philosophical background from whichit arose. Much of that background was supplied not by Russell but by Moore,from whom, on fundamental questions of philosophy, Russell said that he hadderived all the chief features of his position.10 Russell’s later recollection was,more specifically, that the movement of which this book was part (a move-ment which led him to reject the neo-Hegelian idealism espoused by Bradleywhich was then popular in Britain) was born in conversations between himand Moore in 1898.11 The first publications to exhibit this movement areMoore’s articles on ‘The nature of judgment’ and ‘The refutation of ideal-ism’, the central claim of which is that by conceiving of propositions as ob-jective complex entities independent of any knowing mind we can resist thetemptations of idealism. But if the overall shape of the project is clear, the de-tails are not. The targets of Moore’s criticism are broadly spread: although itis Bradley’s post-Hegelian denial that absolute truth is ever attainable which isthe principal target, at times Berkeley’s view that esse est percipi or Kant’s viewthat the relations the objects of experience bear to one another are suppliedby the mind are also attacked.Moore’s conception of a proposition is embodied in two central doctrines.

The first is that the entities of which a proposition is composed (which hecalled ‘concepts’) are themselves the items the proposition is about. Proposi-tions are the objects of judgment, and the concepts that make up the propo-sition are therefore part of what we judge, but the view is nonetheless realistbecause this is ‘no definition of them’; ‘it is indifferent to their nature’, he says,‘whether anyone thinks them or not’.12 Concepts are, that is to say, objec-tive entities, and a proposition consists of such entities somehow related so asto form a complex. Moore opposed this to Bradley’s view that when I havean idea of something, that thing is itself part of the idea. This opposition is9See Hide, ‘Wittgenstein’s books at the Bertrand Russell archive and the influence of scientificliterature on Wittgenstein’s early philosophy’. 10Principles, xviii. 11MPD, 54. 12‘The refutationof idealism’, 4.

)! Finding a problem

plainly not exhaustive of the possibilities, but once he had disposed (no doubtrightly) of Bradley’s view, Moore seems to have seen no need of an argumentfor his own. Moore slid, that is to say, from conceiving of the components ofa proposition as objective (which holds true, for instance, of Frege’s senses) toconcluding that they are the very same as the things the proposition is about(which does not). Nonetheless, the doctrine is central to the refutation of ide-alism as Moore conceived of it.

Once it is definitely recognized that the proposition is to denote not a belief (in thepsychological sense), it seems plain that it differs in no respect from the reality towhich it is supposed merely to correspond, i.e. the truth that I exist differs in no respectfrom the corresponding reality my existence.13

It follows, Moore held, that truth

does not depend upon any relation between ideas and reality, nor even between con-cepts and reality, but is an inherent property of the whole formed by certain con-cepts. . . The ultimate elements of everything that is are concepts.14

The lacuna in Moore’s argument is significant for our present purposes be-cause his conclusion—that a proposition must, if it is to be independent ofthe mind, contain parts of the external world—is one that Russell embracedwholeheartedly. Moreover, Russell did not, any more than Moore, considerat this stage any alternative resembling Frege’s notion of the sense of a name.By the time Russell did come across Frege’s conception, he seems to havebeen too deeply embedded in his own to be able to engage with it. When heconfirmed in response to Frege’s query that ‘in spite of all its snowfields MontBlanc itself is a component part of what is actually asserted in the proposition“Mont Blanc is 4000 metres high” ’, he offered as his reason that ‘if we donot admit this, we get the conclusion that we know nothing at all about MontBlanc’,15 but did nothing to explain why this should follow. This disagree-ment between Frege and Russell is often expressed in terms of names ratherthan sentences. For Russell the part of the proposition that corresponds to theproper name ‘Mont Blanc’ is the mountain itself; for Frege it is not the moun-tain but the sense of the name. Russell’s was, that is to say, what is sometimescalled a one-step, Frege’s a two-step semantic theory.Moore’s second central doctrine was that there are no internal relations

between concepts—no relations between concepts that are part of the natureof the concepts related. What it is for a proposition to be true is just for theconcepts it is composed of to be externally related to each other in a certainway. Once again, it is easy to see what the target is. Bradley had held that allrelations are internal, and had concluded as a result that since in particularknowledge must be conceived of as an internal relation between the knower13‘Truth’. 14Moore to BR, 11 Sep. 1898 (Griffin, Russell’s Idealist Apprenticeship, 300). 15ToFrege, 12 Dec. 1904.

Logicism )#

and the proposition known, the simple act of coming to know the propositionwill turn it into something different from what it was. No truth, according toBradley, is wholly true; truth is ‘subject always to degree’.16

There is room to doubt whether Moore meant by ‘internal’ the same asBradley. And even if Moore was right to reject Bradley’s extreme conclusionthat nothing is ever wholly true, it is much less clear why Moore should havesaid that there are no internal relations between concepts at all: as in the case ofthe first doctrine, he seems to have been oblivious of the need for an argument.Nonetheless, once again Moore’s view was shared by Russell, who as early as1899 confidently asserted that ‘all relations are external’.17

The shadow cast by these two doctrines, that names refer directly to theirobjects without the mediacy of sense, and that there are no internal relations,is long. For they were both not only adopted by Russell but maintained, in acertain sense, by Wittgenstein in the Tractatus. Wittgenstein did not maintainthat propositions contain the parts of the world they are about, but he did sidewith Russell against Frege in rejecting18 the notion that names have sense aswell as reference. And his doctrine19 of the logical independence of elemen-tary propositions can be thought of as a reexpression in Tractarian terms ofMoore’s denial of internal relations between objects. Moreover, it is notablethat the Tractatus contains hardly any argument in support of either claim. In-deed neither of them is discussed in Wittgenstein’s surviving pre-Tractarianwritings. It might seem perverse, therefore, for me to stress these two viewsin a book about the Notes on Logic, in which they do not occur. My groundfor mentioning them nonetheless is that I think the most plausible explanationfor Wittgenstein’s failure to discuss them is that he never saw any reason toquestion them; and indeed they became so embedded in his conception that,like Russell, he found it hard to see the need for argument.

!.$!.$!.$ LogicismThe doctrine that there are no internal relations between concepts runs intoan obvious difficulty in the case of identity statements. If the identity ‘a = a’expresses anything about a—a relation between a and itself—it seems clearthat this must be internal. So if there are no internal relations, we are forcedto conclude that it does not express anything at all. This is perhaps not sobad in itself, but we shall need to say something about the identity ‘Hespe-rus = Phosphorus’, which, apparently at least, expresses genuine astronomicalinformation. And a lot more will have to be said about arithmetic, in whichapparently informative identity statements (such as ‘7 + 5 = 12’) play such acentral role.

16Appearance and Reality, 321. 17CP, II, 143. 183.3. 195.134.

)$ Finding a problem

The work in which this was first attempted was Russell’s Principles. WhatRussell added to Moore’s conception of propositions in order to account forarithmetic (and indeed for mathematics more generally) was the notion of adenoting concept. A denoting concept is what one might call an ‘aboutnessshifter’:20 its task is to enable a proposition to be about something else that isnot itself part of the proposition. On the view that Russell had derived fromMoore, let us recall, the proposition expressed by the sentence ‘I met John’contains me, John, and the universal meeting. The proposition expressed by ‘Imet a man’ will similarly have to contain me, meeting, and a third elementexpressed by the phrase ‘a man’. But what is this third element? It cannotbe any particular man, since it is just the same proposition whichever man itwas that I actually met. We seem forced by this to hold that the third elementis a concept that is somehow related to whatever man I might have met; butthis concept, if a constituent of the proposition, is not one of the things theproposition is about.

The proposition is not about a man: this is a concept which does not walk the streets,but lives in the shadowy limbo of the logic-books. What I met was a thing, not aconcept, an actual man with a tailor and a bank-account or a public-house and adrunken wife.21

Yet there must be some connection between the man with the bank-accountand the propositional component in question. In the Principles Russell callsthe propositional component a denoting concept—elsewhere sometimes a denotingcomplex—and the relation it has to the man that of denoting. ‘A concept denoteswhen, if it occurs in a proposition, the proposition is not about the concept butabout a term connected in a certain peculiar way with the concept.’22

Russell seized on denoting as a central element in the epistemology of math-ematics.

The concept all numbers, though not itself infinitely complex, yet denotes an infinitelycomplex object. This is the inmost secret of our power to deal with infinity. An infi-nitely complex concept, though there may be such, can certainly not be manipulatedby the human intelligence; but infinite collections, owing to the notion of denoting,can be manipulated without introducing any concepts of infinite complexity.23

A proposition about all numbers therefore does not itself contain all numbers;rather it contains a concept which denotes all numbers. The concept is finite,and hence capable of being grasped by our finite intelligence, even thoughwhat the concept denotes is infinite. In the Principles denoting concepts thusact as the bridge between what we are capable of grasping directly and whatwe are not; they enable a proposition to be about something (in this case theclass of natural numbers) which is in a certain sense out of our reach.

20Makin, ‘Making sense of “On denoting”’. 21Principles, §56. 22Ibid. 23Principles, §72.

Russell’s paradox )%

Merely to invoke the notion of denoting is not, plainly, to explain math-ematics. A second element in the development of Russell’s views occurredin 1900 when he attended the International Congress of Mathematicians inParis and learned for the first time of the work of Peano, which demonstratedthe expressive power of symbolic logic in expressing mathematics. The claimRussell made in the Principles was that all the propositions of mathematicscould be reexpressed in the vocabulary of classes and would thereby turn outto be truths of logic.Peano’s work focused largely on the task of expressing the theorems of

mathematics: he was much less concerned with the issue of how to provethem. However, a theory which aimed to achieve this had been developedby Frege, first in his Begriffsschrift and then in more detail in the Grundgesetze.Russell had been given a copy of the Begriffsschrift by Ward (one of the philoso-phers at Trinity), but had not read it.24 He became acquainted with the firstvolume of the Grundgesetze around the beginning of 1901, but wrote the Prin-ciples in ignorance of most of Frege’s writings. Only when the main text wascomplete did he make a study of them and add an appendix summarizing andcriticizing them.25

!.%!.%!.% Russell’s paradox

Shortly before this, however, Russell had discovered a problem not just forFrege but for any prospect of a logicist reduction of mathematics. The prob-lem was that we cannot unproblematically assume that for every propositionalfunction ! there is an extensional entity, the class of !s, corresponding to it.To see why not, consider the class K of all those classes which do not belongto themselves. Then for every class x it is the case that x belongs to K if andonly if x does not belong to itself. In particular, then, K belongs to K if andonly if K does not belong to K. This is a contradiction, and the argument thatleads to it is known as Russell’s paradox.We shall come in a later chapter to the elaborate theory Russell eventually

devised to get round this difficulty. In the Principles he did no more than sketchthe outline of a ‘theory of types’ that might resolve the matter, leaving thedetails as a matter requiring further work.

The totality of all logical objects, or of all propositions, involves, it would seem, afundamental logical difficulty. What the complete solution of the difficulty may be, Ihave not succeeded in discovering; but as it affects the very foundations of reasoning,I earnestly commend the study of it to the attention of all students of logic.26

24Russell, Autobiography, 65. 25See Linsky, ‘Russell’s notes on Frege for Appendix A of ThePrinciples of Mathematics’. 26Principles, §500.

)" Finding a problem

Perhaps it was natural that Wittgenstein would be intrigued by this problemand would take up Russell’s recommendation to attempt a solution. The firstevidence we have of Wittgenstein working on any philosophical problem datesfrom April 1909, when a friend of Russell called Philip Jourdain made thefollowing note in his correspondence book.

Russell said that the views I gave in a reply to Wittgenstein (who had ‘solved’ Russell’scontradiction) agree with his own. . . In certain cases (e.g., Burali-Forti’s case, Russell’s‘class’, . . . Epimenides’ remark) we get what seem to be meaningless limiting cases ofstatements which are not meaningless.27

Jourdain was perhaps a natural person to approach, as he had already pub-lished on the topic,28 but there is also something characteristic of Wittgenstein—his blend of confidence and diffidence—in the fact that he did not write toRussell himself but to someone he may well have known was in contact withhim. Wittgenstein’s letter to Jourdain has not survived: all we have is Jour-dain’s description just quoted of the views of his own which he offered in reply.It is hardly likely, though, that Wittgenstein, a self-taught novice, had comeup with a ‘solution’ to the paradoxes of any interest or subtlety; and Jourdain,whose correspondence does not elsewhere display much grip on the conceptof tact, will no doubt have explained his error to him with clinical directness.Wittgenstein was thin-skinned at the best of times, and a brush-off from

Jourdain might on its own be enough to account for the fact that it was anothertwo years before he felt able to approach Russell in person. However, he mayequally have been influenced by his father, who (at least on Wittgenstein’spresentation of the matter) was ‘disappointed in all his other sons’ and ‘veryanxious this one should do something respectable like engineering and notwaste his time over such nonsense as philosophy’.29 His elder sister’s laterrecollections leave us in no doubt about the strength of the conflict he feltduring this period.

Reflection on philosophical problems suddenly became such an obsession with him,and took hold of him so completely against his will, that he suffered terribly, feel-ing torn between conflicting vocations. . . It shook his whole being. . . During this timeLudwig was in a constant, indescribable, almost pathological state of agitation.30

The issue that so piqued Wittgenstein’s interest was unquestionably im-portant. As soon as it was discovered, Russell’s paradox became the centralproblem in the philosophy of mathematics, a position it held at least untilthe publication of Gödel’s incompleteness theorems in 1931. The attractionfor Wittgenstein, a young man of ambition and talent searching for a field inwhich to make his mark, is therefore easy to understand: the problem was

27Jourdain’s correspondence book, 20 Apr. 1909, quoted in Grattan-Guinness, Dear Russell—Dear Jourdain, 114. 28E.g. ‘On the question of the existence of transfinite numbers’. 29BR toOM, 7 Mar. 1912. 30Quoted in Rhees, Recollections of Wittgenstein, 2.

Russell’s paradox )&

recent, simple, and intriguing; and a satisfying solution to it would be certainto bring its author attention. Not only was this apparently the first philosoph-ical problem Wittgenstein worked on seriously, but the desire to solve it wasprobably what drew him to Cambridge, and therefore to philosophy as a ca-reer rather than a hobby (to the extent that a man of means like Wittgensteinrecognized that distinction). At any rate, it was the philosophy of mathematicsrather than philosophical logic that he stated to Russell as his interest whenhe arrived at Cambridge;31 and there is ample evidence (in the diaries of hisCambridge friend David Pinsent, for example) that he continued throughoutthe two years he spent there to regard solving the paradoxes as one of hisprincipal ambitions.

31BR to OM, 18 Oct. 1911.

Chapter """

First steps

Whatever philosophical study Wittgenstein undertook in Manchester, it wasplainly not something that could be described as a training and, althoughearlier influences on Wittgenstein are no doubt relevant at various points, anystudy of the genesis of the Tractatus naturally begins in earnest with his arrivalin Cambridge to study under Russell. (Later, indeed, he told Ramsey1 thatthe book had taken seven years to write, thus implicitly identifying his arrivalin Cambridge as the point at which his work on the Tractatus really began.)Wittgenstein once advised his friend Eccles of the importance of going to

work in really first-class places.2 In his choice of Cambridge (more particu-larly, Trinity) in 1911, as of Manchester in 1908, he certainly took his ownadvice. Even if it was not the centre of the universe that some of its mem-bers were (and are) inclined to suppose it, a college that boasted Russell,Moore, McTaggart, and Hardy—Whitehead had only recently resigned hisCambridge post and gone to work in London—was plainly the best placeWittgenstein could have chosen to pursue an interest in the philosophy ofmathematics.Wittgenstein’s decision to approach Russell as a possible supervisor also

conforms to a pattern. In 1908 he had sought out Lamb, whose book onHydrodynamics he had been studying; now it was the author of the Principleshe went to see. Russell’s description3 of their first meeting, in October 1911,suggests that Wittgenstein was in as extreme a state of nervousness as he hadbeen when he met Lamb. This time, though, the outcome would be vastlymore fruitful.

".!".!".! CambridgeDuring that first Michaelmas TermWittgenstein’s attendance at Russell’s lec-tures on the foundations of mathematics was evidently something of a trialsubscription. He was still officially a research student at Manchester, had notmatriculated as a member of Cambridge University, and presumably attendedthe lectures only on Russell’s sufferance, staying in rented accommodation in

1Ramsey to his mother, 20 Sep. 1923. 2[July 1912]. 3To OM, 18 Oct. 1911.

Cambridge )(

the town while he did so. Moore was also in the audience for Russell’s lecturesand we have his lecture notes, which give the clear impression that the logicalbackground Russell assumed in his audience was not great. There is hardly alogical symbol in Moore’s notes and what is covered is mainly very elementarymaterial on cardinal and ordinal numbers.At the end of the Michaelmas Term Wittgenstein asked whether Russell

thought he had a future in philosophy. Russell replied that he should writesomething over Christmas and Russell would give an opinion once he hadread it. At some point in this decision-making process Wittgenstein also vis-ited Frege to ask his advice, and it may be that this meeting took place over theChristmas vacation at the end of 1911, since Russell’s later recollection4 wasthat when he arrived in Cambridge Wittgenstein did not yet know Frege per-sonally; another possibility is that Russell was mistaken and the meeting tookplace the previous summer. At any rate, Frege’s advice, whenever exactlyit was given, was that Wittgenstein should study with Russell; and Russell’s,when he had read Wittgenstein’s vacation essay, was the same. Shortly there-after Wittgenstein did matriculate formally as a student at Cambridge.In his second term at Cambridge Wittgenstein attended the continuation

of Russell’s course on the foundations of mathematics, as well as a course of-fered by Moore. The title of Moore’s course was ‘Psychology’, but this hardlyconveys the content to the modern reader. Moore had devoted much of theprevious term to a mind-numbingly careful discussion of how to define psy-chology, and the remainder to a critique of Russell’s multiple relation theoryof judgment; in this second term he spent much of his time on what is nowa-days called the philosophy of mind. Psychology as an empirical science wasrarely mentioned.Russell in his lectures did at least now get as far as discussing the para-

doxes, the subject which had fired Wittgenstein’s interest, but once again thetechnical level was very low. Whatever understanding of the theory of typesWittgenstein did acquire was at any rate not obtained from Russell’s lectures,which were pitched at a level that no doubt matched the technical capabilitiesof most of his audience well enough but could hardly be described as taxing.Quite soon Wittgenstein began to have frequent meetings with Russell to

discuss philosophy outside of lectures. Russell’s letters to Ottoline Morrellfrom this period occasionally mention Wittgenstein’s views, but once we allowfor the licence Russell no doubt took in turning them into amusing stories forhis lover, the views he reports—for example, that nothing empirical is know-able, and hence, notoriously, that he could not be certain there was not arhinoceros in the room5 —give us only the murkiest sense of the problemsWittgenstein was working on. What the letters do convey vividly is the de-mands which he quite soon began to make on Russell’s time and patience,

4CP, XI, 178. 52 Nov. 1911.

!* First steps

demands which Russell treated for the most part with an amused tolerancethat from the perspective of the modern academic seems quite remarkable. Apartial explanation is provided by the nature of Russell’s post at Trinity: beingneither a university lectureship nor a college fellowship, it carried no admin-istrative responsibilities and a strictly limited teaching load (three lectures aweek). One should allow for this, no doubt, and also for the sense one has thatin a university such as Cambridge these were altogether more spacious times.After a while, though, another factor began to play a central role in their rela-tionship. There is ample evidence of the veneration which Wittgenstein quitesoon began to feel for Russell. One of the most touching incidents was whenWittgenstein described hearing Beethoven’s Choral Symphony with Russellas one of the great moments in his life:6 it was not just the music, we are tounderstand, that made the moment great. Perhaps by then Wittgenstein sawRussell as a surrogate father (his own, to whom he was evidently not close,having died in January 1912); Russell certainly began to see Wittgenstein as asurrogate son and said as much to Ottoline.7

"."".""." On denotingIt is plain that during the first of his two years in Cambridge Wittgensteincame to see himself as engaged in a joint project with Russell: his letters referto ‘our’ problems8 and ‘our’ theory.9 However, the direct evidence of Witt-genstein’s own work during that year is scant. To get a sense of what theirjoint project was it will therefore be necessary for us in the next few chaptersto devote a considerable amount of space to Russell’s own work. We shallstart by examining a fundamental change that had come about in his positionsince he wrote the Principles. As we noted in §1.5, what he offered in that bookwas no more than a gesture towards a solution to the paradoxes. It was hisdesire to improve on this gesture that led him to focus his attention on thosedenoting concepts (such as, most famously, that of the present King of France)which do not denote anything. When he wrote the Principles, he had of coursebeen aware that according to his theory there would have to be such concepts;but now he was convinced of the need to gain a better understanding of howthey function, since the paradoxes seem to show that ‘the class of all classesthat do not belong to themselves’ is a denoting concept which does not denoteanything.He had said in the Principles that a proposition in which a denoting con-

cept occurs ‘is not about the concept but about a term connected in a certainpeculiar way with the concept’.10 If the concept does not denote anything,6BR to OM, 9 June 1913. 721 Aug. 1912. 816 Aug. 1912; [Summer 1912] (CL, no. 6). 926Dec. 1912. 10Principles, 53.

On denoting !)

the term in question does not exist (in Russell’s terminology it does not ‘havebeing’), and so the way in which it is connected with the concept will perhapsbe peculiar enough. But the moment of revelation for Russell came when hesaw that the relationship is peculiar even when the term does have being. For ifthere is a relationship between the concept and the thing it denotes, there willbe a true proposition expressing that relationship, and this true propositionwill be about the concept. Yet a denoting concept, let us recall, is defined asone whose job is to occur in a proposition but to point at something else whichthe proposition is about. So how can any proposition be about the denotingconcept itself? What sort of entity should occur in a proposition in order forthe proposition to be about, say, the denoting concept expressed by the phrase‘the first line of Gray’s Elegy’? Not, certainly, the denoting concept itself, sinceif it is doing its aboutness-shifting job properly, it will ensure that the proposi-tion ends up being not about the concept but about what it denotes, i.e. aboutthe sentence ‘The curfew tolls the knell of parting day’. Nor, clearly, is it anyuse to put in the proposition the denoting concept ‘the meaning of the first lineof Gray’s Elegy’, since that would make the proposition be about the meaningof the sentence ‘The curfew tolls the knell of parting day’, which again is notwhat we want. And other attempts, Russell argued, run into similar troubles.There is something in the argument just sketched that is apt to puzzle the

reader on first acquaintance. It is supposed to show that there can be no in-formative proposition about the concept expressed by the phrase ‘the first lineof Gray’s Elegy’. Yet this last sentence seems to express a proposition that isabout just this concept. Russell insists that it is not what he wants. Why? Atthis point he introduces a further constraint. The relationship between a con-cept and its denotation (if any) is not, he says, ‘linguistic through the phrase’.He asserts this without argument, as if it is obvious, but his reason for holdingit depends on his Platonistic conception of logic, according to which logicalrelations, such as that of denotation, are independent of language. Conceptsexist, Russell believed, whether or not we choose to devise means to expressthem in language; so the relationship between the concept and its denotationexists independent of language, and hence so does the proposition express-ing it. Consequently, any sentence in which a linguistic item (such as thephrase ‘the first line of Gray’s Elegy’) is mentioned cannot be what we are after,since the proposition it expresses will be (partially) about language whereasthe proposition we are trying to express would, if it existed, be independent oflanguage.The argument we have just described (which is known as the Gray’s Elegy

argument because of the example he used to make the point) led Russell toreject the theory of denoting he had put forward in the Principles. What hereplaced it with was an account according to which the true structure of the

!! First steps

proposition a sentence expresses is to be revealed by translating it into thepredicate calculus with identity. The sentence ‘I met a man’, for instance,might be translated as (

E

x)(Mx Rax), where ‘Mx’ means that x is a man, ‘Rxy’means that xmet y and ‘a’ denotes me. In the analysed sentence ‘a man’ is nolonger a denoting phrase but occurs only as part of the predicate ‘is a man’:the work of denoting is done by the notation of quantifier and variable. And,as undergraduates learn in their elementary logic course, ‘The present Kingof France is bald’ can be translated as (

E

x)(Kx (y)(Ky ! x = y) Bx), where ‘Bx’means that x is bald and ‘Kx’ means that x is a king of France. Once again,the denoting phrase (in this case ‘the present King of France’) has disappearedin the translation, to be replaced by quantified variables.What was significant about this method of translation was that it showed

how the grammatical form of a sentence might differ from the logical form ofthe proposition the sentence expresses. Thus in the standard example, ‘Thepresent King of France is bald’, the sentence has a subject, ‘The present Kingof France’, which does not correspond to any constituent of the proposition itexpresses. The theory thus avoids the need to appeal to a shadowy realm ofnon-existent objects (often called ‘Meinongian’ although this is a little unfairto Meinong) to explain the meaning of the sentence.This is a general method of considerable power. Wherever in philosophy

we come across linguistic items which appear to refer to entities which are insome way problematic, the possibility now arises that the terms in questionmay be what Russell soon called ‘incomplete symbols’, that is to say expres-sions which have no meaning on their own but which are such that any sen-tence in which the expression occurs can be translated into another in whichit does not. By this means we eliminate reference to the problematic entitieswithout rendering meaningless the sentences which apparently refer to them.It is not known when Wittgenstein first came across the theory of descrip-

tions of ‘On denoting’, but it is certain that he adopted it wholeheartedly andprobable that he did so quite early. As he somewhat patronizingly told Rus-sell, ‘your theory of descriptions is quite CERTAINLYCERTAINLYCERTAINLY correct’.11 And Jourdainwas surely inspired by Wittgenstein when, in a letter to Frege written afterhe had come to know Wittgenstein personally at Cambridge, he referred to‘what seems to be a fact, namely, that Russell has shown that propositions canbe analysed into a form which only assumes that a name has a “Bedeutung”,and not a “Sinn” ’.12 Moreover, it was not just the theory of descriptions thatWittgenstein agreed with: his letters to Russell show that he was soon experi-menting with the method of incomplete symbols as an approach to analysingcompound propositions. Some of the consequences of Wittgenstein’s accep-tance of this method will emerge shortly.

11[Nov. or Dec. 1913] (CL, no. 32). 1215 Jan. 1914.

Sense-data !#

".#".#".# Sense-dataAlthough Russell intended his logic to form a foundation for mathematics, hedid not, of course, conceive of that as its only role. He also hoped to use it toground an account of our knowledge of the external world. In order for histheory of descriptions to be of use in explaining that knowledge, however, hehad to forge a link between logic and epistemology. What he assumed wasthat the direct logical relation between a name in my idiolect and the objectit denotes mirrors a similarly direct epistemological relation (which he calledacquaintance) between me and the object. With this assumption in place,he hoped that his theory would enable him to determine what the objects ofacquaintance are. To understand how it would do this, we need to recallhow it dealt with non-referring expressions. Russell analysed ‘The presentKing of France does not exist’ as !( Ex)(Kx (y)(Ky ! x = y)). (In words: it isnot the case that there is exactly one present king of France.) The key stepoccurred when he decided to apply an analysis of the same form in every casewhere we say that something does not exist. If we say that Homer did notexist, for instance, we should be taken to mean that no one person wrote boththe Odyssey and the Iliad. The apparent predication of a property of existenceis replaced by an existential quantifier, with the result, Russell thought, thatwe avoid the difficulties involved in supposing there to be a person, Homer,with the awkward property of non-existence. ‘Homer’ is thus for Russell anexample of a term that is grammatically a proper name, but not logically so,since the correct logical analysis of ‘Homer does not exist’ reveals ‘Homer’to be really a definite description in disguise. And in the same sort of way‘Sherlock Holmes does not exist’ might be analysed by replacing ‘SherlockHolmes’ with a definite description such as ‘the detective who lived at 221bBaker Street’.Russell used the term ‘logically proper name’ for any proper name which

functions as such not just grammatically but logically—for any name, that isto say, which logical analysis does not reveal to be really a disguised definitedescription. But in ordinary language logically proper names are the excep-tion rather than the rule. For it is not just words for spurious classical poetsand fictional detectives that turn out to be disguised descriptions. The elimi-native doctrine applies in any case where I can say intelligibly, even if falsely,that someone does not exist. Since I can wonder whether Plato existed, ‘Platoexists’ must express a non-trivial proposition fit to be the object of my won-derment. So ‘Plato’ is (at least in my idiolect) a disguised definite description.The same applies to anything whatever of whose existence I can coherentlyentertain a doubt: the term referring to it must on this view be a disguiseddefinite description. It follows that a term ‘A’ in my language can be a logi-cally proper name only if the sentence ‘A does not exist’ is not merely false but

!$ First steps

absurd: the object A must be something of whose existence I am so certainthat I cannot intelligibly doubt it. This is plainly a very demanding criterion:even tables, chairs, and pens do not fulfil it since they might be holograms,tricks of the light, or hallucinations. The only things in the empirical realmthat do fulfil the criterion are what Moore and Russell called ‘sense-data’, bywhich they meant whatever the things are that are immediately given to us inexperience. (The word was coined in 1882 by Josiah Royce and popularizedby William James,13 but it was Moore14 who first used it with the meaning justgiven.) Even if the green table on the other side of the room were an illusion,the patch of green at the centre of my visual field when I look in that direc-tion would certainly exist. It was part of Russell’s view to maintain that if Isay something about the table (that it is oblong, for example), the propositionthat I express does not contain the table itself but may contain various sense-data, such as the green patch just mentioned. And of course I have othersenses too: whatever is presented immediately and indubitably to these othersenses—sounds, pains, feelings of roughness beneath the finger, or bitternesson the tongue—will according to Russell and Moore be sense-data too.What, then, is the relation between me and the green patch in my visual

field? It might reasonably be called a sort of experience, but it is distinguishedfrom other cases of what we normally call experience by its indubitability. IfI think I am experiencing a blow on the back of the head, I may be mistaken:perhaps the pain has some other cause. But I cannot be mistaken that I amin pain. (An amputee who describes a pain in a non-existent leg is mistakenabout the source of the pain, not about the fact of it.) This special sort of ex-perience is what Russell called acquaintance. Acquaintance is thus a relationwhich, when it holds, provides an immediate and indubitable link between themental world of the subject and something else. Moore used the term ‘directapprehension’ for the same relation.In the case of my experience of the contemporary external world, then, the

object of acquaintance is a sense-datum. Russell often talked of acquaintanceas a relation between me and the sense-datum, but he regarded this as a looseway of talking and strictly speaking took the object that is related to the sense-datum by acquaintance to be not me myself but my awareness of the sense-datum, which he called a sensation. Acquaintance thus acts, according to thisview, as a bridge between the mind and the world, relating an event withinmy mind (the sensation) to something non-mental (the sense-datum). In anact of memory, too, acquaintance links the subject to something non-mental:the difference is only that in this case the object of the act is not representedas being simultaneous with the act itself.By taking items of experience as building blocks in this way Russell showed

13See Milkov, ‘The History of Russell’s concepts “sense-data” and “(knowledge by) acquain-tance” ’. 14‘The subject matter of psychology’, 57.

Sense-data !%

evident sympathy with a central strand of empiricism, but he was very far frombeing an empiricist, since he did not think that they are the only constituentsof propositions or that all our knowledge is derivable from them. There are,he thought, other cases in which the object of acquaintance is not a sense-datum, one of which is that of acquaintance with universals. He thus felt ableto maintain a liberal ontology of universals such as love or meeting, which hethought were constituents of propositions such as ‘John met Mary and fellin love with her’. Universals, he somewhat over-exuberantly claimed, are‘unchangeable, rigid, exact, delightful to the mathematician, the logician, thebuilder of systems, and all who love perfection more than life’.15

Russell’s theory has a curious side-effect, however. Recall his argumentfor the identification of the simple entities as those things whose existence itwould be incoherent to doubt. That argument was that if A is a simple entity,then the sentence ‘I doubt whether A exists’ cannot be intelligible, since if itwere intelligible, the Russellian analysis would reveal ‘A’ to be not a logicallyproper name but a disguised description. But we can evidently run an ex-actly analogous argument in the case of the sentence ‘it is possible that A doesnot exist’: if this is intelligible, the Russellian analysis will reveal ‘A’ to be adisguised description once more.But if we simply use this argument to place a further constraint on the

simples, Russell’s theory seems to collapse under the weight, since we nowneed the simples to be entities whose existence is not only indubitable butnecessary, and common sense suggests that even sense-data do not fulfil thiscriterion: I may be sure that there is a patch of green in the centre of my visualfield, but can I not also represent to myself the possibility that there mightnot have been this patch (if, for instance, I had painted the wall a differentcolour)? The only way out if there were to be any simples in the world atall was for Russell to say that despite appearances to the contrary I cannot infact represent the possibility of there not having been precisely this patch ofgreen. Kripke, famously, rejected Russell’s analysis of ‘A does not exist’, butonly at the price of accepting that proper names need not refer to anythingand hence leaving ‘A exists’ unanalysed. Russell, on the other hand, took ourability to represent A’s non-existence as evidence that we conceive of A ascomplex. He held, therefore, that the notions of possibility and necessity areproperly applicable not to propositions but only to propositional functions. Iftalking of propositions as possible was to be legitimate at all, it would haveto be explained as a way of saying something not about how the world couldhave been but about how it actually is.16

15Problems, 57. 16Cf. CP, VI, 194–5.

Chapter ###

Matter

WhenWittgenstein arrived in Cambridge, Russell himself was at a transitionalpoint in his work. The text of Principia Mathematica had been delivered to thepublisher in 1909 (although correcting the proofs occupied him intermittentlyuntil February 1913).1 He wrote his ‘shilling shocker’, The Problems of Philos-ophy, in the summer of 1911 just before Wittgenstein’s arrival. Now he wasready to embark on a new project, and the subject he chose was the problemof matter.

#.!#.!#.! The projectEven if the world contains such things as sense-data, they are plainly not thesame as the objects ordinary language speaks of, nor are they the same asthe particles (electrons, protons, neutrons, or whatever) which physics takes asfundamental. In 1911, when he wrote The Problems of Philosophy, Russell’s viewhad been that sense-data are the only items in the external world with whichwe are directly acquainted, and hence of whose existence we may be quitecertain. He noted, however, that the inference from them to the existenceof matter constituting an independent external world ‘does not lead to anydifficulties, but on the contrary tends to simplify and systematize our accountof our experiences’. For that reason, he thought, ‘there seems no good reasonfor rejecting it’.2 If I look away or leave the room and come back in, or if Ishut my eyes and then open them, the various sense-data I experience have aregularity which is most easily explained by positing a table from which theyare all derived. Or, to put the point in a rather less homely way, the simplestphysical theory that correctly predicts the data of sense is one which quantifiesover the matter of physics and not just over sense-data. This, Russell believed,gives us reason to adopt the hypothesis that matter—‘that thing if any wh[ich]corresponds to sense-data and is independent of perception’3—exists.Russell now made what I shall call the strong constructional conjecture—the con-

jecture, namely, that it might prove ‘possible, by a logical construction, todefine in terms of sense-data alone some object having the properties which

1To OM, 16 Jan., 22 Feb. 1913. 2Problems, 11. 3Russell’s lectures, 1912.

On matter !&

dynamics assigns to matter’.4 A theory of perception might be expected toprovide an account of sense-data in terms of matter: it is because certainpieces of matter are configured in a certain way that I can be acquainted witha certain sense-datum. Russell’s project was, if possible, to do the converse—to provide, that is to say, an account of matter in terms of sense-data—andhence to provide an explanation of how physics can be based on the sense-datum theory.Wittgenstein cannot have played any part in Russell’s formulation of his

project, since Russell discussed the problem, if not the solution, in his Cam-bridge lectures as early as November 1911, when his view of Wittgenstein wasstill that ‘it is really rather a waste of time talking with him’.5 Russell him-self attributed his interest in the problem at least partly to Whitehead,6 whowas working on a related project himself. The constructional conjecture andits variants soon became a central preoccupation for Russell, and in 1912 hetwice delivered versions of a paper on the subject. By then, however, the rela-tionship between him andWittgenstein had changed markedly. Just before hebegan writing his paper on matter, Russell reported that he had ‘got a numberof new technical ideas from him’, which he thought ‘quite sound and impor-tant’.7 So it is natural to look in this paper for signs of Wittgenstein’s influence.What we shall see is that the paper does indeed contain such signs, not all ofthem constructive. Perhaps it is not quite accidental, then, that Russell neverpublished this paper, and that the account he did eventually publish, ‘Therelation of sense-data to physics’, was not written until the beginning of 1914,when Wittgenstein was in Norway.

#."#."#." On matterRussell’s conjecture that matter can be constructed out of sense-data need notin itself be taken to indicate dissatisfaction with his earlier view that the for-mer can be inferred from the latter: a logical construction would, after all,be preferable to an inductive inference. However, in his paper ‘On matter’he did in fact express dissatisfaction with the inference. He now rejected theargument from theoretical simplicity mentioned earlier, saying that we haveno reason to suppose that the universe is simple, and hence no reason a priorito prefer simple explanations to complicated ones. Perhaps this is Wittgen-stein’s voice we hear, since elsewhere in his writing Russell did not hesitate toprefer simple explanations to complicated ones. Moreover, it is notable thatRussell’s tone in the passage is uncharacteristically uncertain.Indeed, this hesitancy pervades the whole paper. Having made his con-

structional conjecture Russell discussed hypothetically what its consequences

4CP, VI, 512. 5To OM, 16 Nov. 1911. 6To OM, 29 Dec. 1912. 7To OM, 23 Apr. 1912.

!' Matter

would be, but then in a brief passage offered an argument to show that theconjecture is in fact false. Having reached this conclusion, he then went backand made a first set of revisions, probably before he delivered the paper atCardiff, altering the passage about the consequences of the strong construc-tional conjecture from a real to an unreal conditional.What Russell had only just realized was that if his constructional conjec-

ture was to have any hope of being correct, it would have to be weakenedso that the constructional base included not only sense-data but things of thesame kind as sense-data that are not in fact sensed by anyone. Let us call thisthe weak constructional conjecture: that the matter of the physicist consists of ‘col-lections of constituents of the nature of sense-data, some actually perceived,some not’.8 Russell’s views were shifting even as he wrote the paper. In thefirst draft he had written, ‘There is, in fact, reason to suppose that what I im-mediately see when I look at an object does not exist when I am not lookingat it.’9 Now he deleted this and said instead,

It is possible that the arguments against naive realism could be met by a theory whichregards a piece of matter as consisting entirely of constituents of the nature of sense-data, by including everything that could be a sense-datum to any possible observer.10

Here, then, we have the first sign that Russell now countenanced, albeit hes-itantly, a new kind of inference, not from sense-data to physical objects, butfrom sense-data to what he now began to call ‘qualities’—things of the samenature as sense-data.Wittgenstein read the beginning and end of Russell’s paper, and thought

they were excellent.11 His view, apparently, was that physics could be reinter-preted directly in terms of sense-data, so that the assumption of matter woulddrop out as unnecessary. ‘If there is no matter,’ he told Russell, ‘physics andastronomy, and all the other sciences could still be interpreted so as to betrue.’12 Presumably, therefore, one of the things he approved of was Russell’sstatement (which he later deleted) of just this view near the end of the paper.

It seems probable that all physical science, in so far as it is verifiable, could be inter-preted in terms of sense-data alone, and might therefore retain whatever truth it canbe known to possess even if there were no such thing as matter; and if this is the case,it follows that we cannot know either the existence or the non-existence of matter.13

But Wittgenstein’s initial approval for Russell’s paper did not last once hehad read the whole of it.14 Russell’s letters speak of minor revisions to thepaper in preparation for reading it in Cardiff, which are probably the minorrevisions to the manuscript to which I have just referred. If so, the versionwhich Wittgenstein read a few days later already concluded that the strong

8CP, VI, 95. 9CP, VI, 512. 10CP, VI, 85–6. 11BR to OM, 22 May 1912. 12BR to OM, 23Apr. 1912. 13CP, VI, 516. 14BR to OM, 26 May 1912.

On matter !(

constructional conjecture was as a matter of fact false. It seems likely, though,that Wittgenstein’s main disagreement lay deeper. What he objected to wasthe conception of metaphysics that lay behind any attempt such as Russell’s toconstruct matter out of things of the same kind as sense-data. Wittgenstein,that is to say, had a conception of analysis that is genuinely eliminative in asense in which Russell’s was not.We shall discuss the difference between their two conceptions of analysis

in the next chapter. The conjecture that the focus of their disagreement layin this difference receives some circumstantial support from the second, moresubstantial round of revisions to the paper which Russell probably made inOctober 1912, in preparation for reading the paper again, this time to theMoral Sciences Club in Cambridge with Wittgenstein in the audience. It wasthen, I conjecture, that Russell deleted a long passage from the middle of thepaper, including the discussion of the metaphysics of constructed entities justmentioned. The passage Russell inserted in the revised version contained amore developed and clearer version of the argument that his strong construc-tional conjecture was false. This new version was based on the observationthat, although the predictions against which physics is tested may be trans-lated into claims about the configuration of sense-data, the theory used tomake the predictions contains parameters which are not determined by thesense-data alone. The example Russell uses to make the point is that of thestars and planets.

If we refer them to polar coordinates with the earth as origin, their angular coordinateswill be observable, but the radius vector will always be a pure inference. . . Thus theradius vector introduces an unobservable distinction between two cases which givethe same sense-data. It is introduced, of course, because it enables us to state simplelaws for the motions of the heavenly bodies; but what is called the verification of theselaws applies only to the angular coordinates, since the radius vector is not amenableto observation. . . We may so choose the coordinates of any physical system as thatsome of them shall have a one-one correspondence with sense-data, while others shallbe entirely independent of sense-data, i.e. different values of them will not correspondto different sense-data. Those that correspond with sense-data may be called verifiablecoordinates, while those that do not may be called hypothetical coordinates. Thus inthe case of the planet, its angular coordinates are verifiable, while its distance from theearth is hypothetical.15

A distant star appears to me as a point of light in a certain position in the nightsky: the distance from me to the object is not part of the sense-datum but isan hypothesis introduced by the astronomer so as to simplify the laws used topredict where in future similar sense-data will appear to me. The inferenceRussell draws is that the relation of the momentary state of the world to oursense-data is many–one, not one–one: there are various ways the world could

15CP, VI, 87–8.

#* Matter

be which would give rise to just the sense-data that we in fact experience. Soany construction of matter in terms of sense-data will have to contain extra hy-pothetical parameters whose purpose can only be to simplify the formulationof physics.Russell’s attempts at constructing matter are not the only things in the

paper Wittgenstein might have disagreed with, however. At one point, forexample, Russell speculates on whether there might be non-logical a prioriprinciples which would license the principle of induction. In support of thispossibility he cites an example: if A is earlier than B, and B is earlier than C,then C is not earlier than A.

If the world were periodic, so that after a certain interval everything became exactlyas it had been at an earlier time, we should still regard the earlier occurrence as nu-merically different from the later occurrence; and so far as I can see, we should do thisin consequence of our a priori certainty that time cannot be circular.16

That some non-logical truths might be knowable a priori is of course some-thing Wittgenstein denied in the Tractatus,17 and it would not be at all surpris-ing if his opposition to it had arisen as early as 1912.Even in its revised form, though, Russell’s paper remains curiously hesitant.

As late as the day of the Moral Sciences Club meeting at which he was due toread it, he wrote to Ottoline, ‘I don’t know yet whether to say there is matteror there isn’t.’18 The claim that physics can probably be interpreted in termsof things of the same kind as sense-data remains, but Russell does not expandon his earlier sketch of how the reinterpretation is to be carried out.

#.##.##.# Dawes Hicks

The position Russell had now argued himself into—in part, surely, atWittgen-stein’s urging—was undoubtedly complicated. He had rejected the inferencefrom sense-data to matter; but he had also shown that his original conjecturethat we can construct matter out of sense-data is false, and conjectured in-stead that we can construct matter out of what he called ‘qualities’, i.e. thingsof the same kind as sense-data which may not actually have been sensed byanyone. But what licenses the new inference that is now required from sense-data to qualities? It is far from obvious that it is any easier to establish thanthe previous inference directly from sense-data to matter.To see why Russell thought the new inference preferable, let us turn to the

concerns expressed by George Dawes Hicks in his 1912 critique ofThe Problemsof Philosophy.19 Russell’s correspondence with Ottoline Morrell has preservedfor us Wittgenstein’s reaction to Dawes Hicks’ article.

16CP, VI, 514. 172.225. 1825 Oct. 1912. 19‘The nature of sense-data’.

Dawes Hicks #)

This morning I made him read two pages on sense-data by a muddle-headed personnamed Dawes Hicks, but the muddle made him quite ill. He declaimed for a longtime and I thought he would murder me!20

But even if Dawes Hicks was muddle-headed, the issue he raised is close toRussell’s central difficulty: if sense-data are objects distinct from all the famil-iar objects of everyday life, how is knowledge about the latter possible? Rus-sell’s theory bears on the surface an obvious risk of falling foul of a problemthat bedevilled early modern philosophy: if physical objects are unknowable(as, for instance, real essences are supposed to be for Locke), then it is hardto see what sense there is in positing their existence nonetheless; Humeanscepticism beckons.In the 1912 paper on matter that we have been discussing Russell acknowl-

edged the point explicitly. ‘Even if we could know that sense-data “corre-spond” to some reality independent of perception, we could know nothingwhatever as to the intrinsic nature of this reality,’21 he said. Now, however,he denied this. To avoid the Humean danger, Russell had to insist that sense-data are just as real as physical objects. Among objects, he was forced to say,the sense-data are merely the ones that are as a matter of fact directly known;other objects, such as tables and chairs, are not known, but this does not, henow said, make them unknowable in principle. ‘I do not know of any reasonwhy the mind should be “disqualified” from knowing the physical thing; thequestion is one of fact, do we know the physical thing or do we not?’22

If, as Russell insisted, sense-data are physical, it is quite difficult to work outquite what they are. We plainly cannot (at least in general) treat the sense-datum as a property of the object perceived, since in some cases (such as mi-rages and other similar illusions) there is in fact no object to be perceived.Russell was therefore surely right to insist, in his reply to Dawes Hicks, thatthe sense-datum is a distinct object and not a mere property of the objectperceived.23 One approach might be to think of the sense-datum theory as acontribution to the psychology of perception. (Sometimes, indeed, one has theimpression that Russell and Moore were trying to do psychology without thebother of actually doing any experiments.) Consider for simplicity, as Russelland Moore most often did, the case of vision. It would then be natural tothink of the sense-datum as some sort of physical occurrence within the visualapparatus. In that case one can ask at which point in the sensory process itis supposed to be located. If we think of the sense-datum as constituted bythe physical configuration of receptors in the eyeball, we explain mirages wellenough but struggle to explain hallucinations and dreams, which do not in-volve the eye at all but are generated within the brain itself. The further up thechain of vision we push the sense-datum, though, the closer the theory comes

205 Sep. 1912. 21CP, VI, 93. 22CP, VI, 186. 23CP, VI, 187.

#! Matter

to treating us as brains in a vat. The sense-datum becomes simply the brainstate that is correlated with the mental act of sensation, so that the relation ofacquaintance is reduced to the mere bridging role of Descartes’ pineal gland.It begins to seem not only wildly over-optimistic to hope that the theory mightbe the foundation of an account of our knowledge of the external world, butstraightforwardly mistaken. The propositions we utter, which we take to beabout the table, turn out on analysis really to be about sense-data. As longas these were interpreted as properties of the table—its solidity, roughness,colour, etc.—that did not seem too implausible; but if they are really brainstates, the proposition simply misses its intended target completely.If sense-data are not properties of the physical object, however, the question

arises whether they exist when no one is looking at them. In The Problems ofPhilosophy Russell had been happy to accept that even if they are not mental,they ‘must be, in part at least, “in” the mind, in the sense that their existencewould not continue if there were no seeing or hearing or touching or smellingor tasting’.24 The greenness I perceive when I look at the wall no longer exists,in other words, when I look away. Why Russell thought this is less clear. Hementions Berkeley, but with only partial approval. ‘His contention,’ Russellsays, ‘was almost certainly valid, even if some of his arguments were not so.’25

But Russell does not develop the point, except to say it depends on ‘variousreasons of detail’,26 and the reader is left uncertain what the valid argumentmight be which establishes that the sense-datum depends for its existence (ratherthan merely for the fact that it is a datum to sense) on the act in which it issensed.The obvious way of avoiding even the strictly limited idealism that this

concession to Berkeley amounts to is to accept that sense-data may exist whenthey are not being experienced, and that there may be things of the same sortas sense-data that no one ever has experienced or ever will experience. On thisview, which Russell now adopted, something becomes a sense-datum when itis apprehended, just as a man becomes a husband when he marries; but inneither case is the object itself brought into existence (or changed intrinsically)as a result. But although Russell now accepted that it is possible for qualities topersist after they cease to be sense-data, he continued to doubt for ‘empiricalreasons of detail’27 whether they do. Once again, though, he did not troubleto explain what these reasons of detail are.We can now see that the advantage of positing qualities rather than matter

lies in the similarity of nature between the entities that are inferred and thosethat are experienced directly. In the Problems Russell had maintained that weinfer the existence of matter from the existence of sense-data. His new viewwas that we infer the existence of unsensed qualities from the existence of

24Problems, 20. 25Ibid. 26Problems, 21. 27CP, VI, 186.

The relation of sense-data to physics ##

sensed qualities (i.e. sense-data). So, although he could not see how to dispensewith inferred entities entirely, now at least the inferred entities had what heclaimed was the virtue that they were all of the same kind as the entities ofwhich I have direct acquaintance. An obvious source for the notion that thisis an advantage was Hertz, who expressly recommended that in a physicaltheory the inferred entities (which he called concealed masses) should be ofthe same nature as the observed masses. Russell had of course read Hertz,but so had Wittgenstein, and it would be no surprise if Russell’s adoption ofthis methodological principle owed something to his prompting.

#.$#.$#.$ The relation of sense-data to physicsRussell continued to think about his project on matter until March 1913,when he put it to one side in order to work on a book on epistemology. He hadreached stalemate because his conception of qualities—things of the same na-ture as sense-data, including those that are not in fact the data of sense—wasnot sufficiently rich to be able to generate matter from them. Wittgenstein’sscepticism about the possibility of carrying out the project remained implaca-ble. ‘I was very interested to hear your views about matter,’ he told Russell inJanuary 1913, ‘although I cannot imagine your way of working from sense-data forward.’Russell did eventually return to the question of matter, and part of the

impetus for him to do so came in October 1913, when he received the manu-script of an article Whitehead had just written on space. He and Wittgensteinread it together.28 This article contained Whitehead’s attempt to show howphysical space could be constructed out of perceptual space. The significanceof the idea does not seem to have struck Russell immediately, however. Itwas not until January 1914 that he came back from a holiday in Switzerlandand dictated most of ‘The relation of sense-data to physics’ to a secretary.29

What he did in this paper was to construct matter out of what he now calledsensibilia.30 A sensibile was to be, just as a quality had been, an entity of thesame kind as a sense-datum, whether or not it is in fact sensed by anyone.31

As we shall see in a moment, however, his conception of what kind this is hadevolved significantly.Russell’s central idea in the paper was to identify a physical object with the

class of all the appearances it presents from various perspectives, and to definethe matter of the thing to be the limit of these appearances as the distancefrom the object diminishes: matter, roughly, is what an object would looklike from all angles if you got very close to it. If there is to be matter in

28BR to OM, 2 Oct. 1913. 29See Blackwell, ‘Our knowledge of “Our knowledge” ’. 30CP,VIII, 7. 31Moore, in ‘The status of sense-data’, preferred the English coinage ‘sensibles’.

#$ Matter

places where no one is looking, as physics (not to say common sense) requires,we evidently have to assume the existence of unsensed sensibilia. In October1912, as we have seen, he had an argument to show that matter cannot beconstructed out of sense-data, because there are in physics hypothetical co-ordinates which do not correspond to sense-data: the example he used was thedistance from us to a distant star, which is not part of the sense-datum since allthat is present to the senses is the direction in which the star appears to us. In1914 he overcame this difficulty by the brashly simple device of supposing thatdistance is part of the sense-datum. According to the new theory, therefore,a sensibile is something like a three-dimensional vector (i.e. it has both lengthand direction), and its origin (the location of the observer) is itself a positionin three-dimensional space. Russell’s conception of the sort of thing that asense-datum is had thus become significantly richer. Perhaps this is why heintroduced the new word ‘sensibile’ for them, rather than continuing with theword ‘quality’ that he had used before. In February 1913 he had spoken ofthe need for a ‘fundamental novelty as to the nature of sensation’.32 Now hehad found (or invented) it.If we succeeded in showing that tables and chairs are logical constructions

out of sensibilia, that would show that the assumption that they exist has nomore explanatory power than the assumption that sensibilia exist, since in anyexplanation involving a table we could simply substitute the logical construc-tion of the table out of sensibilia. It would not on its own show that tables do notexist, however. It would not, for instance, rule out the possibility that sensibiliacould also be constructed out of tables and chairs, in which case some furtherprinciple would be required to settle which should be eliminated in favour ofthe other: Occam’s razor on its own would not be enough. What made Rus-sell try to eliminate matter in favour of sensibilia and not the other way roundwas an epistemological rather than a metaphysical consideration, namely thatwe can have direct acquaintance with sensibilia but not with matter.We should pause here to consider in more detail Russell’s enriched con-

ception of sense-data as involving distance as well as direction. Humans do,it is true, have the ability to judge distance with respect to vision and hear-ing. There are two mechanisms for judging visual distance. The first, whichuses stereoscopic vision to compare the views of a body as observed by thetwo eyes, depends on the fact that the objects we normally observe have aninternal structure, and it is therefore highly fallible when applied to a uniformpatch of colour. The second mechanism, which uses the focusing of the lens inthe eye, is also easily misled in the case of uniform patches and is in any casemuch less accurate than the first. In practice, our ability to judge distance ishighly influenced by contextual factors, as is vividly demonstrated by a wholehost of trompe l’œil illusions. It may be added that both methods are useless

32To OM, 23 Feb. 1913.

The relation of sense-data to physics #%

at large distances from the object, although this is of little import for Russell’saccount since he identifies matter with the limit of the appearances as distancediminishes. What is more troubling, however, is that the methods break downat small distances too. When an object is less than 10 cm from the eye, wehave difficulty focusing on it at all.Similar remarks apply to sound: there is no analogue of lens-focusing, and

our stereoscopic sense of distance is routinely fooled every time we put on apair of headphones. The possibility of optical and auditory illusions is not itselfa difficulty for Russell’s account, of course, but the ease with which we can befooled in our judgments of distance—and know that we are being fooled—addsa layer of implausibility to his decision to treat the distance to an object as partof what is indubitable in experience. It has the curious result, for instance, thatthe discussion one might read in a hi–fi magazine of the breadth and depth ofthe sound stage created by a hi–fi system becomes literally true.In relation to touch, of course, the kinaesthetic sense more plausibly pro-

vides us with a means of judging relative position, so that the distance as wellas the direction from us of the object touched is plausibly part of what weapprehend. It is notable, therefore, that Russell was inclined to downplay thesignificance of touch, perhaps because it is of rather limited use in astronomy.Even by laying claim to an enriched conception according to which dis-

tance is part of the sense-datum, however, Russell could not derive his con-ception of matter from sense-data alone. Although he continued to claim, as‘a probable inference from empirically ascertained causal laws’,33 that sense-data do not persist after they cease to be data, he was forced also to posit theexistence, as sensibilia, of those appearances an object would have from a cer-tain perspective even if there is no being with sense organs positioned so as toexperience these sensibilia as sense-data. This assumption was a practical ne-cessity if he was to identify the physical object with its appearances, but Russellwas plainly uncomfortable with it and hoped that ‘the part played by unper-ceived “sensibilia” could be indefinitely diminished, probably by invoking thehistory of a “thing” to eke out the inferences derivable from its momentaryappearance’.34

However, the notion of an unsensed sensibile is very puzzling. In the casewhere the sensibile is sensed, Russell is willing to grant its

physiological subjectivity, i.e. causal dependence on the sense-organs, nerves, and brain.The appearance which a thing presents to us is causally dependent upon these, inexactly the same way as it is dependent upon intervening fog or smoke or colouredglass.35

What, though, of the case where there are no sense-organs positioned so as tosense the appearance of the thing from the relevant perspective? We might

33CP, VIII, 9. 34CP, VIII, 26. 35CP, VIII, 7.

#" Matter

accept, on grounds of continuity, that in such a case the putative sense-datumwould exist if sense-organs were appropriately positioned, but it seems verystrange to say that it does exist.Russell affected not to recognize this strangeness.

We have not the means of ascertaining how things appear from places not surroundedby brain and nerves and sense-organs, because we cannot leave the body; but conti-nuity makes it not unreasonable to suppose that they present some appearance at suchplaces.36

The ontology Russell conjures up is thus extravagantly rich. For every objectand every position there exists a distinct thing which is the appearance thatobject would have if viewed by us from that position. Moreover, if appear-ances are, as Russell says, dependent on the sense-organs, nerves, and brain,presumably there will be for each such object and position various differentappearances corresponding to different sorts of sensory apparatus. Russellclaimed explicitly that

if per impossibile there were a complete human body with no mind inside it, all thosesensibilia would exist, in relation to that body, which would be sense-data if there werea mind in the body. What the mind adds to sensibilia, in fact, is merely awareness:everything else is physical or physiological.37

It is hard to see, in that case, why we should not say the same about cameras.But the appearance an object presents to infra-red film, for instance, may bevery different from what it presents to ordinary film. Russell was in gravedanger of populating the world with (among very many others) such physi-cal entities as the appearances things present to an ASA 800 film exposed ataperture f8.Yet he continued to say that there is no reason to think a sense-datum per-

sists after I have experienced it. If I stare fixedly at the brown stain on thewall for four seconds, therefore, the sense-datum I perceive persists, presum-ably, for just four seconds. What, then, of the case in which I do not look atthe wall? The stain still presents, Russell claims, an (unsensed) sensibile to theperspective which in the other case I occupied. How long does this sensibilepersist? No very plausible answer to this question presents itself.

#.%#.%#.% The atomistic assumptionIn the period we have been discussing, then, Russell offered several differ-ent accounts of how physics can be derived from sense-data, or entities ofthe same kind as sense-data. All are hopeless, but for different reasons; andwith accounts as varied as these we need not expect their failures to have a

36CP, VIII, 7–8. 37CP, VIII, 8.

The atomistic assumption #&

common source. However, there does seem to be one key move in Russell’sthinking that is worth isolating. It occurred when he identified sense-data asthe simplest items we may attend to.

When I speak of a ‘sense-datum’, I do not mean the whole of what is given in sense atone time. I mean rather such a part of the whole as might be singled out by attention:particular patches of colour, particular noises, and so on.38

Russell then supposed without argument that all our knowledge of the worldderives from such items. ‘Sense-data at the times when they are data,’ he said,‘are all that we directly and primitively know of the external world.’39 But itis very far from clear that this is true. Russell never seriously considered thepossibility that the act of singling out an item of experience by attention some-how reconfigures it. It seems quite possible that there may be knowledge of theworld that we acquire directly but not by means of any item we are capableof singling out by attention. Or, to put it another way, there might be somesorts of experience we have which are such that attending to their constituentsis inherently distorting. In the case of an object in motion, for instance, at-tention might have the effect of freezing the object and hence eliminating themotion. A series of static images on a cinema film can give us the illusion ofmotion, but Russell was much too swift to think that this shows the cinema tobe ‘a better metaphysician than common sense’,40 since it certainly does notfollow from this that motion can be reconstructed out of those components ofour experience of it that we are capable of attending to. This, one feels, is oneof the points in Russell’s work at which the dangers of armchair psychologyare most acute.The central point to note for our purposes, though, is how Russell changes

his conception of what the basic entities are in response to an epistemologi-cal requirement. He took it as given, that is to say, that we do know manyfacts about the external world. Having also made the atomistic assumptionthat there are objects in the world with which we are directly acquainted inperception, his project was then to explain the former in terms of the latter.In order to do so, he was prepared to enrich to the point of implausibility thestructure that he supposed the immediate objects of perception to possess.Logical atomism is perhaps the issue on which Wittgenstein’s debt to Rus-

sell is most apparent; the point of difference comes in their different responsesto the epistemological problem just noted. While Russell was pulled in onedirection or another by the technical constraints of his project, Wittgensteinwas not. Russell’s common-sense instinct led him in the Problems (1911) totreat the existence of matter as a ‘probable inference’; in ‘On matter’ (1912)he attempted to construct matter out of sense-data, and concluded, hesitantly,that it could not be done, but conjectured instead that it can be constructed

38CP, VIII, 6. 39Ibid. 40CP, VIII, 77.

#' Matter

out of things of the same nature as sense-data; in ‘The relation of sense-datato physics’ (1914) he enriched his conception of this nature in such a way as tomake the construction feasible, but with the same goal as before of providinga philosophical licence for our common-sense belief in the existence of matter.Wittgenstein, on the other hand, seems to have been resolute throughout inresisting the need to provide this belief with a grounding of any kind. Whatis striking about the few scattered remarks of his about Russell’s project thathave been preserved from this period is their consistency: they show no hintthat he felt any temptation to pursue Russell’s route towards an enriched con-ception of the internal structure of sense-data. And that is surely because henever felt the tension which had drawn Russell down this path.

Chapter $$$

Analysis

Whatever sympathy Wittgenstein may have had with the basic presumptionsof Russell’s project, we saw in the last chapter that he was sceptical of its goalof constructing matter out of sense-data or things of that kind. He thoughtthat physics could be explained without proceeding via a logical constructionof matter. What this exposes is a fundamental difference between the two menin their attitude to constructed entities and hence in their conception of themethod of analysis.

$.!$.!$.! Inference or construction?

The advantage of Russell’s method of analysis as he advanced it up to 1912was that it explained how we can understand the false sentence ‘The presentKing of France is bald’ without supposing there to be a shadowy, non-existentKing of France, because analysis reveals that the proposition the sentence ex-presses does not contain any component corresponding to the denoting phrase‘the present King of France’. An exactly similar analysis shows that the presentPresident of France is not part of the proposition expressed by the sentence‘The present President of France is bald’. But although we can understandthis sentence without being required to suppose that France has a president, itwill not be true unless she does. The analysis reduces dramatically the ontolog-ical demands of understanding, because the proposition does not contain thePresident as a component, but it leaves the demands of truth unchanged. Aproposition contains as components only entities with which I am acquainted,but its truth may require the existence of various other entities (in this case,the present President of France).Then, as we have seen, Russell’s conception changed radically. In 1912 he

made, and then in 1914 he purported to establish, the weak constructionalconjecture, that matter can be defined as a logical construction out of thingsof the same kind as sense-data. Bolstered, perhaps, by his success in this en-deavour, Russell now began to recommend the method of construction moregenerally: he adopted, that is to say, the methodological slogan that ‘wherever

$* Analysis

possible, logical constructions are to be substituted for inferred entities’.1 Theidea was not Russell’s: he attributed ‘the suggestion and the stimulus for itsapplication entirely to [his] friend and collaborator Dr Whitehead’.2 Nor wasthe idea itself new:3 within mathematics, it had already been current for someyears, and had been exploited throughout Principia. Rather than assuming theexistence of the real numbers, for instance, we devise a logical constructionwhich we can show to have the properties that we believe the real numbersto have. Having done this, we then use the logical construction in place of thereal numbers. What was new was only the application of this method in thecase where the entities to be replaced by logical constructions are not num-bers but people. If any assemblage of matter (such as the President of France)can be represented as a logical construction out of sensibilia, the truth of theproposition that the President of France is bald will not require the existenceof an inferred physical entity, a flesh-and-blood president, but only of variousclasses of sensibilia.Instead of a host of disparate empirical inferences to the existence of tables,

chairs, electrons, quarks, or interest rates, we make a single logical inference tothe existence of the classes required to make the axiom of reducibility true. Fora time, indeed, Russell thought that the success of the hypothesis of reducibil-ity in explaining empirical phenomena constituted a justification for assumingit. This has the virtue of unification, no doubt, but one might wonder whetherit also exhibits the advantages of theft over honest toil.Where does that leave common sense? It does not in itself show that ordi-

nary objects do not really exist alongside their logically constructed surrogates.This is, of course, an issue that Russell had already faced in the mathemati-cal case. If we construct logical objects with all the mathematical propertiesof the real numbers, say, do the real numbers themselves, as we previouslyconceived of them, drop away as unnecessary?In the first draft of ‘On matter’ in May 1912 Russell explicitly contrasted

two attitudes one could take to constructed entities.

If it is possible, by a logical construction, to define in terms of sense-data alone someobject having the properties which dynamics assigns to matter, the cautious theorizerwill adopt such a definition as the basis of his dynamics. In so doing, he insure[s]the certainty that he is not assuming the existence of something which perhaps doesnot exist, and he obeys one of the greatest principles of all philosophizing, I meanOccam’s razor: ‘Entia non multiplicanda praeter necessitatem.’ His task as logician,mathematician, and cautious man of science is then completed, and he may leave tothe philosopher the task of inquiring whether, by means of some a priori principle, heis able to give to the symbols another interpretation in which they convey knowledgeconcerning things not given in sense. We, as philosophers, must pursue this inquiry,

1CP, VIII, 11. 2CP, VIII, 12. 3See Miah, ‘The emergence of Russell’s logical construction ofphysical objects’.

Inference or construction? $)

knowing that mathematics and science can no longer help us, and that our conclusion,if affirmative, is forever incapable of empirical verification.4

I mentioned in the last chapter that there were parts of ‘On matter’ whichWittgenstein disagreed with. This intriguing passage, I shall suggest, may wellbe one of them. For here Russell somewhat curiously treats Occam’s razoras a methodological principle applicable only by the ‘cautious man of science’and not by the metaphysician. If by the method of logical construction wesucceeded in eliminating mention of some kind of entity (quarks, perhaps)from our scientific discourse, that would show that science has no need ofsuch entities; and these entities would, if they exist, be unknowable. Russell,however, leaves open the possibility that the metaphysician would nonethelesswish to assert that they exist. In relation to such entities he thus recommendsagnosticism rather than outright scepticism.When he revised the paper on matter, he dropped the passage just quoted,

but his sympathy for agnosticism did not die. In his 1918 lectures, for instance,he considered the possibility that my desk is a logical construction out of itsappearances.

If you can get on without assuming the metaphysical and constant desk, you have asmaller risk of error than you had before. You would not necessarily have a smaller riskof error if you were tied down to denying the metaphysical desk. That is the advantageof Occam’s Razor, that it diminishes your risk of error. Considered in that way youmay say that the whole of our problem [of logical constructions] belongs rather toscience than to philosophy.5

Russell’s way of drawing his distinction between the attitudes of the scien-tist and the metaphysician is curious, because it is the opposite of what weobserve in practice: it tends to be metaphysicians who doubt the existence oftables and chairs; men of science, however cautious, show no such inclination.And even on his own terms Russell’s attitude of agnosticism towards the tableis surely a little puzzling. According to his Principle of Acquaintance, all theentities that occur in the propositions we understand are ones with which weare directly acquainted. It is far from clear how on this view we can even un-derstand the hypothesis that a real, metaphysical table exists over and abovethe construction out of sense-data that Russell recommends to the cautiousman of science.Wittgenstein would no doubt have applauded the idea of drawing a distinc-

tion between the cautious man of science and the metaphysician. Later he wasoften concerned to distinguish philosophy sharply from the natural sciences,but the concern is already visible in the Notes on Logic, where he remarked that‘the word “philosophy” ought always to designate something over or underbut not beside, the natural sciences’.6 He would, however, have been much

4CP, VI, 512–3. 5CP, VIII, 243. 6B67.

$! Analysis

more likely to draw the distinction between the man of science and the meta-physician the other way round from Russell. One of the methodological prin-ciples that is prominent in his later philosophy is that the philosopher shouldnot attempt to change the practice of other subjects (mathematics, science, etc.):they should carry on just as before. His inclination was therefore to endorsethe fact that scientists talk about such things as electrons and protons, whileallowing nonetheless that the philosopher might have a very different view ofwhat is going on.The issue arises not only in relation to the problem of matter, of course, but

in all the cases in which we apply the method of incomplete symbols whichRussell developed after ‘On denoting’. Wittgenstein shows no trace of Rus-sell’s desire to say that any of the eliminated entities might ‘really’ exist—behind our backs, as it were. Indeed, what little evidence we have of hisearly philosophical views suggests that this habit of ontological parsimony wassomething he acquired very early. He seems to have arrived at Cambridgequite at ease with sparse ontologies—Russell described him as ‘the only manI have ever met with a real bias for philosophical scepticism’7—and in theirdisagreements it was always Russell who was more inclined to feel the drawof common-sense realism: it was Wittgenstein who had made him ‘more ofa sceptic’.8 Wittgenstein’s already mentioned refusal to accept that anythingempirical is knowable might, it is true, be the sort of naive scepticism one oftenmeets in philosophical novices, but it might also, more narrowly, be the ineptexpression of a refusal to infer from the immediate objects of experience theexistence of anything else.At base, the disagreement between Wittgenstein and Russell has to do with

our understanding of the variable. As we noted in §1.4, Russell had originallyhoped that his notion of a denoting concept would somehow explain our abil-ity to talk about objects we are not directly acquainted with. As Moore wasquick to notice, however, his theory of descriptions did not eliminate denotingconcepts completely, but only reduced them all to one, namely the variable;the problem of explaining how we talk about objects with which we are notacquainted remained, but the bridging role which Russell had previously as-signed to denoting concepts now fell wholly on the variable.

You say ‘all the constituents of propositions we apprehend are entities with which wehave immediate acquaintance’. Have we, then, acquaintance with the variable? andwhat sort of entity is it?9

Russell’s response was to admit the problem Moore had pointed out.

I admit that the question you raise about the variable is puzzling, as are all questionsabout it. The view I usually incline to is that we have immediate acquaintance with

7To OM, 2 May 1912. 8To OM, 20 May 1912. 9To BR, 23 Oct. 1905.

Wittgenstein’s conception $#

the variable, but it is not an entity. . . I only profess to reduce the problem of denotingto the problem of the variable.10

Russell’s problem was to explain, without merely lapsing into formalism, howwe succeed in being acquainted with the variable even though its domainincludes objects with which we are not acquainted. Russell’s shift from themethod of inference to the method of construction changed the nature of theepistemological gap the variable had to bridge, but it did not explain how tobridge it.

$."$."$." Wittgenstein’s conceptionWittgenstein’s response, characteristically, was not to solve Russell’s problemdirectly but to adopt a different conception of analysis which obviated theneed for a solution. Russell’s programme aimed to construct out of sensibilialogical proxies for the ordinary objects of the world such as the President ofFrance. If it had succeeded, it would have permitted us, in principle at least, toreinterpret sentences mentioning the President as being about his proxy, andtherefore in the end, if we carry through the analysis in full, about various sen-sibilia. Russell’s programme, that is to say, is one way of trying to reinterprettalk about the world as talk about sensibilia; but it is only one way. In his wayof proceeding, matter is replaced in the first stage of the analysis with a proxyconsisting of various classes of sensibilia. Since classes are in turn according toRussell logical fictions, to be eliminated in the second stage of analysis, thereis therefore a sense in which matter can be said to have been eliminated com-pletely. But because the first stage replaces matter with what appears to be alogical surrogate, the analysis invites—indeed seems designed to encourage—the speculation that there is matter nonetheless. Russell’s discussion does notshow any clear awareness of, or interest in, the possibility of a more radicalreinterpretation in which matter is eliminated in such a way that no apparentproxy for it is provided.Wittgenstein, on the other hand, was much readier than Russell to say not

only that there is no present President of France but that no single entity doesduty for him either. What logical analysis tells us is that the present Presi-dent of France is a complex made up out of components related in a certainmanner, but Wittgenstein denied that the complex is a further entity over andabove the components. When we say that France has a president, we shouldbe interpreted as saying that there are some components related in a certainway, a way which ordinary language speaks of as constituting a president, notthat any one object on its own is President of France. Wittgenstein’s concep-tion of analysis is therefore more radical than Russell’s. The fundamental idea

10To Moore, 25 Oct. 1905.

$$ Analysis

of his eliminative programme is that no proposition is genuinely about a com-plex: the apparent complex will always disappear on analysis to be replacedby a statement about its components.

Every proposition which seems to be about a complex can be analysed into a proposi-tion about its constituents and the proposition which describes the complex perfectly.11

This latter proposition, Wittgenstein explains in a gloss, is ‘that propositionwhich is equivalent to saying the complex exists’.12

In order to illustrate the sort of analysis Wittgenstein had in mind, consideranother example. My computer is complex. It is in fact very complex, but letus for current purposes assume (as I myself generally do when part of it doesnot work and needs replacing) that its immediate components—main unit,screen, keyboard, and mouse—are its ultimate constituents. In that case thestatement that my computer is on the desk can be analysed as the conjunctionof the statement that each of these constituents is on the desk and the state-ment that they are connected together so as to form a computer rather thanjust a set of parts awaiting assembly. In this analysis we do not require a quan-tifier ranging over complexes: the analysis is genuinely reductive in a sense inwhich Russell’s was not. Propositions which appear to be about matter turnout on analysis to be about simples, and the variable is no longer required toact as a bridge between the parts of the world with which I am acquaintedand those with which I am not.To express the point more generally we need to invent a notation: let us

write [p] for the complex, if there is one, consisting of the objects referred toin the proposition p, combined in such a way that p. So Caesar’s death, forinstance, is just [Caesar died]. And in one of Wittgenstein’s own examples abroommay be thought of as a complex consisting of the brush and the broom-stick: in the notation just introduced the broom is the complex [the broom-stick is attached to the brush]. On the other hand, there is no such complexas [Charles I died in bed], since Charles I died on the scaffold. Wittgenstein’sidea, then, was that a proposition !([p]) apparently about the complex [p]should be analysed as the conjunction of a proposition !!(a1, . . . , an) about theconstituents a1, . . . , an of the complex and the proposition p which describesthe complex, i.e. which ‘is equivalent to saying the complex exists’.13 In thecase of a relational proposition aRb, for instance, Wittgenstein’s proposal wasthat we analyse

!([aRb])" !!(a,b) aRb,

where !! is a propositional function appropriately related to the original func-tion !.

11C25. 12Ibid. 13At B17 Wittgenstein describes the analysis as a ‘logical sum’, but this is plainlyjust a slip: he meant logical product.

Practicalities $%

Notice, though, that the apparently innocuous issue of deriving from ! the‘appropriately related’ function !! masks a considerable degree of complex-ity. In the case of the first example considered earlier, of course, the answeris simple: my computer is on the desk just when its components are, so !! issimply the conjunction of the relevant instances of !. Similarly, in Wittgen-stein’s own example, ‘My broom is in the corner’ could perhaps be replacedby ‘The broomstick is there, and so is the brush, and the broomstick is fixed inthe brush’.14 But even a casual consideration of examples such as ‘The armysurrounded the castle’ and ‘My computer weighs 30 kg’ suggests that the linkis not always nearly as simple as this.In the Notebooks Wittgenstein tried repeatedly to ignore this issue, appar-

ently proposing that we should always define !([aRb]) to mean the same as!(a) !(b) aRb.15 This is perhaps one instance (among many) of the remark-able insouciance which he manifested throughout the period we are studyingtowards the details of analysis. Although he was willing to concede that howthe analysis is to be done is ‘an important question’, nonetheless, ‘its answeris not unconditionally necessary for the construction of logic’.16 Not untilJune 1915 did he finally face up to the difficulty, conceding that a propositionsuch as ‘The watch is lying on the table’ obscures a large (and highly context-sensitive, not to say indefinite) degree of complexity. Yet the concession wasonly temporary: the finished text of the Tractatus betrays puzzlingly little traceof it.

$.#$.#$.# Practicalities

Wittgenstein seems to have been confident that physics can be interpreted soas to be about sense-data but plainly had not the least idea how the reinterpre-tation was to be carried out. What we see emerging here is a clear differencein conception between him and Russell. According to Wittgenstein, logicis never concerned with the details of which resources we in practice needin order to represent the world around us; rather is it concerned with thosestructural features which must be present in those resources simply in orderthat it should be possible for them to represent a world at all.That is not to say that he had no interest in identifying the simple entities

that analysis would reveal as the ultimate constituents of the world. This isa recurring theme in the Notebooks, where there is, for instance, the lengthydiscussion of the notion of simplicity which I referred to a moment ago. Theabsence of firm evidence that he was interested in the question while at Cam-bridge need show no more than that he did not yet see any reason to doubt

14PI, §60. 155 Sep. 1914. 16B17.

$" Analysis

the account favoured (with minor variations) by both Russell and Moore atthe time.One of the striking features of the Tractatus, then, is Wittgenstein’s lack of

curiosity about how the reduction of various sorts of discourse to talk aboutsimples is actually to be carried through. The discussion of simplicity in June1915 resonates powerfully with Wittgenstein’s later philosophy, but not withthe Tractatus, where the obvious examples which create problems for his con-ception of analysis (such as ‘The army surrounded the castle’) are quite absent.Contrast this with Russell’s recommendation thata logical theory may be tested by its capacity for dealing with puzzles, and it is awholesome plan, in thinking about logic, to stock the mind with as many puzzlesas possible, since these serve much the same purpose as is served by experiments inphysical science.17

Examples of Wittgenstein’s failure to test his theory by its ‘capacity for deal-ing with puzzles’ abound in the Tractatus. For instance, he treated the in-compatibility of ‘This is red’ and ‘This is green’ as showing that redness andgreenness cannot be simple and require further analysis,18 but there is no signthat he spent much (or indeed any) time thinking out how that analysis mightgo. And he acquired this casual attitude to the practicalities of analysis early,remarking to Russell in 1912 that matter is a ‘trivial problem’.19

What is harder, perhaps, is to see whyWittgenstein took this attitude. As weshall see in the next chapter, he had from very early a conception of logic asbeing of ‘a totally different kind than any other science’,20 and would thereforeno doubt have disliked the analogy Russell drew between logical puzzles andscientific experiments. He thought, that is to say, that nothing in logic canhave a status even analogous to that of an experiment. But something morethan the observation that logic is different from physics is plainly needed ifwe are to show that logical theories cannot be confirmed or refuted by logicalpuzzles.We should recognize the possibility of a non-philosophical explanation for

Wittgenstein’s behaviour. The interest he began to take in the practicalities ofanalysis when he returned to philosophy at the end of the 1920s was stronglyinfluenced by his interaction with members of the Vienna Circle, who tooka decidedly scientific approach to philosophy and certainly saw themselvesas undertaking a project of analysis by which the discourse of natural sci-ence could be represented in a formal logical language. It was probably underpressure from them that Wittgenstein came to see the genuine difficulty whichthe incompatibility of ‘This is red’ and ‘This is green’ poses for his theory.21

Russell, at the time when Wittgenstein worked with him, was engaged in thebeginnings of a project that was similar in many ways to the Vienna Circle’s.17CP, IV, 420. 186.3751. 19BR to OM, 30 Apr. 1912. 20To BR, 22 June 1912. 21Cf. ‘Someremarks on logical form’.

Practicalities $&

So was Whitehead. Yet their effect on Wittgenstein seems to have been theopposite of the Vienna Circle’s fifteen years later: he simply left the technicalwork to them. The explanation cannot be merely that they were both so muchbetter at it than Wittgenstein, although this is no doubt true. This is a matterof symbolic facility and technical ingenuity: Russell and Whitehead were bothhighly competent mathematicians, which Wittgenstein was not. Rather is itconnected with the desire, or even need, which Wittgenstein evidently had atthis early stage in his life to find a field in which he could make a distinctivecontribution. Some of the things Wittgenstein did—his jet engine patent, ex-periments on the psychology of rhythm—have in retrospect a certain resem-blance to the dabblings of a wealthy dilettante. But they also betray signs of hisambition, a desire to find a new field in which he could make his mark. Whenhe arrived in Manchester, there were no doubt other available engineeringprojects less speculative (and probably, in the short term, more fruitful) thantrying to develop a jet engine; but because avionics was a new science, anysuccess he might have had in it would have had that much more impact. Andin the same way one of the things that attracted him to experimental psychol-ogy was surely that it too was an unploughed field: the Cambridge laboratoryin which Wittgenstein did his experiments was only just being set up at thetime.The need to find a creative equilibrium with Wittgenstein is a recurring

theme in Russell’s letters to Ottoline, and he often seems to have conceivedof this as much in terms of marking out his own territory as of negotiatinga productive means of collaboration. Wittgenstein no doubt felt the point atleast as keenly. His efforts to respond to it, and hence find a contribution thatwas distinctively his own, perhaps encouraged him in what became one ofhis most characteristic philosophical techniques, that of seeing past a piece offormalism so as to produce a criticism that is independent of its details.Something of this technique we have perhaps already seen in embryo in his

marginalia to Lamb’s Hydrodynamics, but it is at its most intense in his applica-tion of it to Principia Mathematica. Wittgenstein certainly studied Principia. Weknow this, because when Russell tried to sell Wittgenstein’s copy of VolumeI (presumably left behind in Cambridge when he departed for Norway andtherefore part of the job lot Russell bought from him after the war), it ‘still haddecorations by Wittgenstein’.22 Until Wittgenstein’s copy resurfaces, it canonly be a matter of speculation whether the decorations have the somewhateccentric character of Wittgenstein’s earlier annotations of his copy of theHydrodynamics, or whether they indicate a serious engagement with the formaltechnicalities of the logic Whitehead and Russell developed. In the meantime,though, it is surely worth drawing attention to the curiously hesitant qualityof Wittgenstein’s criticisms of Principia: again and again he gestures towards

22BR to Ogden, 23 June 1922.

$' Analysis

errors in Russell’s presentation but stops just short of saying exactly what theyare or, more importantly, quite what ought to be done about them.Perhaps the most startling fact to note in relation to Wittgenstein’s concep-

tion of analysis, though, is how early he came upon it. Both the idea itselfand his insouciant attitude to the details of its implementation seem alreadyto have been in place by April 1912, when he told Russell—without, surely,any idea how it could actually be done—that ‘if there is no matter, . . . physicsand astronomy, and all the other sciences could still be interpreted so as to betrue’.23

23To OM, 23 Apr. 1912.

Chapter %%%

The fundamental thought

Later, not just in the period leading up to the Tractatus but throughout hisphilosophical career, Wittgenstein was an inveterate keeper of Tagebücher—journals—in which he recorded day-by-day his philosophical thoughts, andthere is no reason to think that his method of working was any different whilehe was at Cambridge. It may well be that he was carrying on a habit he hadlearned as an engineering student: it bears an obvious similarity to the prac-tice scientists have of keeping lab books. However, the notebooks devoted toWittgenstein’s Cambridge period have not survived: he left them in storage inCambridge when he left for Norway in 1913, and after the war asked Russellto have them destroyed.1 So the only evidence we have from Wittgenstein’sown pen of his philosophical progress consists of letters he wrote to Russell,usually during vacations when one or other of them was away from Cam-bridge. Our aim in the next few chapters will be to find out what these letterscan tell us about the development of his thought.

%.!%.!%.! Why logic?When he arrived at Cambridge in October 1911, Wittgenstein told Russellthat he wanted to work on the philosophy of mathematics. Yet the surviv-ing letters make it clear that from quite early on the focus of his work wasthe philosophy not of mathematics but of logic: he was trying to construct atheory of the symbolism, by which he meant an account of the structure ofpropositions which would both characterize the notion of logical truth andexplain its nature. With remarkable stability of purpose, moreover, this re-mained the focus of his work until at least the end of 1915: the book which hewas then beginning to compile was originally, it seems, to be called Der Satz(‘The Proposition’).2

But this was not really a change of focus from Wittgenstein’s original in-tention to work on the philosophy of mathematics. By the time he arrived inCambridge Whitehead and Russell had published volume I of Principia Math-ematica, the book in which they aimed to make good Russell’s claim in the

11 Nov. 1919. 2Bartley,Wittgenstein, 45.

%* The fundamental thought

Principles that mathematics is part of logic. Quite soon, it seems, Wittgensteindecided to subject the philosophical foundations of Principia to a detailed cri-tique. One impetus to this will no doubt have been his meetings with Frege,whose views on Principiamay be fairly gauged from the letter he wrote to Jour-dain in 1914. This expounds over several pages Russell’s confusions over theuse of the word ‘variable’, but nowhere in the letter do Frege’s criticisms ad-vance beyond the first fifteen pages of the Introduction. (His first draft of theletter makes a brief comment on the explanation of the ramified hierarchyon pages 52–4.) Frege had never been the most sympathetic reader of otherphilosophers’ work, but by this stage in his life he evidently found it hard toturn to the second page of a book until he had corrected all the errors of ex-pression on the first. It is easy to see how a conversation with him might havegiven Wittgenstein the idea of writing an extended critique of the Introduc-tion to Principia, with the aim of putting the logic presented there on a firmphilosophical foundation.It is clear, too, that Russell himself soon accorded with Wittgenstein’s

project of criticizing and revising the early pages of Principia: from April 1912onwards he repeatedly spoke of abandoning technical philosophy to Wittgen-stein;3 in February 1913 he remarked, ‘Wittgenstein has persuaded me thatthe early parts of Principia Math[ematic]a are very inexact, but fortunatelyit is his business to put them right, not mine’;4 and by August Pinsent waswriting in his diary, ‘It is probable that the first volume of the “Principia” willhave to be rewritten, andWittgenstein may write himself the first eleven chap-ters. That is a splendid triumph for him!’5 The first eleven chapters constituteprecisely the parts of Principia that deal with logic proper, Section A on ‘Thetheory of deduction’ and Section B on ‘Theory of apparent variables’: it wasnot proposed, it seems, that Wittgenstein should revise Section C on ‘classesand relations’.Russell recognized quite early that Wittgenstein’s approach involved above

all a concern with the most fundamental philosophical issues. ‘He doesn’twant to prove this or that, but to find out how things really are.’6 It probablytook him some time, however, to realize just how true this was. When heinvited Wittgenstein to rewrite the first two sections of Principia, the Tractatuswas presumably not quite what he had in mind: when he eventually realizedthat Wittgenstein was not going to make the kind of revisions he envisaged,he had to make the revisions by writing an Introduction to the second editionhimself.Throughout, Wittgenstein’s approach to the task evidently shared some-

thing with Frege’s, mentioned earlier. His concern was, as Frege’s had been,overwhelmingly with the very earliest parts of Principia. There is very little evi-3To OM, 30 Apr., 1 June, 24 July, 4 Sep., 14 Oct. 1912, 22 Jan., 7 Feb., 23 Feb. 1913. 4ToOM, 23 Feb. 1913. 529 Aug. 1913. 6To OM, 8 Mar. 1912.

Logical constants as incomplete symbols %)

dence requiring us to suppose that he ever got much beyond the Introduction,and the vast majority of his comments are directed at issues that crop up inthe first twenty pages of that. Moreover, the issues which concerned him weregenerally those that one might notice if one focused on Russell’s manner ofexpression rather than on the general drift of his exposition. To the mathe-matician’s eye the most singular move in the whole Introduction, which criesout for more justification than Russell could give it, is surely the Axiom ofReducibility. Yet we have no evidence that Wittgenstein was concerned withthis problematic axiom until the summer of 1913, and even then he seems atfirst to have recognized only dimly what makes it controversial.

%."%."%." Logical constants as incomplete symbolsSuppose for a moment that we start, as Wittgenstein evidently did, with Rus-sell’s conception of propositions as complexes. If p and q are two such com-plexes, what is there in the world that would make the conjunction ‘p and q’or the disjunction ‘p or q’ true? Russell’s earlier view had been that these maybe explained as more elaborate complexes: ‘p and q’, for instance, would bea complex consisting of the complex p, the complex q and the universal con-junction. In his lectures during February 1912, however, we find Russell saying(according to Moore’s notes), ‘logical constants are not the sort of constantswh[ich] can be substituted: e.g. or, not, true, 0, 1, 2 etc. All of these are in-complete symbols (I think, but am not sure).’ We know that Wittgenstein likedto read Russell’s lectures before he gave them, in order (presumably) to get hisobjections in early.7 So it may be that a trace of Wittgenstein’s influence is tobe found in what Russell had to say here, but whether in the suggestion or inthe hesitation it is hard to be sure.There are two parts to Russell’s suggestion. The first part, that occurrences

of logical constants in a proposition cannot be replaced by variables, is some-thing he had long held, at any rate in application to constants such as ‘or’and ‘not’: he stated it explicitly in the Principles, for example. Moreover, it issomething Wittgenstein himself later held: he stated it in the Notes dictated toMoore;8 and in the case of numbers it is a view he expounded in the Tractatusand probably held by the summer of 1914, if not before.As we saw in the last chapter, the second part of Russell’s suggestion, that

logical constants are incomplete symbols, need not on Russell’s lips have com-mitted him to denying that there are such things as logical constants. It would,on the other hand, have committed him to what we would nowadays call aresearch programme, namely to provide contextual rewriting rules for all thecontexts in which logical constants can meaningfully occur. And there is evi-

7BR to OM, 7 May 1912. 8¶47.

%! The fundamental thought

dence that Wittgenstein may have been engaged during the summer of 1912in a variant of such a programme. In one letter he suggests to Russell thatuniversal generalizations might be incomplete symbols:

Will you think that I have gone mad if I make the following suggestion?: The sign‘(x) !(x)’ is not a complete symbol but has meaning only in an inference of the kind:from ! !x !x " x !(a) follows "a. Or more general: from ! (x) !x #0(a) follows !(a).I am—of course—most uncertain about the matter but something of the sort mightreally be true.9

The caution with which Wittgenstein presented his ideas here presumably hassomething to do with the distance he was now going beyond what Russell hadsuggested in his lecture. The idea then had been that the logical constantsare incomplete symbols. In the current case that would amount to holdingthat any sentence involving a universal quantifier should be rewritten in sucha way that the universal quantifier disappears. But now Wittgenstein wasflirting with something much more like the idea that the sense of a logicalconstant is given by the inference rules that govern its application. In anotherletter10 around this time Wittgenstein introduced rather similar speculationsconcerning the meaning of disjunctive propositions, but once again he offeredthem only hesitantly and we have no evidence that he developed them veryfar (nor any reason to think that he would have been well advised to do so).The proposals Wittgenstein makes in these letters are thus experimental at

best. Moreover, it is hard for us now to fill out his sketches in anything likea satisfying manner. This is at least partly because Wittgenstein himself soonabandoned them (in many cases wisely), with the result that his later, muchmore detailed writings contain no further clues as to what he had in mind.

%.#%.#%.# There are no logical constantsLet us turn now to an earlier letter Wittgenstein wrote to Russell. It dates fromearly in the summer of 1912, when Wittgenstein had not yet left Cambridgefor the long vacation, and the philosophical remarks it contains are the earliestfrom Wittgenstein’s pen that we have. In contrast to the letters from laterthat summer, this one contains thoughts that turned out to be of fundamentalimportance for his future work: what chance has preserved for us as his firstphilosophical remarks are among his most significant.

Logic is still in the melting pot but one thing gets more and more obvious to me: Thepropositions of logic contain ONLYONLYONLY apparent variables and whatever may turn out to bethe proper explanation of apparent variables, its consequences must be that there areNONONO logical constants. Logic must turn out to be a totally different kind than any otherscience.11

91 July 1912. 10[Summer 1912] (CL, no. 4). 1122 June 1912.

There are no logical constants %#

Wittgenstein’s most immediate target in saying that there are no logicalconstants was of course Russell, whose conception entailed that items of dis-tinctively logical vocabulary such as ‘not’ and ‘or’ refer to entities. But theremark also opposes him to Frege, whose inventory of logical objects includednatural numbers, real numbers and truth-values, all of which he reduced inthe Grundgesetze to a single kind of logical objects which he called value-ranges.If Wittgenstein had by this time adopted the view described in the last chap-

ter, that there is nothing in the world to which an incomplete symbol refers,his pronouncement that there are no logical constants might be read at firstsight as no more than a redescription of the research programme he had em-barked on, apparently at Russell’s instigation, of providing elimination rulesfor logical constants. Even if so, however, it is plain that the motivations ofthe two men were different. For Russell the idea that logical constants are in-complete symbols was a technical conjecture to be tested: the test is whetherwe can find appropriate rewriting rules that allow us to eliminate the logi-cal constants from all the contexts in which they occur meaningfully. ForWittgenstein, on the other hand, the reason to think that there are no logicalconstants is not a technical one but springs from his conception of the natureof logic. The completion of the research programme would therefore at bestprovide confirmation of what he already thought he knew.This is characteristic of a difference in philosophical method which runs

right through their work: Russell, it is clear, enjoyed technical challenges fortheir own sake; but more than that, he thought that technical work could leadto philosophical conclusions. Wittgenstein, by contrast, was inclined to in-dulge in technical projects only in order to fill the gaps in his system which hisphilosophical speculations had already identified, and usually with the greatestreluctance. Russell once confessed it to be one of his dreams ‘to found a greatschool of mathematically-minded philosophers’. Wittgenstein, he thought,was ‘exactly my dream’.12 But it is hard to believe that Wittgenstein was everquite what Russell was hoping for: his work was never really mathematicallyminded in the way that Russell meant.It is worth noting, too, that in Russell’s usage the phrase ‘logical constant’

was of wide range: he applied it, for instance, to what he called propositionalforms, such as the general form of all subject-predicate propositions or thegeneral form of all relational propositions.13 No doubt Wittgenstein intendedhis remark to apply to all such supposed entities as well. His usage did laterwaver a little: in the wartime Notebooks he was willing to grant that there isone ‘logical constant’, namely the general form of proposition.14 However, itwould take a lot of work to represent this as a significant retreat from his 1912view, since (as Wittgenstein’s use of scare-quotes hints) whatever the sense is in

12To OM, 29 Dec. 1912. 13Cf. McGuinness, Approaches, 104. 14NB, 5 May 1915.

%$ The fundamental thought

which the general form of proposition can be thought of as a logical constant,it is at some distance from the uses Russell and Frege made of the term.On the other hand, perhaps the exact scope Wittgenstein meant the term

to have is not especially important, because he does not seem to have sup-posed that he had a general argument that there are no logical constants. Thusin the Notes on Logic we find not the general claim that there are no logical con-stants but only particular instances of it: in various places he targeted Russell’sconception of ‘not’ and ‘or’ and of variables, and Frege’s conception of truth-values, and he offered piecemeal arguments against these different targets. Hedid not think it sufficient simply to appeal to the general claim. Perhaps thereason is that the general claim had for him the character more of a guidingmethodological principle—he later called it his ‘fundamental thought’—thanof a firm conclusion for which he could offer a persuasive argument.

%.$%.$%.$ There are no real variables

On Russell’s understanding variables, too, were (somewhat confusingly) log-ical constants. This is because, at least when he wrote the Principles and forsome time thereafter, he tended to use the word ‘variable’ to refer not to thesigns ‘x’, ‘y’, etc. that are used in his logical notation, but to what these signsexpress. Even in Principia Russell still often speaks this way: for instance, hetalks of a logical assertion as concerning a variable x in the same way that anon-logical assertion concerns Socrates or Plato.15 So when Wittgenstein saidin his letter to Russell that the propositions of logic do not contain real vari-ables, this may be read as a particular case of the general claim just discussed:if there are no logical constants, there are no real variables in the sense of thePrinciples, and hence no propositions that are about them. It is worth observ-ing, though, that (in conformity with the pattern we noted) Wittgenstein doesnot present his expulsion of real variables from logic as a consequence of thegeneral claim. Rather he holds out the hope that a correct account of thevariable might somehow have the general claim as a consequence.If we are to follow Wittgenstein’s thought processes here, we should start

by considering his claim that the propositions of logic contain only apparentvariables; or, as we should say nowadays, that any variables in them may onlyoccur bound and not free. Wittgenstein’s complaint therefore has somethingto do with Whitehead and Russell’s practice of stating theorems in such a waythat they involve real variables as well as apparent variables. ‘There is,’ ashe later put it, ‘no proposition which is expressed by “x = x”, for “x” has no

15PM, I, 93.

There are no real variables %%

signification; but there is a proposition “(x) x = x” and propositions such as“Socrates = Socrates” etc.’16

But if this were Wittgenstein’s only point, it would surely not go at all deep,since the practice just mentioned seems to be only a harmless abbreviatingconvention. After all, very many theorems in logic and mathematics havethe logical form of universal generalizations. So it would be a convenience toadopt the convention that if we assert a formula involving an apparent vari-able ‘# !(x)’, we should be taken to mean to assert the universal generalization‘# (x) !(x)’. All the convention does, it seems, is to make the formula shorterand hence somewhat easier to read.That leaving off initial universal quantifiers is best thought of as no more

than a convenient abbreviating convention is indeed a rather obvious point,and not one that originates with Wittgenstein: the insistence that whatevercan be asserted cannot contain a free variable is ubiquitous in Frege, for in-stance. What is capable of truth or falsity is a complete thought, he says, andwhat expresses a complete thought is a sentence. If we replace an occurrenceof a name in a sentence with a variable, the resulting formula does not expressa thought. So although in the Begriffsschrift17 Frege adopted the abbreviat-ing practice just mentioned, he made it clear that it was only an abbreviation,and therefore harmless. (Indeed he adopted at the same time, presumablyto emphasize his awareness that it is only an abbreviating device, the furtherconvention that variables whose binding is supplied only implicitly by the ab-breviating device should be written with Latin letters, whereas he always usedGerman letters for ordinary bound variables.)On the face of it, then, Wittgenstein was merely advocating a piece of

elementary logical hygiene which he had learnt from Frege; a piece of hy-giene, moreover, with which Whitehead and Russell would have agreed read-ily enough. They stated quite clearly, after all, that ‘ “x is hurt” really makesno assertion at all, till we have settled who x is’.18 They made use of the notionof the assertion of a propositional function in Principia, therefore, only becausethey found it technically convenient to do so.But it is convenient, and the hygiene involved in eliminating it, although in

a sense elementary, is by no means trivial. It is far from obvious, that is to say,how to amend the formal presentation of logic in Principia so as to eliminatethe need to assert propositional functions rather than just propositions. Thisis because in Principia the notion of asserting a propositional function, as whenwe write ‘# !(x)’, is taken not as an abbreviation for ‘# (x) !(x)’, as in effect itis by Frege, but as a distinct primitive idea. Because of this, various primi-tive propositions are then needed to govern the transitions between the twosorts of assertion. For example, they had to adopt the rule, ‘In any assertion

16B13. 17§11. 18PM, I, 14.

%" The fundamental thought

containing a real variable, this real variable may be turned into an apparentvariable of which all possible values are asserted to satisfy the function in ques-tion.’19 The result of all this is that it is rather hard to determine from a casualinspection how easy it would be to recast $9 of Principia so as to remove theneed to take assertion of a propositional function as a primitive notion. Thecomplexity of the task may be judged, indeed, from Russell’s introduction tothe 1925 second edition of Principia, where he attempted to complete it by of-fering a new $8 to replace $9 in which assertion of a propositional functionwas no longer taken as a primitive.Note, moreover, that in his letter to Russell, Wittgenstein did not make the

general claim that propositions contain only apparent and not real variables(even though that is certainly true), but restricted himself to the special caseof the propositions of logic. This is at first sight odd since, as we have alreadyobserved, there is an obvious argument leading directly to the conclusion thatno proposition can contain a free variable. (An expression containing a freevariable can only be a formula, not a sentence. And a formula cannot expressa proposition; only a sentence can do that.) This suggests rather strongly thatthe general point was at this stage a peripheral issue, and that what reallyconcerned Wittgenstein was a specific point about the nature of logic.One reason Wittgenstein limited himself to the claim that the propositions

of logic do not contain real variables is no doubt that what motivated himto make it was a conception of logic as a subject without a subject matter.On Russell’s conception a proposition containing a sign for a real variablesays something about that variable. But according to Wittgenstein the propo-sitions of logic are not about anything; and that is why they cannot containreal variables. Russell’s use of real variables, which ought to have been (asit was for Frege) no more than a convenient abbreviation, had on Wittgen-stein’s diagnosis misled Russell into imagining that logic has a subject matterof which these variables form a part. ‘By employment of variables instead ofthe generality-sign, it comes to seem as if logic dealt with things which havebeen deprived of all properties except thing-hood.’20

But there is another point Wittgenstein is making here. When he says thatthe propositions of logic contain only apparent variables, he is ruling out notonly real variables but constants too. The reason is, once again, his con-ception of logic as a subject without a subject matter. He does not herestate the converse, that the propositions of logic are the only true proposi-tions which do not contain any constants, but he may for a time have be-lieved it. At any rate, it was not until he was in Norway that he developeda criterion—tautologousness—to distinguish between logical truths and thosethat are wholly general but true only by accident.

19PM, !9.13. 20B71.

Logic as a special science %&

%.%%.%%.% Logic as a special scienceAccording to Wittgenstein, the reason that the propositions of logic are notabout anything is that ‘logic must turn out to be a totally different kind than anyother science’. He explained the point in a little more detail in the Notes, wherehe said that a reason against the existence of logical constants ‘is the generalityof logic: logic cannot treat a special set of things’.21 His view contrasts starklywith that of Russell, for whom the world has a physical aspect, which is thebusiness of physics, and a logical aspect, which is the business of logic. Inorder for Russell’s view of logic to make sense, there has to be something, asubject matter, that logic may be said to be about. Over the decade since thePrinciples Russell had been forced, in his various responses to the paradoxes,to revise several times what he thought this subject matter was, but he neverseems to have given up the view that it has a subject matter of some kind. Eventhe conjecture he made in his 1912 lectures that the logical constants may beincomplete symbols is not really a denial of this view. He hoped, perhaps, thatall the contexts in which ‘!’ and ‘v’ occur meaningfully could be analysed insuch a way that these logical constants do not occur in the analysis. But in thatcase other notions with some title to be regarded as logical would presumablyoccur in the analysans. These other notions would then turn out to be the reallogical constants. There is no suggestion that there might turn out to be nological constants at all.Wittgenstein’s view, on the other hand, is just that: when language is cor-

rectly analysed, it must turn out that there is nothing in the world correspond-ing to any of the logical words in our vocabulary. His reason was that Russell’sassumption of the existence of a logical realm existing somehow alongsidethe physical one rendered it inexplicable why the laws obeyed by the formershould be applicable to the latter. His alternative explanation, discernible al-ready (at least in outline) in his letter of June 1912, is that logic applies inreasoning about anything whatever just because it has no subject matter.Wittgenstein said in the Notes both that philosophy does not exist alongside

the natural sciences22 and that ‘philosophy consists of logic and metaphysics:logic is its basis’.23 It follows in particular that logic cannot exist alongsidethe natural sciences either. One way of expressing this would be to press thequestion of how Russell’s notion of acquaintance could apply in the case oflogic—to ask the question how it would be possible to apply logic if therewere such a thing as logical experience. ‘We could say: if there would bea logic, even if there were no world, how then could there be a logic, sincethere is a world.’24 This way of making the point is of a later date, and reflectsthe fact that by then Wittgenstein had a developed account to offer of thenature of logic: logic is an expression of the structure of the world, and so

21C9. 22B67. 23B61. 245.5521.

%' The fundamental thought

if there were no world, there would be no logic. But the underlying pointremains the one he hadmade in 1912, namely that Russell’s conception makesthe propositions of logic into substantive facts about a logical realm whichexists independent of the physical world (and would exist even if there wereno world); far from explaining the universality of logic, Wittgenstein thought,this makes it utterly mysterious.

%.&%.&%.& Logic as contentlessLogic ‘cannot treat a special set of things’, because if it did it would not bewholly general. The generality of logic that Wittgenstein is appealing to herestands in contrast to the other sciences. We might think of it as the businessof scientists to formulate laws of general applicability, but in saying this weassume implicitly that the range of applicability of these laws will not be com-pletely general. The laws of supply and demand may apply quite generally toeconomic transactions between people, but they are of no use whatever if weare studying the flow of water on a river bed. The laws of fluid dynamics willbe relevant in this case, but they in turn are of little use in studying the motionof the planets. If we think of logic as a science, however, we are immediatelystruck by its special character: it is applicable in reasoning about anythingwhatever. There seems to be something radically incoherent about supposingthat there could be anything not subject to the laws of logic.In previous chapters of this book it has almost exclusively been Russell’s

influence on Wittgenstein that we have been discussing. From now on, how-ever, it will be increasingly necessary to take into account the influence, inwriting and in person, of Frege. I have already mentioned a visit Wittgen-stein made to Frege in 1911. The only other visits that we know of were atChristmas 1912 and Christmas 1913, and lasted a few days at most. Althoughthe personal contact between them was thus very limited, the effect Frege’sthinking had on Wittgenstein was nonetheless profound. The letter we arenow discussing already shows signs of his influence, for the very question thatnow engaged Wittgenstein—how to explain the nature of logic—was one thathad preoccupied him.The laws of logic, Frege said, have a special title not possessed by the laws of

geometry and physics because ‘they are the most general laws, which prescribeuniversally the way in which one ought to think if one is to think at all’.25 No-tice how Frege here advances beyond the thought just adumbrated that logicis the maximally general science—the thought, that is to say, which might becaptured by saying that its propositions contain only apparent variables—inorder to make the stronger claim that it represents a normative constraint on

25Gg, I, xv.

Logic as contentless %(

thought itself. The generality of logic, he might have said, is not accidental gen-erality.26 And when he came to offer an explanation for logic’s universality,his thought took a correspondingly unexpected turn. Geometry and physicshave truth as their goal, he said, but only logic has truth as its subject matter.Logic is concerned with the predicate ‘true’ in a quite special way, namely in a wayanalogous to that in which physics has to do with the predicates ‘heavy’ and ‘warm’or chemistry with the predicates ‘acid’ and ‘alkaline’.27

For Frege, the point of conceiving logic in this manner was that it allowedhim to resist psychologism: the laws of logic are not laws of thought but lawsof truth.I understand by ‘laws of logic’ not psychological laws of takings-to-be-true, but lawsof truth. If it is true that I am writing this in my chamber on the 13th of July 1893,while the wind howls out-of-doors, then it remains true even if all men should subse-quently take it to be false. If being true is thus independent of being acknowledged bysomebody or other, then the laws of truth are not psychological laws: they are bound-ary stones set in an eternal foundation, which our thought can overflow, but neverdisplace.28

When Wittgenstein found himself in a prisoner of war camp at the end of thewar, he quoted passages of Grundgesetze from memory to another prisoner inorder to convince him of the glories of its preface.29 It is hard not to imaginethat this was one of the passages he recited; hard, too, to disagree with hisassessment of it.When Frege tells us that logic is concerned with the predicate ‘true’ as

physics is with the predicate ‘heavy’, we may read him as saying that logicis about truth just as physics is about heaviness. But this should give us pausewhen we unite it with his further claim that truth is not a property as heavinessis. The peculiarity of truth, that is to say, is not so much that it ‘is obviouslysomething so primitive and simple that it is not possible to reduce it to any-thing simpler’, but rather that it makes no specific contribution to the senses ofsentences in which it occurs. ‘What distinguishes it from all other predicatesis that predicating it is always included in predicating anything whatever.’30

The special character of logic therefore consists for Frege in the peculiarlyevanescent quality of its subject matter.The word ‘true’ is not an adjective in the ordinary sense. . . The sense of the word‘true’ is such that it does not make any essential contribution to the thought. If I assert‘it is true that sea-water is salty’, I assert the same thing as if I assert ‘sea-water issalty’.31

The quotation is from a note in Frege’s Nachlaß of uncertain date headed ‘Mybasic logical insights’, but Wittgenstein could have found the same idea inFrege’s published writings if he read them carefully enough.26Cf. 6.1232. 27PW, 128. 28Gg, I, xvi. 29McGuinness, Young Ludwig, 270. 30PW, 129. 31PW,251.

"* The fundamental thought

One can, indeed, say: ‘The thought, that 5 is a prime number, is true.’ But closerexamination shows that nothing more has been said than in the simple sentence ‘5 isa prime number’. The truth claim arises in each case from the form of the declarativesentence, and when the latter lacks its usual force, e.g. in the mouth of an actor uponstage, even the sentence ‘The thought that 5 is a prime number is true’ contains onlya thought, and indeed the same thought as the simple ‘5 is a prime number’.32

Truth’s peculiarity, in other words, is that it is not properly to be conceivedof as a property at all.33 But if logic is about truth and truth is in this senseredundant, it is a short step to conclude that logic is not about anything.Wittgenstein’s conclusion that logic is contentless thus derives from two

tenets central to Frege’s thinking. But however short a step it is from Frege’sviews to this conclusion, it is one that Frege himself never took. Why not? Theshort answer is that he did not take the redundancy just mentioned to showthat the word ‘true’ has no sense. Early and late, he always shied away fromsaying that truth is redundant and emphasized instead not its redundancy butits indefinability. ‘What true is, is indefinable’,34 he wrote in an early note ofhis views on logic; and in 1918 he still maintained that truth is ‘sui generis andindefinable’,35 before wondering ‘whether it can be called a property in theordinary sense at all’.36 But he always stopped short of saying that there is nosuch property.This short answer, though correct, immediately invites the further question

why Frege resisted at just this point. After all, if we grant that the sense of ‘It istrue that sea-water is salty’ is just the same as the sense of ‘Sea-water is salty’,this might tempt us (as Frege himself granted) ‘to think that the word “true”has no sense at all’. Frege’s response, however, was to resist this temptationbecause

in that case a sentence in which ‘true’ occurred as a predicate would have no senseeither. All one can say is: the word ‘true’ has a sense that contributes nothing to thesense of the whole sentence in which it occurs as a predicate.37

Rather than treating the absence of the word ‘true’ from the second sentenceas showing that it contributes nothing to the sense of the first, therefore, heconcluded instead that the sense of the word somehow lurks in the locality ofevery sentence, implicated in, but not quite contributing to, its sense.The position to which Frege was driven in order to avoid admitting the

redundancy of truth is unquestionably puzzling: it is hard to see how a wordcan have a sense which makes no contribution to the sense of sentences inwhich it occurs. But if Frege’s stance is so very implausible, there must havebeen something driving him to adopt it. Perhaps, then, part of his reason forresisting the redundancy of truth was that he saw just where it would lead.

32‘Über Sinn und Bedeutung’, 34. 33Cf. PW, 129 and 233. 34PW, 174. 35‘Der Gedanke’, 60.36‘Der Gedanke’, 61. 37PW, 251–2.

The fundamental thought ")

Since inventing the Begriffsschrift in 1879 Frege had been keen to emphasizehis logical system’s fruitfulness, its ability to ‘bring forth judgments that at firstsight appear to be possible only on the basis of some intuition’.38 In contrastto the old logic, which only takes out of the box what has just been put intoit, the polyadic logic of the Begriffsschrift yields a content sufficiently rich thatmathematics can be founded on it. If that content rests ultimately on noth-ing but the concept of truth, then that concept, even though sui generis andindefinable, cannot, Frege thought, be wholly redundant.

%.'%.'%.' The fundamental thoughtI mentioned earlier that Wittgenstein called his insight that there are no log-ical constants his Grundgedanke, his ‘fundamental thought’. The descriptioncomes from the Notebooks,39 but Wittgenstein was sufficiently impressed by itto think it worth repeating in the Tractatus itself.40 Why did he regard it asfundamental? Some commentators, understanding ‘fundamental thought’ tomean something like ‘central claim’, have found the remark puzzling. Andindeed it is not completely straightforward to present the claim that logicalconstants do not represent as the argumentative lynchpin of the Tractatus. Butthe rather lowly place (4.0312) Wittgenstein gives the remark in the number-ing system of the Tractatus suggests that this is not quite what he meant. Ifinstead we interpret Wittgenstein’s remark rather more autobiographically,as meaning that his insight concerning logical constants was what started himon the journey away from Russell’s conception of logic towards the accountgiven in the Tractatus, it is much less puzzling. Indeed the remainder of thisbook will provide ample confirmation of the central role Wittgenstein’s funda-mental thought played in the process which led him to the Notes on Logic, andthence to the Tractatus.What might be taken at least as a central motivation of the Tractatus is per-

haps not so much the claim that logical constants do not represent, but ratherthe reason he offers for thinking that this should be so, namely that logic mustturn out to be a totally different kind than any other science. Russell’s con-ception put logic alongside the natural sciences and hence rendered its specialnature inexplicable. Wittgenstein’s, on the other hand, offered the hope ofdissolving the epistemological problem because if logic is contentless there isnothing there to know.The centrality of this thought in the Tractatus is to be explained by the use to

which Wittgenstein puts it in subverting the transcendental idealism of Kant’sCritique. By the time of the Tractatus, that is to say, Wittgenstein saw the empti-ness of logic as obviating Kant’s need for a transcendental deduction of the

38Bs, §23. 39NB, 25 Dec. 1914. 404.0312.

"! The fundamental thought

categories. No deduction is needed because there are no categories to requirevalidation in their application to the world. And without a transcendentaldeduction there is no pressure towards transcendental idealism either. Butthere is nothing to suggest that he understood that until 1916. What is strik-ing is how the view that logic is empty is fully present in 1912, although thelater application of it is not. And it is plain that this emptiness is somethingwhich, for Wittgenstein, flowed from his fundamental thought. But the flowwas autobiographical at least as much as it was strictly logical.

Chapter &&&

The symbolic turn

One of the most profound themes in the Tractatus is its exploration of theharmony between the structure of the world and the structure of the symbolswe use to talk about it. This theme began to emerge in Wittgenstein’s writingwhile he was at Cambridge. The ‘symbolic turn’ in his thought is in its wayas significant a moment as the fundamental thought we discussed in the lastchapter. What led him to it was his emerging concern, already mentioned,with understanding the nature of propositions.

&.!&.!&.! PropositionsThe account of propositions which Moore proposed in 1898 and Russell ad-vocated in the Principles was that a proposition is a complex which may haveamong its components parts of the actual world. John, for instance, is ontheir account a constituent of the proposition that John loves Mary. ‘That’sright, John himself, right there, trapped in a proposition.’1 Not all the com-ponents of propositions are parts of the empirical world, of course: loving,for instance, is not. And, as we saw in §2.3, even John was soon ousted fromthe proposition on account of not being simple enough. But what replacedhim, whether sense-data or sensibilia, were in turn conceived of as parts of theworld. Central to Wittgenstein’s symbolic turn was his abandonment of thisconception. Wittgenstein conceived of propositions instead as symbols havingnames of objects, not the objects themselves, as constituents.To say that a proposition has symbols as components rather than the parts

of the world those symbols refer to is so far to disagree with Russell only ona point of terminology, namely how to use the word ‘proposition’. What issignificant is not the change of terminology but the change of focus it betokens.Wittgenstein regarded the study of propositions, understood now in his senseand not Russell’s, as the goal of his work. But what exactly was his sense of‘proposition’? Saying that a proposition is a symbol is not yet to identify it.What the symbolic turn amounts to is the recommendation that in identifyinga proposition we should pay attention to those features of it that are relevant

1Kaplan, ‘Dthat’, 13.

"$ The symbolic turn

to its ability to say what it does. This, admittedly, is still programmatic—it will be some time before we are in a position to ascertain which are thefeatures Wittgenstein regarded as relevant—but even at this programmaticstage we can recognize here a distinctive turn in Wittgenstein’s thought awayfrom Russell’s.Certainly the decision to focus on symbols entails that we should treat

names as types, not tokens. Wittgenstein himself made this point in his char-acteristically oblique manner. ‘It is to be remembered that names are notthings, but classes: “A” is the same letter as “A”. This has the most impor-tant consequences for every symbolic language.’2 If names (and other words)are types, not tokens, a sentence will then also be a type. But it will not bewhat Wittgenstein meant by a proposition. This is because the propositionconsists only of those features of the sentence that symbolize, i.e. those thatare essential to the sentence’s ability to say what it does. That certainly entailsthat a proposition is a type, not a token, but it entails quite a lot more, sincea sentence of ordinary language may have features that are not relevant towhat it expresses, and any such irrelevant features will have to be ignored inidentifying the proposition involved.We cannot now reconstruct just when Wittgenstein decided to take this

course. The first sign we have of it is in a letter to Russell of January 1913,where he appealed to the idea that a thing just is ‘whatever can be symbol-ized by a simple proper name’. We shall be discussing various aspects of thisremarkable letter in the course of the next three chapters, but the point to fo-cus on now is the casual manner in which Wittgenstein moves here from talkabout ‘things’ to talk about ‘whatever can be symbolized by a simple propername’. Wittgenstein wrote the letter only a few weeks after one of his visits toFrege, and it is natural to wonder, as we explore the contours of the symbolicturn for which it is the earliest written evidence, what role this visit might haveplayed in his decision to take it.

&."&."&." The rejection of psychologismTo approach this question, it will be helpful to compare the symbolic turnwhich I am attributing to Wittgenstein with what has become known as thelinguistic turn. The moment when Frege took the linguistic turn—the mo-ment, indeed, when it was born—occurred in the Grundlagen when, havingstated the context principle, that ‘only in the context of a proposition do wordsmean anything’,3 he then used this principle to transform the Kantian ques-tion, ‘How are numbers given to us?’ into the linguistic question, ‘How arenumber words used?’ Frege explicitly advertised one of the purposes of the

2B27; cf. 3.203. 3§62.

The rejection of psychologism "%

context principle as being to resist psychologism. If we ignore this princi-ple, he said, we are ‘almost forced to take as the meanings of words mentalpictures or acts of the individual mind’,4 and hence, he thought, to descendinto psychologism. The linguistic turn, in Frege’s hands at least, was thus acomponent part of his war on psychologism.Already this is enough to open up some distance between Frege’s linguis-

tic turn and Wittgenstein’s symbolic one, since for Wittgenstein psychologismwas never the target. This is not because he had any more sympathy withit than Frege did, but because he regarded this as a battle already won. Hehad probably never read Lotze as Frege had, and he therefore had no needto object to that author’s treatment of thoughts as distinctive segments of ourstream of ideas. Nor, presumably, had he read the psychologistic work ofErdmann which incited Frege to such lengthy vituperation.5 In 1913 Witt-genstein wrote a scathing review6 of a not very distinguished textbook by alogician called Coffey, and this may indeed be the only book belonging to theearlier tradition in logic that he ever read. (Later he claimed not even to knowwhat a logic textbook contains,7 and one wonders whether he had actuallyread this one right through: all the examples he mentioned in his book revieware from the first fifty pages of Volume I.)One of the most prominent features of Wittgenstein’s symbolic turn is the

implicit assumption that we have no direct access to the world and must there-fore deduce its features from features of the symbols we use to represent it.This does not, certainly, amount to a denial of metaphysics—logical posi-tivism avant la lettre—but it does amount to a consciousness of its limits. Inthis respect Wittgenstein’s attitude bears an obvious similarity to Frege’s, butfor Frege it was thought that should be regarded as being out of reach of di-rect philosophical discussion: we analyse the structure of language as a meansto analyse the structure of thought. We have to accept that thinking, for ushumans at least, involves language essentially. As Frege put it, ‘We thinkin words. . . Without symbols we would scarcely raise ourselves to conceptualthought.’8 For Wittgenstein, on the other hand, it is the world that is in a cer-tain sense out of reach: we are compelled to approach features of the worldby means of features of the symbols we use to represent it.Notice, though, that whether either project, Frege’s or Wittgenstein’s, has

any hope of success depends on what degree of fit there is between languageand whatever language expresses, whether that is in the first instance takento be thoughts, as by Frege, or the world, as by Wittgenstein. Notice, too,that even if we assume that there is such a fit, that does not commit us to aview about the direction of fit. For all that has been said so far, it remainedopen to Wittgenstein to say that the world might owe its shape to language,4Gl, Intro. 5Gg, I, xv–xvii. 6‘Review of Coffey, Science of Logic’. 7Parak, ‘Wittgenstein in MonteCassino’, 149. 8‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’.

"" The symbolic turn

so that what it is for an object to be simple would be for it to be the referenceof a simple symbol; or to say that language owes its shape to the world, sincea symbol would not be capable of expressing anything about the world if itdid not conform to the requirements which the world imposes. The first ofthese two views we might label a sort of idealism, the second a sort of real-ism. Nothing Wittgenstein said in the Notes committed him to either view, oreven to seeing them as alternatives to be arbitrated upon. One of the mostintriguing features of the Tractatus is that it resists any attempt to impose onedirection rather than the other (probably because by then Wittgenstein hadcome to regard the very notion of a direction of fit between language andworld as misconceived).

&.#&.#&.# The reliability of languageIf the linguistic turn is to have any prospect of success, we have to assume thatthere is some correspondence, even if not a perfect one, between the structureof language and the structure of what it represents. This assumption is animportant strand in Frege’s thought. He observed in a letter to Russell, forexample, that ‘to the decomposition of the sentence there corresponds a de-composition of the thought’.9 Presumably, though, Frege should not be takenas claiming that the sentences of ordinary language are already structured soas to reveal with perfect clarity the thoughts they express. His purpose musthave been only to point to a certain sort of linguistic analysis, the decomposi-tion of a sentence into logically relevant components, as capable of revealingthe structure of the underlying thought.Even so, it remains that if we have to rely on language as a guide to the

structure of what language represents, we are in an obvious difficulty becauseof the unreliability of language in this regard. This, too, is a recurring themein Frege, whose desire to expel psychologism from logic encouraged in him asuspicion of ordinary language.

It cannot be the task of logic to investigate language and determine what is containedin a linguistic expression. Someone who wants to learn logic from language is like anadult who wants to learn how to think from a child. When men created language,they were at the stage of childish pictorial thinking. Languages are not made so as tomatch logic’s ruler.10

This, incidentally, is a strand of thought that can also be found in Russell: he,too, protested against the confusion that results from ‘the notion that wordsoccur in propositions, which in turn is due to the notion that propositions areessentially mental and are to be identified with cognitions’.11

But now there is an evident tension in the position. To avoid psycholo-gism we must recognize both that all our thoughts are expressed in words and

928 July 1902. 10To Husserl, 30 Oct. to 1 Nov. 1906. 11Principles, §51.

The reliability of language "&

that ordinary language constantly misleads us as to the true structure of thosethoughts. In principle, one way to resolve this tension would presumably beto adopt a language in which sentences do correspond to thoughts, but thosewhich express the same thought, and which are therefore for logical purposesequipollent, need not be distinguished.Only now that logical analysis proper has become possible can the logical elementsbe recognized, and we can see the clearing in the forest. All that would be neededwould be a single standard sentence for each system of equipollent sentences, and anythought could be communicated by such a standard sentence.12

Presumably Frege was here alluding to what he would later call a logicallyperfect language.13 Perhaps he thought at first that his Begriffsschrift came closeto being such a language. He recognized, of course, that his notation wasincomplete: he noted, for example, that it would have to be extended if wewanted to use it to talk about geometry. And he granted, too, that it might‘fail to reproduce ideas in a pure form’;14 but he does not seem to have meantby this that his Begriffsschrift might inevitably distort the thoughts it expressed.Indeed, he claimed that ‘we can restrict the discrepancies to those that areunavoidable and harmless’.15

But, after Russell’s discovery of the contradictions and his own failure torepair the damage they caused to his formal system, Frege seems increasinglyto have thought of logical perfection as an unachievable aspiration rather thanan ambitious but in principle attainable goal.If our language were logically more perfect, we would perhaps have no further needof logic, or we might read it off from language. But we are far from being in such aposition. Work in logic just is, to a large extent, a struggle with the logical defects oflanguage, and yet language remains for us an indispensable tool.16

Frege seems here to be on the verge of saying that language might be inevitablydistorting, and the goal of a logically perfect language in principle unattain-able. That is not to say, though, that Frege ever gave up on the idea that theroute to thought is via language. As Dummett succinctly puts it, ‘Languagemay be a distorting mirror: but it is the only mirror we have.’17

It is worth noting how close what Frege says here is in spirit to one ofthe central themes of the Tractatus. Wittgenstein displayed much the samehesitancy as Frege concerning the significance of the features of sentences inordinary language. Sometimes he claimed that the presence of some featurein an ordinary language sentence is an indication that the feature must beplaying some symbolizing role. He thought, for instance, that in ‘not-p’ theoccurrence of p is necessary.Ordinary language would not contain the whole propositions if it did not need them:However, e.g., ‘not-p’ may be explained, there must always be a meaning given to thequestion ‘what is denied?’18

12To Husserl, 30 Oct. to 1 Nov. 1906. 13PW, 256. 14Bs, Preface. 15Ibid. 16PW, 252. 17OAP,6. 18B21.

"' The symbolic turn

Elsewhere, though,Wittgenstein insisted that ‘distrust of grammar’—by whichhe presumably meant the grammar of ordinary language—‘is the first requi-site for philosophizing’.19 And he specifically warned against thinking that wecan read off the structure of the proposition from the structure of the ordinarylanguage sentence. ‘Ordinary language conceals the structure of the prop-osition: in it, relations look like predicates, predicates like names, etc.’20 Inordinary language, that is to say, sentences sometimes contain features thatdo not play a symbolizing role: ‘Language disguises the thought.’21 It is thetask of the philosopher to identify and eliminate (or at any rate see beyond)these features.So Wittgenstein’s criterion, according to which a thing is whatever can be

represented by a simple symbol, is not to be applied straightforwardly to ordi-nary language. Or perhaps we would be better to say that it cannot be appliedto ordinary language, because ordinary language does not contain any simplesymbols. For this reason it is a constant theme of Wittgenstein’s work fromnow on to seek the level at which the structure of the proposition—the struc-ture which does correspond precisely to the simplicity of the parts of the worldbeing represented—is revealed.At once this makes it clear how distant Wittgenstein’s notion of a symbol is

from that of a sign as used in linguistics: anyone who imagined that the struc-ture of the world might be revealed simply by the linguistic analysis of sentences,in any ordinary sense of that word, would plainly be beyond the reach of rea-son. Wittgenstein’s propositions are not, as Moore’s were, indistinguishablefrom the worldly facts that make them true; but nor are they sentences in thesense familiar from linguistics; they occupy a third level between these two,where the structure of the world is revealed but how things stand in it is not.

&.$&.$&.$ Conflicting conceptionsWhatever the similarities, then, the symbolic turn in Wittgenstein’s thoughtwhich we have been discussing is not Frege’s. Frege had a clear conceptionof complexity in language, and of the idea that it is logic’s role to explain andreflect that complexity, but he did not have a corresponding conception thatsimplicity in language might be a mirror or proxy for simplicity in the world.The reason is that his conception of the relationship between language andthe world did not allow for it. For him, every expression in language relatesto the world only mediately via its sense, and his notion of sense, if it was todo the work he wanted it to do, could not leave room for the sort of simplicityWittgenstein appealed to.Consider, for instance, an object picked out by what would have counted

for Wittgenstein as a simple proper name (a demonstrative perhaps). Even

19B63. 20B69. 214.002.

Conflicting conceptions "(

if Frege counted this name as simple, it was central to his conception of anobject to allow that there could be other names referring to the same object,and these other names need not be simple. Accordingly, whatever complexityparts of the world possess cannot on Frege’s conception of sense be straight-forwardly read off from the complexity of the terms we use to refer to them.There could for him be no direct connection between language and the world,and therefore no direct inference in either direction between the simplicity ofan expression in language and the simplicity of the object it refers to.Perhaps, then, Wittgenstein’s conception is best thought of as a synthesis of

two influences, Frege’s and Russell’s. His conception of what it is for objectsto be simple is one which, as we saw in chapter 4, he obtained from Russell:they are what resist elimination by Russell’s method of definite descriptions(and, more generally, the method of incomplete symbols of which it is aninstance). What is not present in Russell, but pervades the writings of Frege,is a recognition of the importance of studying the distinctive role of languagein representing the world. (Russell, on his own admission,22 did not becomesensitized to the value of studying language, rather than using it as a mediumfor the study of thought, until 1918.) It is only by putting these two strandstogether that we obtain the idea, embedded in Wittgenstein’s letter to Russellof January 1913 and from then on a fixed point in his thinking, that a thing is‘whatever can be symbolized by a simple proper name’.In chapter 3 we noted the distance that Wittgenstein seems to have main-

tained from Russell’s oscillations concerning the nature of sense-data. Russelltreated the relation of acquaintance as fundamental, and therefore hoped toidentify the simple objects of experience by means of a sort of informed intro-spection. ForWittgenstein, on the other hand, the simple objects are whateveris represented by the simple symbols. This conception need not in itself con-tradict Russell’s, since for all that has been said so far a ‘symbol’ might be,or be correlated with, a mental event such as the sensation of a sense-datum.That partly explains, perhaps, why there is no suggestion that Wittgensteinactively opposed Russell’s view of sense-data while he was at Cambridge. Butonce Wittgenstein began to fill in more of his conception of what a symbol is,his conception of the simples inevitably drifted away from Russell’s. All theindications are that this happened while Wittgenstein was in Norway duringthe Autumn of 1913. That was when he settled on the Tractarian concep-tion of propositions as expressing possibilities, and of objects as the constantelements on which these possibilities hinge. Hence his comment in a letterto Russell that ‘the individual primitive signs . . . are not at all the ones youthought’.23

22MPD, 98. 23[Nov. or Dec. 1913] (CL, no. 32).

Chapter '''

Simplicity

Wittgenstein took the symbolic turn, I have suggested, when he used it as acriterion for the simplicity of an object in the world that the object shouldbe symbolized by a simple name. We now need to explore what the ideaof simplicity that is at work here amounts to. We shall do this by compar-ing Wittgenstein’s account with Frege’s. Both authors offered explanations ofhow names contribute to the meaning of sentences in which they occur, butwhere Frege distinguished three distinct items—the referent of the name, itssense, and the idea I associate with it—Wittgenstein had only one, the ob-ject which is the name’s referent. One way of gaining some understanding ofWittgenstein’s conception, then, is to ask what Wittgenstein’s reasons were forcollapsing the distinctions Frege had drawn.

'.!'.!'.! Realism

I said in §1.3 that Russell had a semantic theory for names according to whichthey refer directly, whereas Frege had a theory according to which names (andsingular terms generally) refer only mediately via their senses: the primarypurpose of the notion of sense was to allow for different names of the sameobject to make different contributions to the thoughts expressed by sentencesin which they occur. A quick way of putting Wittgenstein’s understanding ofthe semantic role of names would (for the time being, at least) be to say thatin this matter he sided with Russell’s theory against Frege’s. When he cameto this view is uncertain: it is implicit in Wittgenstein’s letter of January 1913,and the way in which the letter alludes to this conception suggests that it wasby then already an accepted part of his thinking. But there is no hint eitherthere or in the Notes of his reason for adopting it. As we noted in §2.2, oneof Jourdain’s questions to Frege in January 1914 was whether he acceptedRussell’s demonstration that the distinction between sense and reference isunnecessary, but if it is Wittgenstein’s view he is there reporting, Jourdainleaves us tantalizingly short of a hint as to his reason for holding it.Another natural place to look for enlightenment on this dispute would be

Realism &)

the writings of Russell and Frege themselves, but what makes their disagree-ment disappointing as a moment in the history of philosophy is that neither ofthem ever really engaged with the other’s arguments. For instance, I know ofno evidence that Frege ever actually read ‘On denoting’, and his response toJourdain’s enquiry does not take up the implicit invitation to address Russell’sviews directly. Frege’s famous argument for the notion of sense on the basis ofthe non-trivial informational content of the identity ‘Hesperus = Phosphorus’is hardly persuasive on its own, since it is so plausible to think of ‘Hesperus’and ‘Phosphorus’ as disguised definite descriptions, for which Russell’s analy-sis provides a rather convincing explanation of the informativeness of identitystatements. If we accept that analysis, an application of Occam’s razor thenallows us to remove the senses of the terms ‘Hesperus’ and ‘Phosphorus’ fromour ontology.The underlying point which Frege’s example is intended to highlight, of

course, is that the names ‘Hesperus’ and ‘Phosphorus’ have a perspectivalcharacter: they both refer to the same object, namely the planet Venus, butthey do so in different ways, under different modes of presentation. In orderfor Frege’s argument to be persuasive against Russell, he would have to offera reason to think not merely that some singular terms have this perspectivalcomponent but that all do—that none simply refers to an object raw andunmediated.The Gray’s Elegy passage, discussed in §2.2, offers what Russell claimed

was an argument against Frege’s notion of sense. However, he originally for-mulated that argument as an attack not on Frege but on his own earlier theoryof denoting. If a denoting concept and the object it denotes are at the samelevel, as Russell had believed, it is indeed reasonable to ask about the proposi-tions expressing the relationships between them. But against Frege the argu-ment shows at most that there is a regress: a further step would be required toshow that the regress is vicious.So if we want to find Russell’s real objection to Frege’s two-step semantic

theory, we must look elsewhere. Russell did not examine Frege’s writings inany detail until 1902, by which time the conception of propositions as contain-ing parts of the world, which he adopted during conversations with Moore in1898, was probably too embedded in his thinking for him to question it. Butwhat had led them to adopt this conception was a desire to embrace realism,a desire which they had expressed by way of an identity theory of truth: whatit is for a belief of mine to be true is for it to be the case. If we add to thisthe view that what is the case, a fact, can be a fact about the world only ifit contains parts of the world, we arrive at the conclusion that the object ofmy belief must contain parts of the world too. Frege, who also adhered to justsuch an identity theory, had therefore, in consistency, to deny this last premise

&! Simplicity

and conceive of facts as belonging to the realm of sense, not to the world it-self. ‘A fact,’ on this account, ‘is a thought that is true.’1 For Russell, on theother hand, facts are as worldly as it gets, so it is easy to see how it would haveseemed to him as if Frege’s account fell fatally short of the world.

I believe that in spite of all its snowfields Mont Blanc itself is a component part of whatis actually asserted in the proposition ‘Mont Blanc is more than 4000 metres high’. Wedo not assert the thought, for this is a private psychological matter: we assert the objectof the thought, and this is, to my mind, a certain complex (an objective proposition,one might say) in which Mont Blanc is itself a component part. If we do not admitthis, then we get the conclusion that we know nothing at all about Mont Blanc.2

Quite soon Russell adopted views which were incompatible with the iden-tity theory, at least in this naive form, and eventually he himself realized thisand abandoned it. (So, incidentally, did Moore.)3 But Russell’s response wasto change the complexes to be analysed from propositions to judgments, not toremove the objects from the analysis. So his old rejection of idealism survived,recast only slightly to say that objects with which I am acquainted occur ascomponents in my judgments, rather than, as before, in the propositions thatI judge. Frege’s notion of sense continued, that is to say, to seem to him toerect a needless veil between us and the world.But to anyone outside the argumentative loops of Russell’s theorizing, this

way of expressing a commitment to realism is apt to seem like no more thanan unmotivated metaphysical dogma. What we need is an independent ar-gument for the idea that realism requires a one-step semantic theory. In theTractatus the so-called ‘argument for substance’4 is supposed to fulfil this role.And if Frege’s notion of sense is seen as mediating between language and theworld in such a way as to contradict this requirement, then we reach Witt-genstein’s desired conclusion that names do not have sense but refer directlyto their objects. However, the argument for substance, at least in the formin which it occurs in the Tractatus, probably dates from some time after Witt-genstein had left Cambridge: the most natural way of explaining it makes itdepend on assumptions which, though part of the Tractarian system, Witt-genstein probably did not work out until he was in Norway.So if we are to discern in the argument for substance any clue to what origi-

nally ledWittgenstein to adopt Russell’s one-step semantic theory, we must tryto extract its essence by disengaging it from its Tractarian framework. Whatunderlies the argument, surely, is the naive thought that sometimes languagemust just latch directly onto the world since, if it did not, the process of rep-resentation by which we succeed in expressing ourselves would never be ableto get started: each attempt at saying something would depend for its mean-ingfulness on the meaningfulness of something else. In order to have some

1‘Der Gedanke’, 74. 2To Frege, 12 Dec. 1904. 3SMPP, 308. 42.021–2.0212.

Solipsism &#

intuitive plausibility, this simple thought need not be turned into a specificproposal involving Russell’s theory of descriptions and his distinction betweeninternal and external negation. Instead, we can present it at a much simplerlevel as an expression of the realist’s instinct that language’s success in refer-ring to things in the world is ultimately due to the fact that we knock againstthem from time to time. If we think of the argument in this light, its keypremise is the claim, already stressed in the Notes, that ‘propositions have asense which is independent of their truth or falsity’,5 and therefore ‘we mustbe able to understand a proposition without knowing whether it is true orfalse’.6 If we press that view back to its base, we obtain the conclusion thatthere must be propositions whose sense we can understand even if we knownothing at all about how things stand in the world. In other words, the real-ist’s instinct (familiar enough from the more recent philosophical literature infavour of semantic externalism) is that sometimes in language we do not say,but simply presuppose, that there is a world for our words to refer to.

'."'."'." SolipsismThat, or something very like it, is the realist’s argument for identifying thereference of a name with its sense. In arguing for his notion of sense Fregegenerally preferred to emphasize the distinction between the sense of a nameand the idea associated with it in my mind. The reason he repeatedly offeredfor this distinction was that it was required in order to explain communication.When I tell you something you did not know, there is a single thing we bothgrasp, namely a certain thought. That thought, and by extension the senseswhich are its component parts, cannot be ideas, since these, being mentalentities, are intrinsically private to the mind. Neither of us has the thought,but both of us grasp it.For Russell, on the other hand, it was not really part of the task he was

engaged in to explain communication. On his view, indeed, the fact that wecommunicate at all emerges as a kind of miracle. For the sense-data experi-enced by me are not the same as those experienced by anyone else. Even ifyou are in the room with me, the angle at which you look at the table, andhence the exact sense-data you obtain from it, will be different. As a con-sequence, the logically proper names in your idiolect never mean quite thesame as those in mine.7 The only entities that are common to the proposi-tions that you express and the ones I express are universals. Now the propo-sitions of mathematics and logic, Russell thought, have no components thatare not universals; so they can be grasped by more than one person. Ordi-nary propositions about the world, however, always contain, when completely

5B9. 6C5. 7Russell, CP, VIII, 174.

&$ Simplicity

analysed, simple objects of acquaintance, whether sense-data or ideas. Since,as a matter of fact if not of necessity, no two people have ever been acquaintedwith exactly the same object, it follows that no such proposition has ever beengrasped by more than one person. This does not absolutely rule out the pos-sibility that we genuinely communicate with each other; but if we somehowsucceed in doing so, the proposition is not in any straightforward sense thevehicle of communication. It is not, as it is in Frege’s account, an entity whichembodies what it is that is communicated.A fundamental difference between Frege’s theory of reference and Rus-

sell’s, therefore, is that Frege aimed to explain how communication is possible,whereas Russell treated communication only as an observed fact from whichit might be possible to draw conclusions (for instance, about the existence ofother minds). Wittgenstein, on the other hand, was quite happy to circum-vent the issue entirely by tolerating solipsism. ‘He admits,’ Russell reported,‘that if there is no matter then no one exists but himself.’8 And if we embracesolipsism, the problem which led Frege to distinguish ideas from senses, andRussell to make communication a kind of miracle, simply dissolves.Wittgenstein’s response to this disagreement between Russell and Frege is

characteristic in several respects: it is characteristic that he should have beenunwaveringly determined to accept the consequences of his views, howeverunpalatable or frankly implausible they might have been; characteristic thathe did not feel the draw of Russell’s robust (if somewhat selective) commonsense; characteristic, too, that he should adopt a view one of whose effects wasnot so much to argue against an opposing view but rather to undermine it byreconfiguring the project to which it was intended to contribute.

'.#'.#'.# Idealism

In the last two sections I have traced out the connections between realismand the identification of sense and reference, and between solipsism and theidentification of idea with sense. I want now to examine the consequences ofidentifying idea with reference. In particular, I want to mention a particularfeature that ideas are normally taken to have, namely that they do not haveunexpected aspects. If an idea is presented to me in a certain way, it is naturalto think of that as the only way it could be presented to me: if it were presentedin a different way, it would just be a different idea. Indeed talking of how anidea is presented to me is already misleading. The transparency of ideas totheir bearers consists in this, that ideas cannot be distinguished from how theyare presented: an idea just is its presentation.

8To OM, 23 Apr. 1912.

Idealism &%

Now Russell certainly did not wish to identify a sense-datum with a men-tal entity such as my idea of it. Yet various readers of his work thought thathe did, and as a result he was required to insist indignantly that he had beenmisunderstood.9 The issue on which Russell was so often misunderstood nodoubt has a connection with the discussion of solipsism in the last section: partof the reason for the misunderstanding was that no two people have ever ex-perienced precisely the same sense-datum. In order to counter the inferencethat sense-data are private, and hence mental, it was necessary for Russell toclaim that this is only an empirical fact about sense-data and not part of theirnature. But another component in the explanation for the misunderstandingis that on Russell’s conception sense-data do have the feature I have just al-luded to, namely that there is no gap between how they seem and how theyare. If they are objects, we might say, they are so only in a thin sense whichrenders them transparent to us. We may contrast them in this respect with theordinary objects of common-sense experience, which are opaque: they mayhave various aspects to them, sometimes unknown and unexpected ones.It is helpful here to consider Frege’s response when Jourdain asked him

whether there was really any need for a distinction between sense and refer-ence. Frege asked Jourdain (or would have done if he had sent the letter hedrafted) to imagine two explorers who come upon the same mountain, but seeit from different perspectives in different valleys. If one gives this mountain thename ‘Afla’, the other the name ‘Ateb’, it is then a genuine discovery, whichwe require the notion of sense to explain, that these are names of the samething.In the form in which Frege tells the story,10 it does no more than to make

once more the point we discussed in the last section, that we need the notionof sense if we are to explain communication. But it is easy to see that just thesame issue arises if we alter the story so that there is only one explorer, nottwo: the explorer travels to both valleys in turn and sees the same mountainfrom different sides without realizing it. In its new form, the point the storymakes cannot any longer be one about communication, but must rather beabout our conception of objects. When the explorer learns that Ateb is thesame mountain as Afla, this no doubt counts as a discovery, but not one thatoverturns anything he formerly believed or that requires him to revise hislanguage. It causes him, we might say, to add some information to his map ofthe world, but not to delete anything.What this transposition of Frege’s example demonstrates is that although

Frege repeatedly offered the need to explain communication as a motivationfor the notion of sense, it is not the only one. His notion of sense also seeks tocapture the realist’s belief that objects are, in the sense just mentioned, opaqueand not transparent: the way in which they are presented to us need not

9CP, VIII, 88. 10PMC, 80.

&" Simplicity

exhaust what there is to them. Now on Russell’s account the opaque objectsare to be constructed logically out of the transparent ones. One way of puttingthe realist’s point would be to doubt whether it is even possible to perform sucha construction. How can logic on its own, using transparent objects as its onlybuilding blocks, deliver opaque ones? Where is the opacity supposed to comefrom? Another way of putting something like the same point would be to notethat Russell’s elaborate 1914 construction of matter from sensibilia still seemshopelessly far away from an explanation of how matter has mass, for example,when sensibilia (one assumes) do not.Of course, to press this concern is so far only to insist that objects may have

unknown aspects, not to allow that they have unknowable ones. Nonetheless,the two thoughts tend in the same direction. They are both ways (the secondof course much stronger than the first) of responding to realism’s sense of theworld as something to be explored, not constructed. Anyone who supposesthat the surface of the world is the whole of it runs, the realist thinks, a constantrisk of unpleasant surprises. The difficulty this realist conception of the worldposes for atomism, whether Russell’s or Wittgenstein’s, is that their methodof analysis equates transparency with logical simplicity and is thus at a loss toexplain how it can be that our current conception of the world already makesroom for it to have a complexity to which we are not yet privy.

'.$'.$'.$ ReconciliationMy aim in this chapter has been to draw out some connections between Witt-genstein’s understandings of realism, solipsism, and idealism. In explainingthe contribution a name makes to determining the sense of any sentence inwhich it occurs, Frege distinguished between three elements, the idea I asso-ciate with the name, the sense of the name, and its reference. Wittgenstein’sconception of objects certainly does not coincide neatly with any of the three,but what I have been suggesting here is that it shares something with each.Depending on which we emphasize, his view will seem to be characterizableas idealism, solipsism, or realism.Of course, something like this thought is present in the Tractatus,11 but the

route by which Wittgenstein reached it there is one he did not travel until1916. First he connected solipsism with realism.

Here we can see that solipsism coincides with pure realism, if it is strictly thought out.The I of solipsism shrinks to an extensionless point and what remains is the reality

coordinate with it.12

Then he connected idealism with solipsism as well.

11Cf. 5.64. 12NB, 2 Sep. 1916.

Reconciliation &&

This is the way I have travelled. Idealism singles men out from the world as unique,solipsism singles me alone out, and at last I see that I too belong with the rest of theworld, and so on the one side nothing is left over, and on the other side, as unique, theworld. In this way idealism leads to realism if it is strictly thought out.13

Wittgenstein’s route to the identification is therefore intimately connectedwith his later understanding of the metaphysical subject, an understandingof which not a glimmer is to be seen in the Notes. That would have to wait notonly for the removal of the empirical subject from the analysis of judgment,which he arrived at in Norway, but, more importantly, for the emergenceof transcendental reflections as a theme in his philosophical thinking duringthe summer of 1916. It is striking, though, how short a distance there is stillto be travelled. If Wittgenstein was to reconcile the Russellian conception ofsimplicity with a Fregean conception of thought, he had little choice but tocollapse the traditional distinction between realism, idealism, and solipsism;and once he had done that, he could do no other than to reduce the self to anextensionless point, leaving only the reality coordinate with it.

13NB, 15 Oct. 1916.

Chapter (((

Unity

Merely to take the symbolic turn—to conceive of a proposition as symbolizingwhat it expresses, rather than being identical with it—is not yet to go very fartowards uncovering the structure of propositions. And in his letters to Russellduring 1912 Wittgenstein was still operating with a Russellian conception ofthat structure. A proposition, that is to say, he still thought of as a sort ofcomplex. Two questions preoccupied him. What are the components of thecomplex? And what is the manner of combination by which these componentsunite to form a proposition?

(.!(.!(.! The copula

It is a familiar requirement of English grammar that every declarative sen-tence should contain a verb. We have already noted Russell’s fitful tendencyto regard features of the sentences of our language merely as proxies for thefeatures of the propositions they express. In accordance with this tendency,therefore, he made the obvious move of projecting this linguistic requirementonto the complex which the declarative sentence expresses and insisting thatthat complex must contain a corresponding element: this essential ingredientof a proposition he also, for want of a better word, called a ‘verb’.But Russell was also impressed by the devices which languages such as En-

glish have for converting verbs or verbal phrases into verbal nouns. We canderive hitting from hits, dying from dies, mortality from is mortal. Russell thereforeheld that verbs have a ‘curious twofold use’: they can occur in propositionsas verbs or as verbal nouns, but although the mode of occurrence is different,he claimed that the entity that occurs is the same in the two cases. His ar-gument to this effect1 is based on the assumption that if the verb really weredifferent from the verbal noun, we could make a statement to that effect: wecould say, for instance, ‘the verb is mortal differs from the noun mortality’. Butin that statement is mortal occurs as a noun, not as a verb. So the suppositionis self-refuting.

1Principles, §49.

The copula &(

This is not a very good argument. Just the same problem occurs if we tryto state the contrary: in the statement ‘the verb is mortal is the same entityas the noun mortality’ the verb is mortal occurs as a noun, not a verb. But letus leave that to one side and for the moment accept Russell’s conclusion—accept, that is to say, that the proposition ‘Socrates is mortal’, in which ismortal occurs as a verb, is equivalent to the proposition ‘Mortality belongs toSocrates’, in which it occurs as a verbal noun. Since the mode of occurrenceof mortal is different in the two cases, Russell was compelled to admit that theyare different propositions. Nonetheless, they are logically equivalent, and itis therefore hard to see why we should not always substitute the latter for theformer. In that case, though, it is a natural further thought that if a verbcan sometimes be replaced by a verbal noun, then it always can. There will still(presumably) need to be a verb to bind the disparate elements of the complexinto a proposition, but it will now be only a bare verb without any specificmeaning, i.e. what grammarians traditionally called a copula.This, then, is the sort of analysis that Wittgenstein was working with dur-

ing 1912. The number of elements in the analysis will vary according to thetype of proposition that is in question, and so there will have to be differentcopulae for the different types. An atomic subject-predicate proposition !ais analysed as #1(a,!): the subject a and the predicate ! are here bound to-gether by the copula #1. The proposition ‘Socrates is mortal’, for example,contains ‘Socrates’ and ‘mortality’, bound together by #1. This copula is thusthe form which all subject-predicate propositions have in common. Similarly,an atomic relational proposition aRb is analysed as #2(a,R,b): here a, b, and Rare bound together by a different copula #2. Wittgenstein also countenancedas a degenerate case the one-place copula #0, enabling him to analyse ‘a exists’as #0(a).We shall look later at how Wittgenstein proposed to analyse these copulae.

In the meantime let us focus on the idea that we can convert the apparent verbof the proposition into a noun and hence regard mortality as an object. Thisis evidently problematic. If we can turn verbs into nouns at will, for instance,one might wonder what objection there could be to also turning the copula isinto the verbal noun being and hence dissolving the proposition entirely. Onemight also wonder what conception Wittgenstein had of the pure copulae #n

that he thought was consistent with his fundamental thought of June 1912 thatthere are no logical constants.Given these difficulties, it is perhaps no great surprise that Wittgenstein

quite soon abandoned the idea of explaining propositional unity by means ofthe copula. What is suggestive, however, is the timing of the abandonment,which occurred just after Wittgenstein had visited Frege during the Christmasvacation at the end of 1912. In chapter 6 I mentioned a letter Wittgensteinwrote to Russell soon after this visit. It provides, I said, the first clear ev-

'* Unity

idence of the symbolic turn in Wittgenstein’s thought. In the same letter,however, we also find Wittgenstein reporting that he had abandoned the ideathat what supplies unity to a proposition is a copula distinct from its meaning-contributing components.

I have changed my view on ‘atomic’ complexes: I now think that qualities, relations(like love) etc. are all copulae! That means I for instance analyse a subject-predicateproposition, say, ‘Socrates is human’ into ‘Socrates’ and ‘something is human’, (whichI think is not complex). The reason for this is a very fundamental one: I think thatthere cannot be different Types of things! In other words whatever can be symbolizedby a simple proper name must belong to one type. And further: every theory oftypes must be rendered superfluous by a proper theory of symbolism. For instance ifI analyse the proposition Socrates is mortal into Socrates, mortality and (

E

x,y)#1(x,y) Iwant a theory of types to tell me that ‘mortality is Socrates’ is nonsensical, because ifI treat ‘mortality’ as a proper name (as I did) there is nothing to prevent me to makethe substitution the wrong way round. But if I analyse (as I do now) into Socratesand (

E

x)!x it becomes impossible to substitute the wrong way round because the twosymbols are now of a different kind themselves.2

The essence of Wittgenstein’s new proposal, then, was to be that ‘Socratesis mortal’ is analysed not into two names, ‘Socrates’ and ‘mortality’ connectedby a copula, but into one name ‘Socrates’ and a second component of a kindwhich he soon began to call a form; there was no need to appeal to a separatecopula #1 to explain the unity of the proposition because that function wastaken over by the form.

(."(."(." There cannot be different types of things

Wittgenstein gives two reasons for rejecting the copula. The first is that therecannot be different types of things. It is obvious straightaway, of course, thatif this is right, it will put paid to any theory that treats Socrates and mor-tality as things, for it then entails that they are of the same type, and hencethat ‘mortality is Socrates’ is just as grammatical as ‘Socrates is mortal’. It isclear enough, too, how a fundamental grammatical distinction such as the oneWittgenstein offers might suffice to explain the ungrammaticality of ‘mortalityis Socrates’. But that does not prove Wittgenstein’s claim, since it does notdemonstrate that there are not other cases of ungrammaticality for which adistinction of type among things is necessary.So why didWittgenstein think that there cannot be different types of things?

In the letter he gives no reason for the claim (which in any case he later aban-doned). However, it is certainly a view that is to be found in Frege. When hewas considering how to deal with Russell’s paradox—one of the most obvious

2To BR, 16 Jan. 1913.

There cannot be different types of things ')

reasons that might have pressed him to recognize different types of thing—Frege briefly entertained the proposal that classes might be of a different kindfrom other objects. (He suggested calling them ‘improper’ objects.) But heimmediately insisted that there would have nonetheless to be some functionswhich could have both kinds of objects as arguments.

At least the relation of equality (identity) would be a function of this sort. (An attemptmight be made to escape this by assuming a special sort of equality for improperobjects. But that is certainly ruled out. Identity is a relation given to us in such aspecific form that it is inconceivable that various kinds of it should occur.)3

But Frege seems here to be doing little more than restating in differentlanguage his claim that all objects are of the same kind. If there are indeeddifferent kinds of objects, then there will be different kinds of identity relationthat can hold between them: Frege regards this as ‘inconceivable’, but he doesnot explain why. We may suspect, then, that his argument from the universal-ity of the relation of identity cannot really be what motivated his view. Afterall, he was quite content to accept a distinction of type between objects, first-level concepts, second-level concepts, and so on. Moreover, concepts are forFrege objective entities, and there must therefore be a relation between themanalogous to that of identity between objects. This seems to be only termi-nologically different from saying that there are different kinds of identity, onekind that may hold between objects, another between first-level concepts, andso on. His real reason for thinking that there cannot be different kinds of ob-ject seems rather to be that any difference of type between entities must stemfrom a fundamental difference of grammar. Russell’s paradox could on itsown at most indicate the existence of such a difference; it could not constituteit. This is thus an instance in which Frege’s deep instincts, if not his logicalpractices, were in harmony with what I have called Wittgenstein’s symbolicturn.Frege’s view that there is only one kind of object is of course equivalent

to the claim that the universal quantifier is genuinely universal. It was hisdeep-seated adherence to this doctrine that was responsible for his inability todeal with the effect of Russell’s paradox. It is interesting to compare Frege’sresponse to this difficulty with Russell’s. Frege’s argument that there is onlyone relation of identity would hardly have swayed Russell, who would simplyhave accepted that functions such as identity are typically ambiguous: wetalk as if there is one relation of identity that applies to objects of all types, butstrictly there is a whole family of such relations, one for each type. (For Russellidentity is a defined relation, but that does not affect the point at issue here.)The danger inherent in the device of typical ambiguity, though, is that we donot take it seriously. Taking the type distinctions seriously would involve using

3Gg, II, 254.

'! Unity

for each type i a distinct equality sign ‘=i’. Moreover, it would be important,if we did this, to recognize that here ‘=i’ is a complete symbol, not a functionof the subscript i.But Whitehead and Russell were not so careful. Throughout Principia they

did not use subscripts to indicate differences of type. So the same sign ‘=’ isused for equality in every type. But, as Wittgenstein would shortly point out,

it can never express the common characteristic of two objects that we designate themby the same name but by two different ways of designation, for, since names are ar-bitrary, we might also choose different names, and where then would be the commonelement in the designations? Nevertheless, one is always tempted, in a difficulty, totake refuge in different ways of designation.4

Although he here makes the point about names, it evidently applies also toexpressions for relations. We must not pretend, by using the same sign indifferent cases, that we have expressed any similarity in what the sign expressesin each case.

(.#(.#(.# The theory of types is superfluousLet us turn now to Wittgenstein’s second reason for rejecting the idea thatpropositions consist of names related by a copula, namely that the theory oftypes is superfluous. When Wittgenstein introduces this second reason withthe word ‘further’, what he means is presumably that this is to be understoodas an additional reason for the same conclusion, since if the first reason iscorrect, so that all names are of the same type, no theory of symbolism will beable to prevent us from swapping ‘Socrates’ and ‘mortality’ in the old analysisso as to obtain the nonsensical proposition that mortality is Socrates. As withhis first reason, though, Wittgenstein does not in the letter to Russell supply anargument for his claim. What he does is only to offer an example of how, bydistinguishing between names and forms, we can explain without any appealto the theory of types why ‘Socrates is mortal’ makes sense but ‘mortality isSocrates’ does not.Wittgenstein’s account here is evidently little more than a rewording of

what he would have gleaned from Frege. Frege’s recommendation was thatwe should analyse ‘Socrates is mortal’ not into ‘Socrates’ and ‘mortality’ butinto ‘Socrates’ and ‘x is mortal’. The proposition is what results if we substi-tute the first of these expressions into the argument place in the second. If,on the other hand, we try to substitute the second expression (‘x is mortal’)into the first (‘Socrates’), what we get is not so much nonsense as perplexity:since ‘Socrates’ has no argument place, there is simply no sort of substitutingthat can be done in it. Of course, the more examples we supply in which our

4B3, cf. 3.322.

The theory of types is superfluous '#

‘proper theory of symbolism’ explains away apparently problematic combi-nations of signs, the more superfluous the theory of types will seem, but suchindividual cases fall some way short of explaining why the theory of types mustbe superfluous.The notion that a correct theory of symbolism will render the theory of

types superfluous is in fact one that recurs throughout Wittgenstein’s thoughtfrom this point on. On the face of it, though, one might wonder what the con-trast is that Wittgenstein is drawing. After all, one might think that a theory oftypes just is a proper theory of symbolism, in which case it would be mystify-ing how the latter could render the former superfluous. Such, evidently, wasRussell’s view when he wrote to Wittgenstein just after he read the Tractatusfor the first time.

The theory of types, in my view, is a theory of correct symbolism: a simple symbolmust not be used to express anything complex: more generally, a symbol must havethe same structure as its meaning.5

Even after all this time, though, Russell had still not quite got things right.

That’s exactly what one can’t say. You cannot prescribe to a symbol what it may beused to express. All that a symbol can express, it may express. This is a short answerbut it is true!6

It is hard to see what Wittgenstein could have found to object to in Russell’sapparently innocent remark. One can only presume that in Russell’s (surelyinnocuous) use of the word ‘must’ he discerned an expression of the view hewas concerned vehemently to oppose, namely that the theory of types is some-thing that can be superimposed on language after it is operational. The caseof ordinary language exemplifies this point quite well. The words we use canbe categorized into grammatical types such as noun, verb, adjective, adverb,etc. But if I understand a word, then it already belongs to a grammatical cat-egory; and if I define a new word, the definition must be such as to determineits category. I cannot later decree the word’s grammatical type when I alreadyknow what it means. It is incoherent to conceive of grammar as somethingthat can be imposed on words when they already have a sense, since grammaris no more than a systematization of how words contribute towards the sensesof the sentences that they form.There has been a tendency for commentators to misunderstand this, espe-

cially in the formulation Wittgenstein dictated to Moore. ‘A THEORYTHEORYTHEORY of types isimpossible. It tries to say something about the types, when you can only talkabout the symbols.’7 Wittgenstein was not denying that propositional func-tions fall into a hierarchy of levels—something he evidently maintained in theTractatus and would in any case have found it hard coherently to deny. Nor

513 Aug. 1919. 6LW to BR, 19 Aug. 1919. 7NdM, ¶18.

'$ Unity

was he denying—as would, once again, be hard—that it is possible to drawup, post facto, rules describing which combinations of signs make sense andwhich do not. Such a set of rules could no doubt be called a theory of types.What is a mistake, according to Wittgenstein, is to regard such a theory asdetermining, rather than merely describing, which combinations make sense.

The names ‘Socrates’ and ‘Plato’ are similar: they are both names. But whateverthey have in common must not be introduced before ‘Socrates’ and ‘Plato’ are intro-duced. The same applies to a subject-predicate form, etc. Therefore, thing, proposi-tion, subject-predicate form, etc., are not indefinables, i.e., types are not indefinables.8

Wittgenstein’s point is that it makes no sense to introduce a new grammat-ical category into language in advance of having any words to put into it.Grammatical categories such as ‘proper name’ or ‘verb’ are not buckets intowhich we cast words before they have meanings; they are descriptions of re-semblances between the meanings which various existing words possess. Weintroduce the words ‘Socrates’ and ‘Plato’, and the way we introduce themmakes it the case that they are both proper names. ‘Proper name’ is ourgrammatical term to express what it is that such words have in common; butwhat they have in common they owe to the manner in which they were givenmeaning, not the other way round.Wittgenstein is here drawing on an analogy between the indefinables of a

logical system and the unknowns of a physical system. Applied mathemati-cians talk about the number of degrees of freedom in a system, meaning bythis the number of unknowns in the equations which may be determined in-dependently of each other. If we have a set of equations in five unknowns withfour degrees of freedom, for instance, we can choose values for any four of theunknowns freely, but the value of the fifth will then be fixed by the choices wehave made for the other four. When we set up a formal system, Wittgensteinthought it important to ensure that there are no more indefinables than thereare degrees of freedom in the system. This idea is something he evidentlyobtained from Hertz, who remarked that we have

accumulated around the terms ‘force’ and ‘electricity’ more relations than can be com-pletely reconciled amongst themselves. We have an obscure feeling of this and want tohave things cleared up. Our confused wish finds expression in the confused questionas to the nature of force and electricity. But the answer which we want is not really ananswer to this question. It is not by finding out more and fresh relations and conse-quences that it can be answered; but by removing the contradictions existing betweenthose already known, and thus perhaps reducing their number. When these painfulcontradictions are removed, the question as to the nature of force will not have beenanswered, but our minds, no longer vexed, will cease to ask illegitimate questions.9

8C28. 9Prinzipien der Mechanik, Introduction.

The theory of types is superfluous '%

The methodological principle which Hertz offers in this passage is that philo-sophical perplexity often arises because our system of beliefs has more inde-finables than degrees of freedom. It is a principle that remained central toWittgenstein’s thinking long afterwards. In 1939, for instance, he quoted thepassage from Hertz at a meeting of the Moral Sciences Club and said, as theminutes report, ‘that he must confess that this passage seemed to him to sumup philosophy’.10

Expressed in these terms, Wittgenstein’s point was that if, after introducingthe names ‘Socrates’ and ‘Plato’, we regarded ‘proper name’ as a further in-definable, the system would have more indefinables than degrees of freedom.He concluded that differences of type between linguistic items can arise onlythrough differences in their grammatical roles. This is to be distinguishedfrom Wittgenstein’s further view, expressed in the June 1912 letter but laterrejected, that there cannot be different types of things. The latter will followonly if name is assumed to be a single grammatical category. As I have said,it is tempting to detect Frege’s influence in Wittgenstein’s adoption of this lastassumption. The reason that Frege held name to be a single category was thathe believed the only possible source for such differentiations to be his distinc-tion between saturated and unsaturated expressions. For him, therefore, allnames, being saturated, belong to the same category. But Wittgenstein doesnot offer any argument to explain why we should follow Frege in this belief.

10Quoted in Klagge and Nordmann, Public and Private Occasions, 379.

Chapter )))

Fregean propositions

One of the central doctrines of Frege’s logical theory from 1890 onwards wasthat a proposition is a name of a truth-value. The main task of this chapterwill be to analyse the argument Wittgenstein offers in the Notes to show thatthis doctrine was misconceived. His argument turns on the use Frege made ofthe sign ‘#’ as a symbol to represent assertion.

).!).!).! Frege’s notion of assertionIt is worth emphasizing straightaway that although it was Frege who devisedthe turnstile sign ‘#’, his use of it to signify assertion is not at all the same as itswidespread modern use to signify logical derivability. This is partly becausethe modern usage gives a sense to occurrences of the sign with sentences bothbefore and after it, as, for instance, when we say that

if A# B, then C # D.

In Frege’s usage, by contrast, the turnstile cannot be embedded in this man-ner, since it is intended to signal the act of assertion, not to be part of what isasserted. (For that reason it is incorrect to read ‘# A’ as ‘A is asserted’: the for-mer, unlike the latter, does not say that A is being asserted but actually assertsit.) And even in the simple usage with a single sentence

# A,

the modern understanding of the turnstile is different, since it is taken to assertnot merely that A is true but that it is provable by logical means. In practice, ofcourse, Frege asserted sentences in Grundgesetze only when he believed them tobe provable, but that is merely because Grundgesetze is a work of logic. It wouldbe quite correct, in Frege’s symbolism, to use the turnstile to assert anythingthat is true, whether or not the ground of its truth is logical.When Frege originally introduced the turnstile in his Begriffsschrift, he had

not yet formulated his view of sentences as names of truth-values. In the

Frege’s notion of assertion '&

Begriffsschrift he conceived of the turnstile as a compound sign: the function ofthe horizontal part was to ‘combine the signs that follow it into a totality’,1

so as to make them capable of being judged; the vertical part of the sign,which he called the ‘judgment stroke’, then expressed the assertoric force ofthe judgment. Frege’s explanation does not really clarify what the purposeof the horizontal stroke is, but suggests that he saw it as a sort of copula,conferring propositional unity on the signs that follow it. If so, then by thetime of the Notes Wittgenstein had come to regard it as unnecessary: whatfollows it must already possess the requisite unity. More to the point, though,Frege himself must soon have come to see that it is unnecessary.In any case this account of the turnstile plainly could not survive Frege’s

adoption around 1890 of his mature semantic theory, according to which ev-ery sentence is a name of one or other of the two truth-values, which he called‘the True’ and ‘the False’. From then on he continued to use the sign ‘#’,and to represent it as compound, but the parts were now different. The hori-zontal part of the sign no longer served as a sort of copula; instead it had therather curious function of converting names which are not already sentencesinto names of the False; names that are already sentences it left untouched.Since ‘2’ is a name of a number, not of a truth-value, for instance, ‘%2’ is inFrege’s post-1890 notation a name of the False. But of course, no one wouldin any normal circumstances think of writing that. (Frege himself did, butonly as an intentionally outlandish example to demonstrate how he intendedhis notation to operate.)Frege now used the vertical part of the turnstile, the judgment stroke, to

indicate that the thought indicated by the name following it was being utteredassertorically. Of course, he did not place the turnstile in front of anythingother than a name of a truth-value: to do so would always be an error. So thehorizontal part of it is in practice eliminable: if we already have before us aname of a truth-value, adding the horizontal in front of it simply produces aslightly more elaborate name for the same truth-value. Since the horizontalacts in this way as a kind of identity function, adding two horizontals has thesame effect as adding one: perhaps Frege intended the way the signs mergeinto each other visually to mimic this.For present purposes, then, let us concentrate on the judgment stroke. If

sentences are names of truth-values, some device is needed merely to copewith the grammatical difference between uttering a name and expressing aproposition. When I assert a name of the True, I speak truly; when I asserta name of the False, I speak falsely. The effect of the judgment stroke is toconvert a name of a truth-value into the assertion that what that name refersto is the True. Now ordinary language contains a device which fulfils the

1Bs, §2.

'' Fregean propositions

grammatical role of converting a name of a sentence into something meet tobe asserted, namely the phrase ‘is true’: by writing

‘Snow is white’ is true

I express the thought that snow is white; and if, more generally, ‘A’ names anyproposition, by writing

A is true

I express something not expressed by simply writing the name ‘A’. Frege’sjudgment stroke can be thought of as combining two roles, then: first it con-verts such a name into something of the right form to be asserted, just as the‘is true’ operator does; then it asserts that.

).").")." Propositions are not names of truth-valuesIt is plausible enough that

‘Snow is white’ is true

succeeds in expressing the thought that snow is white; but it is much less plau-sible that anything analogous holds for other sorts of names. If I assert

Gödel’s theorem is true,

I no doubt commit myself to the truth of Gödel’s theorem, but have I ex-pressed it? In an obvious sense I have not. The example in which we usethe quotation-name of a sentence is special just because there is in this case ameans of semantic descent more direct than adding ‘is true’, namely to deletethe quotation marks. In essence, Wittgenstein’s objection to Frege’s doctrinethat sentences are already names even before we add quotation marks tothem is that it deprived Frege of the resources to explain why expressionslike ‘Gödel’s theorem’, which name propositions, can only ever be parasiticon sentences, which express them.In order to formulate his objection, Wittgenstein in effect decomposes the

judgment stroke into two components in the manner I sketched in the lastsection. He then points out a difficulty for the first of the two components, theoperator ‘is true’ which converts a name into an assertoric sentence. Wittgen-stein expresses his objection to this operator by means of an analogy whichmay well have been inspired by one of Frege’s own. In Grundgesetze2 Frege hadobjected, against one formalist account of arithmetic, that it was like definingsigns to be white if they belong to white objects. This is illegitimate because it

2§59.

Propositions are not names of truth-values '(

only makes sense if we already know what the word ‘white’ means, in whichcase we are not free to define afresh what it is for a sign to be white. To makethe point vivid, Frege noted that if the definition were in good order, we couldmake a black patch on a sheet of paper white by the simple device of agreeingto use it as a sign for a white sheet of paper.In the Notes Wittgenstein adapts this example in order to turn it against

Frege himself. The focus of the example is still a black patch on a sheet ofpaper, but the question at stake is now what is involved in saying that a par-ticular point on the sheet is white or black. In Frege’s theory, if I merely uttera sentence A, I am only setting up an assumption for consideration. If I wantto say that the sentence is true, I must utter it with assertoric force: we wouldnormally represent that by writing ‘A is true’ (or, in his formal notation, bywriting ‘# A’). In something like the same way, if I indicate a point on thesheet of paper and name it P, I am not yet saying anything. If I say ‘P isblack’, on the other hand, I have made an assertion which may be true orfalse.But, as Wittgenstein points out, the analogy goes wrong at just this point.

I can indicate a point of the paper, without knowing what white and black are; butto a proposition without sense nothing corresponds, for it does not designate a thing(truth-value), whose properties might be called ‘false’ or ‘true’.3

There are, Wittgenstein is saying, two stages involved in saying that a par-ticular point on a sheet of paper is black: the first, naming the point, doesnot involve knowledge of the difference between white and black; the second,predicating blackness of the point indicated, does. And if Frege’s account wereright, there would similarly be two stages involved in asserting something: firstwe would construct a name of a truth-value; then we would predicate truthof it. It should therefore be possible, by analogy with the case of the sheet ofpaper, for the first stage, that of constructing a name which expresses whatwe wish to assert, to be done by someone who does not know what truth andfalsity are. And that is ridiculous. In order to express a thought, I have to real-ize that thoughts aim at truth. It is incoherent to imagine someone coming toan understanding of language as a device for picking out one or other of thetruth-values while still ignorant of what the point of this practice is. In contrastto the case of naming a point on a sheet of paper, there is nothing analogousto blind pointing by which I could, as it were, succeed in expressing a thoughtby accident. Frege’s mistake, in other words, was that he had left himself noresources with which to restrict our freedom of action in constructing namesof truth-values in such a way that their usefulness in making assertions is builtinto them.

3B10; cf. 4.063.

(* Fregean propositions

).#).#).# Whose influence?Wittgenstein’s argument against Frege has attracted much scholarly attention.Ever since Anscombe,4 it has been a commonplace for commentators to noteRussell’s influence on the way the argument is formulated. In one place Fregenotes in passing that the proposition A on its own is an expression for a truth-value but does not make any assertion; if we did not make this distinction,he says, we could not express something that is a mere assumption (Annahme),‘the putting of a case without a simultaneous judgment as to whether or notit obtains’.5 Russell, when he summarized Frege’s views in an appendix tothe Principles, saw in his use of the word Annahme an echo of Meinong’s,6 andincorrectly supposed that Frege was using it in something recognizably likeMeinong’s way, as a technical term for any use of a sentence not prefixed byan assertion-sign.Anscombe is quite right to note that when Wittgenstein criticizes Frege in

the Notes, he does indeed use the word ‘Annahme’ in this Meinong-like sense.And it is quite possible, of course, that Wittgenstein picked up this usage fromRussell. There is surely nothing strange in this: it is, after all, easy to imaginethat if Wittgenstein and Russell ever discussed Frege’s views, as presumablythey did, they might have resorted to Russell’s preferred terminology in doingso. But that is not the only route by which Wittgenstein might have picked upthe usage. Hemight equally have got it fromMoore, who discussedMeinong’sidea of an Annahme in his lectures during Lent Term 1912.The point to stress, though, is that the fact that Wittgenstein used termi-

nology in ways then current in Cambridge is not a sufficient reason to takethe further step of supposing that Wittgenstein saw Frege’s works through thelens of Russell’s summary of them. After all, if we are to discuss Frege’s con-ception of propositions, it is convenient to have a word for the occurrence ofa proposition unasserted, and for that purpose ‘Annahme’ is surely as good aword as any, even if it is not really Frege’s. Wittgenstein himself, at any rate,evidently thought so, since he continued to use the word in this sense throughto the Investigations.7 But to use a word that is not Frege’s is not yet to showthat one misunderstands him. Russell’s discussion8 of Frege’s conception ofsentences as names of truth-values is riddled with confusions and misunder-standings; Wittgenstein’s, as I have outlined it above, is not. That is in itselfenough to cast doubt on the idea that Wittgenstein approached Frege’s writ-ings via Russell: if he had, one would surely expect his criticisms of them tohave been equally confused.I think, though, that we can go further. Not only does Wittgenstein’s crit-

icism of Frege show an informed understanding of Frege’s own views, ratherthan with Russell’s summary of them, but it is itself based on a profound en-

4IWT, 105–6. 5Funktion und Begriff, 21. 6Über Annahmen. 7PI, §22. 8Principles, §477.

Propositions as articulate ()

gagement with Frege’s conception of logic. Wittgenstein’s argument againstFrege depends on a certain understanding of the relationship between thoughtand truth, and hence between propositions and truth. It is essential to athought that it should be capable of being true, and essential to a propositionthat it should express a thought. Two things follow from this: first, truth is nota property of thoughts as redness is a property of flowers; and second, it is es-sential to a proposition that it should have a structure that enables it to expresssomething capable of being true. Wittgenstein’s argument that propositionsare not names of truth-values is merely an application of these two principles.I noted earlier that the exampleWittgenstein uses to make his point is based

on one Frege himself uses against a kind of formalism, but of course that is atbest evidence only of superficial borrowing, not of deep influence. What doesmake Wittgenstein’s argument profoundly Fregean is that the two guidingprinciples it relies on are ones Frege himself throughout his working life madecentral to his conception of logic. The first, that truth is not to be thought of asa property, we have already discussed. As for the second, that it is essential toa proposition that it should express something capable of being true or untrue,this is explicit in a pre-Begriffsschrift sketch9 of Frege’s logical doctrines.

A criterion for whether a mode of connection constitutes a thought is that it makessense to ask whether it is true or untrue. Associations of ideas are neither true noruntrue. . . The expression in language for a thought is a sentence. We also speak in anextended sense of the truth of a sentence. A sentence can be true or untrue only if it isan expression for a thought.10

Here, then, we have an instance of Wittgenstein’s tendency to think, in Kien-zler’s nice phrase, ‘with Frege against Frege’.11 We have met several examplesin earlier chapters where Wittgenstein exploited Fregean principles, but whatmakes his employment of them here especially notable is that he directs them,to devastating effect, against Frege himself. What he displays is an inconsis-tency in Frege’s own views, and he does so in a manner which makes it all themore surprising that Frege did not spot it himself.

).$).$).$ Propositions as articulateHere is another, quicker way of seeing what is wrong with Frege’s idea thatsentences are names of truth-values. According to that doctrine, I speak trulywhenever I assert a name of the True. In particular, since ‘the True’ is aname of the True, I speak truly when I assert ‘the True’. But what am I thenasserting? If I represent my assertion as ascribing a property to an object, thenby saying that the True is true I seem indeed to have said something—trivial,admittedly, but correct. However, if I insist, as Frege did, that there is no such

9See Dummett, FOP, 77. 10PW, 174. 11Wittgensteins Wende, 227.

(! Fregean propositions

property as truth for me to be ascribing here, and no sense to which the phrase‘is true’ corresponds, that explanation cannot be right. At this point, though, itbecomes difficult to see what else I can be saying if not that. When I assert ‘theTrue’, it seems that I am performing the incoherent feat of telling the truthwithout there being any particular truth I am telling. On this way of puttingthe point, therefore, what Frege had failed to do by treating propositions asnames of truth-values was to build into them the requirement that they shouldbe complex. ‘Propositions,’ Wittgenstein said, ‘can never be indefinables, forthey are always complex.’12 The names of the two truth-values, ‘the True’and ‘the False’, contravene this dictum.It is important to understand clearly why a proposition cannot be simple.

Consider, for example, two notations for postal addresses in a town: in thefirst system, each address consists of a number followed by a street name; inthe second, every house has a name of its own. It is not hard to see whichnotation a postman would prefer. The compositional structure of the atomicproposition is, we might think, grounded only in this point about economy ofexpression.That, however, is not Wittgenstein’s point at all. To see that it mistakes

his intentions, it is enough to note that the degree of economy effected by thedevice under discussion, although sometimes substantial, is always finite. It isnot a device which enables us by learning a finite number of items to expressinfinitely many different senses. It can therefore only make a difference to therelative feasibility of the notation, not to its absolute possibility. But this issueof relative feasibility was far from Wittgenstein’s concerns. Later in his life hecame to address seriously the issue of what it amounts to to say that some-thing is possible even though it would be quite impracticable for any humanto do it, but throughout his early philosophy there is a consistent tendency forhim to regard issues of feasibility almost as missing the point. One sometimessuspects that he thought the gap between possibility and feasibility was justwhat servants were there to bridge: he would not have regarded the views ofthe postman as relevant. Wittgenstein’s point was rather that a proposition isessentially articulate. The reason that the notation with street names is prefer-able to the one where each house has its own name is not that there are fewersigns in it, but rather that the first notation is capable of telling the postmanwhere each house is: the composition of the address corresponds to the actualarrangement of the houses.Wittgenstein makes the obvious point that the single-word Latin sentence

‘ambulo’ (which means ‘I walk’) is not a counterexample to this, because theexistence of other related words such as ‘ambulas’ and ‘ambulat’ with thesame root and different terminations shows it to be really hiding the articulatestructure ambul|o within the single-word form.13 However, there is another

12B64. 13Ibid.

Propositions as articulate (#

fairly obvious kind of apparent counterexample to the thesis that propositionsare essentially articulate that needs to be considered. ‘How is it possible,’Wittgenstein asked,for ‘kilo’ in a code to mean ‘I’m all right’? Here surely a simple sign does assert some-thing and is used to give information to others.—For can’t the word ‘kilo’, with thatmeaning, be true or false?14

To answer his question we have to recall our earlier discussion of the sym-bolic turn. The task of a sentence of ordinary language is to bring to mind asymbolizing fact. But, as was stressed before, not everything in the sentenceis relevant to this task. We are used, when we read English sentences, to ig-noring a host of irrelevant features in discerning what symbolizes in them.What Wittgenstein’s example shows is that sometimes in ordinary languagethe opposite is true. Some signs have in certain respects less complexity thanthe facts that we read into them. The most obvious examples of this are, likeWittgenstein’s in the above quotation, abbreviations; and in such cases thenatural thought is that the abbreviation simply inherits the complexity of theexpression it abbreviates. The obvious hypothesis would be that abbreviationsare the only such cases, but there is nothing in Wittgenstein’s writing to showthat he was committed to this hypothesis, at any rate if we limit ourselves toexplicit abbreviations.The irony is that the articulacy of propositions is once again something

Wittgenstein could have learnt from Frege. For it was above all Frege whostressed that thoughts are essentially complex. He found it ‘extraordinary’, forexample, ‘that some linguists have recently viewed a “sentence-word”, a wordexpressing a whole judgment, as the primitive form of speech and ascribe noindependent existence to the roots, as mere abstractions’.15 Yet Wittgenstein’saccusation is, in effect, that if sentences were names of truth-values, it would beimpossible to explain what was extraordinary about this view. What is wrongabout Frege’s assimilation of sentences to names is that it offends against aprinciple which Frege himself held dear.

14NB, 4 Oct. 1914. 15PW, 17.

Chapter !*!*!*

Assertion

!*.!!*.!!*.! The judgment stroke as force indicator

As we noted in the last chapter, the judgment stroke is not interchangeablewith the ‘is true’ operator. For a sentence of the form ‘A is true’ may, like anyassertoric sentence, be used for purposes other than simple assertion. It may,for instance, be used to form compound sentences, such as

If A is true, then I am a Dutchman

orA is not true.

Frege’s judgment stroke, on the other hand, cannot be embedded in this man-ner, because it combines two roles: it both converts a name of a truth-valueinto a sentence and indicates the assertoric force of the utterance, which thephrase ‘is true’ cannot do on its own. Nor does this point to some weakness ofthe word ‘true’: it is just as wrong to read ‘# A’ as ‘A is asserted’ as it is to readit ‘A is true’, since, once again, ‘A is asserted’ can be embedded in compoundsbut ‘# A’ cannot. What this shows is that even if we accept Wittgenstein’s rec-ommendation to absorb the verb ‘is true’ into the proposition, there is still afunction left for Frege’s turnstile to perform, namely that of indicating asser-toric force.Natural language, as Frege was fond of pointing out, has no sign for asser-

tion, relying instead on a variety of means (especially context) to indicate whenan assertion is being made. Frege regarded this as a defect of natural languageand introduced the turnstile in order to correct it, but it is far from clear thathe was right to do so: as we shall see shortly, the turnstile has been responsiblefor considerable confusion. At the very least, its grammar takes some gettingused to. If a text contains the assertion ‘# p’, then what the author asserts isthat p; it is an easy slip to disquote and represent him, incoherently, as assert-ing that # p. Because it is an indicator of force, and has no semantic content,the turnstile must not be thought to modify what p expresses. I may assert,doubt, suppose, or entertain the bold conjecture, that p; and in each case itis the same proposition p to which I adopt an attitude. As Wittgenstein ob-

Asserted and unasserted propositions (%

served, ‘ “#” . . . belongs as little to the proposition as (say) the number of theproposition.’1

In an appendix to the second volume of Grundgesetze Frege discussed variousways of avoiding the paradox which Russell had recently pointed out to him.Because of their tentative nature Frege did not prefix his proposals with theassertion-sign. Geach has called this ‘hypocritical’,2 but surely it is worse thanthat. Once Frege had adopted the convention that a sentence of the Begriff-sschrift must be prefixed with an assertion sign if it is to count as an assertion,simply writing a sentence without such a sign was not merely hypocritical butungrammatical. He could, of course, have introduced a further sign to markthe force of tentative assertion, but even tentative assertion is a kind of asser-tion and therefore a linguistic act.There is perhaps even something in Wittgenstein’s much later suggestion3

that the assertion-sign should be thought of as a punctuation mark. Therewas a time when books on English usage insisted that the exclamation markshould be used only to terminate an exclamation or command, and not, as isnowadays common, as a sort of generalized emoticon. If the stricter conven-tion were followed universally, we could equally consider using the full stoponly to terminate assertions. But then there might also be a point in havinga sign to mark the start of an assertion, to fulfil the function, like cinema ads,of reassuring us that we have not arrived halfway through. In ordinary writ-ten English, of course, capital letters usually fulfil this role well enough, but inlogic, where we cannot capitalize willy-nilly, we might feel the need of anotherconvention, such as Frege’s.Whatever the practical merits of such a proposal, however, the real point

Wittgenstein was making, both early and late, was of course a negative one:the assertion sign must at any rate not be thought of as more than a punctuationmark. This is not to doubt its role as a mark of force, but only to proscribeany further role for it.

!*."!*."!*." Asserted and unasserted propositionsWittgenstein’s recommendation to treat the assertion sign as a punctuationmark thus has some purchase as a correction to Frege, but it applies witheven greater urgency to Russell, who had struggled to explain the differencein role between occurrences of a proposition asserted and unasserted in logicalarguments.Russell shared Frege’s anti-psychologistic view of logic. Consequently, he

thought that expelling psychology from logic required him to treat the turn-stile, when it occurs in logical inferences, as logical and not psychological.

1B32. 2‘Saying and showing in Frege and Wittgenstein’, 63. 3PI, §22.

(" Assertion

Moreover, since he did not in the Principles have Frege’s distinction betweencontent and force, he supposed there that the difference in role between ‘p’and ‘# p’ must reflect a difference between the entities these expressions referto. He therefore treated ‘p’ and ‘# p’ as distinct entities, which he called theunasserted and asserted proposition respectively. But since the conception ofassertion in question here was logical, not psychological, an asserted proposi-tion was for him just the same thing as a true one. ‘It is plain,’ he said,

that true and false propositions alike are entities of a kind, but that true propositionshave a quality not belonging to false ones, a quality which, in a non-psychologicalsense, may be called being asserted.4

But what is the ‘quality’ which distinguishes true propositions from falseones? ‘There are,’ he had to admit,

grave difficulties in forming a consistent theory on this point, for if assertion in any waychanged a proposition, no proposition which can possibly in any context be unassertedcould be true, since when asserted it would become a different proposition. But this isplainly false; for in ‘p implies q,’ p and q are not asserted, and yet they may be true.5

He could have added, indeed, that the view would also require him to aban-don Moore’s simple identity theory of truth. When I believe something, theobject of my belief cannot be the asserted proposition, both because I canbelieve falsely and because if it were, I could hope to discern its truth by in-trospection on the object of my belief. But if what I believe is the unassertedproposition, then in the case when my belief is true, what is true, the assertedproposition, is not the same as what I believe.Yet it was Russell himself who insisted in a letter to Frege that whether I

judge a proposition or only imagine it, ‘the object is the same in both cases. . .The judgment stroke therefore means a different way of being directed to-wards an object.’6 In truth, Russell had got himself into a tangle, as is almostvisible on the page.

Assertion does not seem to be a constituent of an asserted proposition, although it is,in some sense, contained in an asserted proposition. If p is a proposition, ‘p’s truth’is a concept which has being even if p is false, and thus ‘p’s truth’ is not the sameas p asserted. Thus no concept can be found which is equivalent to p asserted. Yetassertion is not a term to which p, when asserted, has an external relation; for anysuch relation would need to be itself asserted in order to yield what we want. Also adifficulty arises owing to the apparent fact, which may however be doubted, that anasserted proposition can never be part of another proposition: thus, if this be a fact,where any statement is made about p asserted, it is not really about p asserted, butonly about the assertion of p.7

If the view that asserted propositions are different entities from unassertedones leads to so many difficulties, one might expect Russell to have abandoned

4Principles, §38. 5Ibid. 624 May 1903. 7Principles, §478.

Asserted and unasserted propositions (&

it rapidly, but he did not. Instead he adopted the rather desperate strategy(which he often used when he found himself in a hole) of reaffirming the viewwhile granting that he had no answer to the difficulties it leads to. ‘Leaving thisproblem to logic,’ he neatly advised, ‘we must insist that there is a differenceof some kind between an asserted and an unasserted proposition.’8

That was in the Principles. Was he any clearer by the time of Principia?There he probably did not intend to allow the turnstile to embed in largerpropositional contexts, although he contrived to create an appearance to thecontrary by adopting9 a convention according to which ‘# p, # p ! q, so # q’,for instance, is abbreviated to ‘# p ! # q’. Nonetheless, he did still speak of theasserted proposition as a different entity from the unasserted proposition.10

For Russell, then, a proposition is somehow different when it occurs as-serted from when it occurs unasserted. Wittgenstein rejects this doctrine veryfirmly in the Notes: in the Birmingham Notes he points out that ‘the assertionsign is logically quite without significance’;11 and in the Cambridge version hetells Russell that ‘there are only unasserted propositions’.12 He does not offeran argument for this. No doubt, the argument we offered in the last chaptercould be adapted to meet the case: if, as that argument aims to show, a prop-osition, even an unasserted one, ‘must already contain the verb’,13 it alreadycontains everything that is needed to determine what it expresses, i.e. whatis said to be the case when the proposition occurs asserted. But Wittgensteinprobably did not formulate the argument with that in mind: after all, he wasexplicit that its target was Frege, not Russell, and he did not allude to it againwhen he came to make his point against Russell. In truth, though, he did notneed to: the view could easily be attacked on other grounds, not least thoseRussell himself offered in the Principles.One thing that follows from Wittgenstein’s view, of course, is the hope-

lessness of trying to disentangle truth from assertion. ‘Could we not expressourselves,’ Wittgenstein asks,

by means of false propositions just as well as hitherto with true ones, so long as weknow that they are meant falsely? No! For a proposition is then true when it is as weassert in this proposition; and accordingly if by ‘q’ we mean ‘not-q’, and it is as wemean to assert, then in the new interpretation ‘q’ is actually true and not false.14

It is incoherent, that is to say, to suppose that we could lie all the time: lying isparasitic on the practice of telling the truth. This point is again a Fregean one,since it was Frege who stressed the curiously intimate relationship betweentruth and assertion. The word ‘true’, he said, ‘seems to make the impossiblepossible: it allows what corresponds to the assertoric force to assume the formof a contribution to the thought’.15

8Principles, §38. 9PM, I, 104–5. 10PM, I, 92. 11B32. 12C40. 13B10. 14B9. 15PW, 252.

(' Assertion

!*.#!*.#!*.# Assertion as psychologicalRussell’s misbegotten doctrine of asserted propositions arose from two viewsof his: first that the turnstile has, as we might now say, semantic content,rather than being merely an indicator of force; second, that what it indicatesis something logical, not psychological. That the first of these views is wrongis something Frege had been clear about long before he met Wittgenstein. Onthe second, however, Frege’s views were not perhaps as clear as some16 haveheld. What the judgment stroke indicates is for Frege an act, the mental act ofjudging a thought to be true.There is a strand in Fregean commentary17 that presents him as holding

that we cannot judge falsely. That is uncharitable: the few passages in whichhe seems to speak that way—most notably an unsent draft of a letter—aresurely best passed over as no more than loose phrasing.18 But if judgment,for Frege, is a mental act, hence both psychological and fallible, what shouldhe then say about assertion? Judgment and assertion are not the same, as hehad long realized. ‘When we inwardly recognize that a thought is true, we aremaking a judgment; when we communicate this recognition, we are makingan assertion.’19 For Frege, the realm of psychology is the mental, and whatis characteristic of the mental is that it is private: it is a recurring theme ofFrege’s writings that my ideas are intrinsically mine and hence unavailableto anyone else. So if judgment is psychological and assertion is the publicmanifestation of judgment, should we suppose that assertion is in some waypsychological too? Pain, we may agree, is private and hence psychological.Crying out plainly is not: although often a sign of pain, it may be feigned,by actors and malingerers, for their own ends. But crying out in pain has apsychological component, which it inherits from the pain: an actor cries out,but does not cry out in pain. Is assertion in this respect like crying out or likecrying out in pain? In order that I should assert something, is it required thatI should judge it?There are indications that Frege himself was tempted to think of assertion

as entailing judgment. It is consistent with such a view, for instance, thathe should have been so exercised by the case of actors on stage, who uttersentences which they do not, in propria persona, intend their hearers to believe.He held that because they are not really judging as true the sentences theyutter, the actors are not asserting anything either.20

To some extent, no doubt, ‘assertion’ is a philosopher’s term of art; and tothat extent we are free to legislate its use as we please. Dummett is right tomaintain, nonetheless, that there are clear advantages to thinking of assertionas wholly non-psychological. Frege’s view misses the essentially public nature16E.g. Goldfarb, ‘Wittgenstein’s understanding of Frege: The pre-Tractarian evidence’. 17E.g.Anscombe, IWT, 114. 18See Kremer, ‘Judgment and truth in Frege’. 19PW, 139. 20‘ÜberSinn und Bedeutung’, 34–5.

Psychology ((

of assertion. Precisely because we can indicate it linguistically, what is essentialto assertion is the expression of it, not whether we mean it in our hearts. Ifwe had in natural language an explicit device like Frege’s judgment stroketo indicate assertion, then we would expect actors on stage to use it with fullvigour: anything less could scarcely count as a convincing performance. Itthus seems right to say, as Dummett21 does, that the actor is doing more thanasserting, not less. Similar remarks may be made about liars and conmen. Itmay indeed, as Davidson22 claims, be hard to specify the linguistic conventionsgoverning when something is to count as assertion; but that is not sufficientreason to deny that such conventions exist, so that if something I say conformsto them, then it is an assertion. For I can be convicted of error if it turns outto be wrong, just as I can be blamed for failing to keep an insincere promise.It follows from this that assertion (unlike judgment) is not in Frege’s sensepsychological, since it is not private.

!*.$!*.$!*.$ PsychologyAt first sight, this conclusion—that assertion is not psychological—is in flatdisagreement with Wittgenstein’s claim that it is. ‘Assertion,’ he claims inthe Notes, ‘is merely psychological.’23 However, this is a straightforward con-tradiction only if Wittgenstein’s use of the word ‘psychology’ is the same asFrege’s. What should give us pause is that Wittgenstein describes the corre-lation of names with their meanings as psychological.24 What Wittgenstein isdescribing here is the process by which we turn linguistic items into symbols:they acquire membership of the various grammatical categories by virtue ofbeing given symbolizing roles in this way. This is what we might nowadayscall linguistics: Wittgenstein makes no distinction between it and psychology.The point here is one we discussed in chapter 7. By refusing to distinguish

between the subjective and the intersubjective Wittgenstein collapsed the dis-tinction between ideas and senses, and hence that between psychology andlinguistics. When he said that attaching a meaning to a name is a matterfor psychology, he intended to distinguish psychology not from linguistics butfrom logic.Wittgenstein says, moreover, that epistemology is the philosophy of psy-

chology.25 Epistemology studies the process by which linguistic items acquiremeanings, and thereby symbolic roles. This process is not the concern oflogic,26 which studies these symbolic roles themselves. For Frege the distinc-tion between ideas and senses was fundamentally one between what is privateand what is public, between the subjective and the intersubjective. Wittgen-stein denied this distinction.

21FPL, 311. 22‘Moods and performances’. 23C40. 24B43. 25B62. 26B24.

)** Assertion

I mentioned earlier that Wittgenstein does not seem to have been troubledabout whether his views were solipsistic. This is not quite the same as sayingthat he was a solipsist, but at any rate it is clear that he did not, as Fregedid, intend the word ‘psychology’ to demarcate the mental as a private spherein contrast to the public sphere of language. For, as we have just noted, hewas explicit that the correlation of name and meaning is psychological,27 andthat is a matter of linguistic convention if anything is. For this reason wecan, I think, agree with Wittgenstein’s claim that ‘assertion is psychological’,by which he meant only that it is not a matter for logic, without disagreeingwith the conclusion we reached earlier that assertion is not in Frege’s sensepsychological, since it is not private.This question of whether assertion should in any sense be regarded as psy-

chological was for Frege by no means a peripheral one. Long before he metWittgenstein, he had noted the curious connection between the use of theword ‘true’ and the assertoric form of the sentence.

If I assert that the sum of 2 and 3 is 5, then I thereby assert that it is true that 2 and3 make 5. In the same way I assert that it is true that my idea of Cologne Cathedralagrees with reality, if I assert that it agrees with reality. Therefore it is really by usingthe form of an assertoric sentence that we assert truth, and to do this we do not needthe word ‘true’. Indeed we can say that even where we use the form of expression ‘itis true that . . . ’ the essential thing is really the assertoric form of the sentence.28

There is in Frege’s Nachlaß a document headed ‘The following may be of someuse as a key to the understanding of my results’. Perhaps it is not too fancifulto wonder whether Frege wrote this document in preparation for, or as aresponse to, one of his meetings with Wittgenstein. Here Frege repeats hisview, which he had stated elsewhere, that the word ‘true’ indicates the essenceof logic. But then he shifts his ground.

The thing that indicates most clearly the essence of logic is the assertoric force withwhich the sentence is uttered. But no word, or part of a sentence, corresponds to this;the same series of words may be uttered with assertoric force at one time, and not atanother. In language assertoric force is bound up with the predicate.29

I suggested in §5.6 that Wittgenstein’s fundamental thought could be seen asthe combination of two Fregean principles, that the subject matter of logicis truth and that truth is redundant. It seems now as if, in order to resistthis conclusion, Frege was prepared to say that the subject matter of logic isassertion.It is plain that if he were indeed to say that, and not give up on his virulent

anti-psychologism, he had to be clear that assertion is not psychological inhis sense of the word. I have suggested that he may not have been as clearabout this as he should have been. There would, on the other hand, have

27B43. 28PW, 129. 29PW, 252.

Psychology )*)

been no strong reason for Frege to resist the idea that part, at least, of logicis psychological in Wittgenstein’s sense, since this only means that it takescognisance of the realm of sense. Frege was committed to the idea that it isthe richness of this third realm that explains the richness of arithmetic. Heapproved, for instance, of Dedekind’s proof that there are infinitely manythings in the third realm,30 a proof that is central to Dedekind’s way of showingthat arithmetic is part of logic. It follows that Frege’s conception of logic, likeDedekind’s, has a component that is in this sense (but not in Frege’s own)psychological.Wittgenstein, on the other hand, would have insisted that logic is not psy-

chological even in his own wider sense: he used the words ‘logical’ and ‘psy-chological’ in what he plainly intended as contrastive senses. So Wittgenstein,in addition to having a wider conception of psychology than Frege, also hada narrower conception of logic. It is therefore plausible to hear Wittgenstein’svoice in the background when Jourdain asked Frege, ‘whether you now regardassertion (#) as merely psychological’.31 If Frege was, as I have suggested,32 insome confusion over the matter, perhaps it is telling that both his draft reply(never sent) and his actual reply avoid answering Jourdain’s question outright.

30PW, 136. 3115 Jan. 1914. 32Contra Goldfarb, ‘Wittgenstein’s understanding of Frege: Thepre-Tractarian evidence’.

Chapter !!!!!!

Complex and fact

The distinction between complex and fact is easily stated. A complex is anarrangement of things; that they are arranged in this way is a fact. A complexis thus something we can refer to by means of a description; we specify a fact,on the other hand, by means of a ‘that’ clause. This distinction is one of themost important inWittgenstein’s philosophy, and it pervades the Notes on Logic.

!!.!!!.!!!.! A world of facts, not of thingsIn chapter 4 I discussed Wittgenstein’s analysis of a proposition about a com-plex into ‘a proposition about its constituents and the proposition which de-scribes the complex perfectly’. His idea, we saw there, was to analyse

!([aRb])" !!(a,b) aRb.

where !! is a propositional function related appropriately to !. In the case ofa proposition which says only that a certain complex exists, !! is vacuous, sothat the analysis reduces to something like the following:

[aRb] exists" aRb. (1)

So if we take Wittgenstein’s eliminativism seriously, the existence of the com-plex referred to on the left of this biconditional is reduced to the fact expressedon the right. At the last stage of analysis, that is to say, we obtain a world offacts, not of complexes.What recommended Wittgenstein’s eliminative conception of analysis over

Russell’s might be described as a ruthless application of Occam’s razor; or,less charitably, as the triumph of ontological scepticism over common sense.But what reason is there for wielding the razor in this direction? Consider themore general equivalence

[p] exists" p, (2)

and leave to one side whether it could hold in full generality. The question weneed to address is whether, in those cases in which it is true, there is any reason

A world of facts, not of things )*#

to treat one side rather than the other as more fundamental. Note first that itis problematic to try to privilege the left-hand side of the equivalence over theright in all such cases, since that would require us to assume that exactly oneof the two complexes [p] and [!p] exists, without providing any explanationfor this curious coincidence. The view would thus seem to load onto theworld the whole responsibility for logic. Whenever a complex goes out ofexistence, another complex, its logical contradictory, has to come magicallyinto existence in its stead. This makes the connection between p and not-p,according to which not-p is true just in case p is false, a kind of miracle, a giftdonated to logicians by a generous world.But when we restrict ourselves to the very simplest case, where p is an

atomic proposition, it is not so clear that this objection still applies: one mightperhaps think that it is a criterion of the atomicity of p that p and not-p shouldbe related in this manner. (I shall discuss this thought further in §15.2.) Isthere any other reason, in this atomic case, to privilege the right-hand sideof (2) over the left? Russell, at any rate, thought not. In 1910 he outlined atheory according to which what makes an atomic proposition true or false isthe existence or non-existence respectively of the corresponding complex. Butin fact his conception oscillated uncertainly. We have already come acrosshis willingness to turn verbs casually into verbal nouns. He often displayeda similarly relaxed attitude to the process of nominalization which turns thefact that Caesar died into Caesar’s death: one moment he gave as an exam-ple the complex ‘a-in-the-relation-R-to-b’;1 the next he used the fact ‘Socratesis mortal’.2 On occasion, he referred to complexes by means of that-clauses,and he observed that the grammar of such clauses gives them no completemeaning on their own. ‘We feel that the phrase “that so-and-so” is essentiallyincomplete, and only acquires full significance when words are added so asto express a judgment.’3 But, not having a clear grasp of the distinction, heregarded this only as an argument (and even then not a decisive one) againstthe existence of propositions, not as a way of arbitrating between complex andfact.Whatever Russell’s uncertainty, it is clear that he conceived of the denizens

of his world as much more complex-like than fact-like. However, the diffi-culty with this conception of the world as consisting ultimately of complexesarises when we consider our most basic judgments about it. Russell used theterm perception for the experience of an atomic complex which grounds ourindubitable judgment that the complex exists. If I perceive a red circle, for in-stance, that may, according to Russell, ground my judgment that the red circleexists. But as Wittgenstein put the point much later, ‘To say that a red circle

1CP, VI, 10. 2CP, VI, 12. 3CP, VI, 119.

)*$ Complex and fact

is composed of redness and circularity, or is a complex with these componentparts, is a misuse of these words, and is misleading.’4

One way of explaining what Wittgenstein was referring to here is by notinga fundamental difference in structure between complex and fact. How muchstructure Russell supposed a complex to carry is somewhat uncertain: perhapsthe truth is that he supposed it to carry whatever structure it needed in orderto serve the task at hand. At any rate, a complex cannot be simply the mereo-logical sum of its components, since that will exist whenever the componentsdo.5 We may disagree about the precise ingredients of a salad Niçoise (whetherit should contain eggs, for example), but whatever those ingredients are wewill surely agree that they do not constitute a salad until they are put in thesame bowl; I would be a poor host if I simply laid them out for my guests toassemble themselves.Ordinary language sometimes wavers. If I dismantle my computer in the

hope of fixing it, I may prefer to say that my computer is in pieces rather thanthat it no longer exists. But at any rate many of the objects of everyday lifeare, like salads, complexes made up of parts related in certain ways. That theparts are related in the requisite manner is a fact about the complex. We maysay that the complex exemplifies the fact. What is important—or at any rateone thing that is important—is that on this understanding of the distinctionthe correspondence between complexes and facts is one–many. One complex,that is to say, may exemplify several different facts. The salad, for instance,exemplifies the fact that the bowl contains the requisite ingredients (tuna, an-chovies, etc.); but it exemplifies, too, the fact that the dressing coats the otheringredients evenly. The red circle in Wittgenstein’s example exemplifies thefact that it is red; but it also exemplifies the fact that it has a certain radius;and, too, that it occurs in a certain position in my field of vision. It is mislead-ing to say that a red circle is a complex consisting of redness and circularitybecause it privileges a particular fact about it, namely that it is a red circle.But complexes do not speak: they are what they are, and do not present anyparticular facts as salient. Complexes, therefore, cannot be what ground prop-ositional thought, because they do not have the right kind of structure to doso. Only facts can do that.We might wonder once again, though, whether the argument we have of-

fered applies in the atomic case. Perhaps there are complexes of a kind thatonly exhibit a single fact. For such complexes the issue of salience wouldnot arise. But that is not the point. The lack of a one–one correspondencebetween complexes and facts is only a symptom of the difference, not the un-derlying reason to eliminate the former in favour of the latter.

4PG, 200. 5Cf. Russell, CP, VI, 355.

Influences )*%

!!."!!."!!." InfluencesWhen he discussed the distinction between complex and fact in the 1930s,Wittgenstein was quite clear where he had obtained the insight just men-tioned, that a red circle is not a complex consisting of redness and circularity:‘Frege was aware of this and told me.’6 Although the distinction is not some-thing Frege emphasized in his published writings before 1913, he did makethe following observation in his 1918 article, ‘Der Gedanke’.

That the sun has risen is not an object which emits rays that reach my eyes, it is not avisible thing like the sun itself. That the sun has risen is seen to be true on the basis ofsense-impressions. But being true is not a material, perceptible property.7

This shows Frege to have been aware of the distinction between complex andfact, but he cannot8 here have been responding to the Tractatus: he sent thefinal version of ‘Der Gedanke’ to be published a couple of weeks before hefirst saw the Tractatus.For Frege, though, facts never quite played the dominant ontological role

that they did for Wittgenstein. In ‘Der Gedanke’ he remarks that ‘a fact is athought that is true’, but he does not go on to make any great play with thenotion. The offhand way he mentions it suggests, indeed, that his meaningis almost deflationary: a fact is merely a thought that is true. Wittgenstein, onthe other hand, famously chose to make facts salient at the beginning of theTractatus. If he himself was not quite so emphatic in the 1913 Notes, Russell wasexplicit, presumably on his behalf, in the first of his logic lectures at Harvardthe following year. ‘What we analyse,’ Russell said then, ‘is a fact, not athing. A complete description of the real world is not given by enumeratingthe things in it.’Although Frege’s influence would be sufficient to explain the importance

Wittgenstein attached to the distinction between complex and fact, he couldalso have found it in a continental tradition stemming fromBrentano. Stumpf,9

for instance, noted in 1907 that Brentano had in his lectures thirty years ear-lier distinguished between what he called the specific and the presentationalcontent of a judgment. He went on to note that the former is what is expressedlinguistically in a ‘that’-clause and to propose the word Sachverhalt as a techni-cal term for it. Others in the same tradition such as Reinach10 and Marbe11

also discussed the distinction between complex and fact in various guises. Ofthe two aspects to the distinction that we have highlighted—the difference ofidentity conditions and the difference of nature—it is the latter that most ofthese authors emphasize. Only with Reinach12 do we begin to see a stress onthe idea that a variety of facts (that this rose is red, that this rose forms the

6PG, 200. 7p. 61. 8Contra Sullivan, Formal concepts, 110. 9‘Erscheinung und psychis-che Funktionen’. 10‘Zur Theorie des negativen Urteils’. 11‘Beiträge zur Logik und ihremGrenzwissenschaften’. 12Ibid. ch. I, §7

)*" Complex and fact

substrate of redness, that redness is inherent in the rose) might be exemplifiedin a single complex.We do not know which of these authors Wittgenstein read, but it is surely

quite likely that he read some of them: if not, it is at least a curious coincidencethat he later used Sachverhalt in something rather like Reinach’s sense and thathe chose to illustrate his discussion13 with the very same example (‘This roseis red’) as Reinach and Marbe. Stumpf is also someone Wittgenstein is verylikely to have known about. Moore was reading him in the summer of 1912,and Wittgenstein would in any case have been likely to come across the namein the course of his experiments on rhythm at the Psychological Laboratory.McTaggart, who was also at Trinity and first met Wittgenstein in October1912,14 is also known to have taken an extensive interest in Stumpf’s work,and his own writings are notably free of the confusion in question.15

!!.#!!.#!!.# Russell on factsWhatever the influences were that helpedWittgenstein towards the distinctionbetween complex and fact, it is at any rate clear that Russell was not one ofthem. It remains to ask, though, whether there was any influence in the otherdirection, from Wittgenstein to Russell. If we are to answer that question, weneed to say a few words first about the dating of the Notes. I shall explain inAppendix A my reasons for thinking that the three sections of what I therecall the Birmingham Notes are compilations of remarks from three distinctnotebooks in which Wittgenstein had been writing his thoughts day by dayduring various parts of his time at Cambridge. If this is right, then it hassome bearing on the subject of the current chapter to note that the distinctionbetween complex and fact is exploited in all three sections of the BirminghamNotes; yet the letter of January 1913 does not betray a glimmer of it. Thissuggests quite strongly that the Birmingham Notes do not contain very muchmaterial drawn from before January 1913.On the other hand, I do not think that Wittgenstein could have started

writing the notebooks from which the Birmingham Notes are derived muchlater than January 1913. By saying this I contradict the popular view that theNotes are in their entirety a report of Wittgenstein’s work in the summer of1913 (or, even more impressively, just during his three-week holiday in Nor-way with Pinsent). This view does admittedly originate in a remark by Russellhimself, who seems to have been under the impression that much of the workthat Wittgenstein was reporting to him had been done in August and Septem-ber16 but, although I cannot refute the notion conclusively, the sheer number

13B7. 14BR to OM, 29 Oct. 1912. 15Geach, ‘Saying and showing in Frege and Wittgenstein’,67. 16To Lucy Donnelly, 19 Oct. 1913.

Russell on facts )*&

of substantial philosophical ideas the Notes contain surely makes it wildly un-likely. But it has contributed to the more general impression occasionallyvoiced that Wittgenstein worked in extremely concentrated spurts before col-lapsing exhausted. Another strand in the same thread, now disproved, used tohold that Wittgenstein wrote the whole Tractatus in the summer of 1918: themore mundane truth is that he had by then been working on the book itself(as opposed to the notebooks from which much of it was derived) for almostthree years. Indeed, there is every reason to think that his working methodswere not at all impulsive, but patient and methodical.One reason the myth took hold was no doubt that it resonates with some-

thing in our conception of the genius as someone possessed by an externalforce. It is not sufficient that Mozart wrote wonderful music: we express hisgenius by emphasizing the astonishing speed with which he wrote it. In thecase of the Notes, though, there is another, much more prosaic reason whysome have been tempted to suppose that they were written on holiday in Nor-way, namely that many of the ideas contained in them were evidently new toRussell when Wittgenstein explained them to him in October 1913. If, as Iam claiming, some of them may date from as early as the previous January,why did Russell not know them already?The conclusion we must draw, I believe, is that this is a sign of a marked

change in the working relationship between the two men. Russell alludedto something of this in his letters to Ottoline, where he confessed that he nolonger toldWittgenstein about his own ideas, because he ‘treats infant theorieswith a ferocity which they can only endure when they are grown up’.17 Per-haps Wittgenstein was now in his turn becoming selective about which of hisideas he shared with Russell. If so, then it may be an instance of this emergingintellectual gap that in May 1913 (when, if the dating I have been urging isright, Wittgenstein may already have been using his new conception of factsfor several months) Russell could still write:

It may be questioned whether a complex is or is not the same as a ‘fact’, where a ‘fact’may be described as what there is when a judgment is true, but not when it is false.(I do not suggest that this is a proper definition of a ‘fact’, but merely that it serves topoint out what sort of object is in question.) However this may be, there is certainlya one–one correspondence of complexes and facts, and for our present purposes weshall assume that they are identical.18

By asserting that the correspondence between complex and fact is one–one,Russell demonstrated his continuing failure to adopt Wittgenstein’s concep-tion: he was still using the word ‘fact’ for what Wittgenstein now called acomplex, namely what there is when a judgment is true, whereas Wittgen-stein used it for what is the case if the judgment is true. Indeed the passage

1724 Apr. 1913. 18ToK, 79–80.

)*' Complex and fact

just quoted reads rather as if Wittgenstein had already pressed on Russell adistinction between complex and fact, but Russell had not yet really graspedwhat it was. One reason Russell did not immediately grasp the distinctionwas no doubt that he had no need of it in order to make the point aboutstructure. That is to say, he could, if he wished, construct structured entitieshaving the unique parsing into constituents that was the main structural featureof Wittgensteinian facts.One point of interpretation is perhaps worth noting here. Since Russell

was slow to grasp Wittgenstein’s distinction between complex and fact, andsince the distinction is never explicitly explained in the Notes, one might won-der how reliable a guide Russell’s translation of these notes is to Wittgenstein’sintentions. Can we be sure, in other words, that ‘fact’ here really does meanfact and not complex? I think that we can. Both words occur in the Cam-bridge notes, dictated in English; both occur, too, in the Birmingham notes astranslated by Russell. Moreover, the usage throughout is consistent with thedistinction in the form in whichWittgenstein later made it explicit. This wouldbe an odd coincidence unless Russell was consistently translating Tatsache as‘fact’ and Komplex as ‘complex’.It is worth mentioning, too, that however slow on the uptake he may have

been, Russell did eventually embrace Wittgenstein’s view, at least to the ex-tent of adopting his terminology: his logic lectures at Harvard in 1914, whichbegan by saying that what we analyse is a fact, not a thing, refer repeatedlythereafter to facts, not complexes; in the Lowell Lectures, meanwhile, he de-fined a fact as ‘that a certain thing has a certain quality, or that certain thingshave a certain relation’;19 and again in 1918, still before he had read the Trac-tatus, he used the language of facts in claiming, for instance, that he did notbelieve in complex entities of the kind for which we habitually use propernames.20

19OKEW, 51. 20CP, VIII, 170.

Chapter !"!"!"

Forms

At the centre of Wittgenstein’s project was the task of explaining the unity ofthe proposition. We have already noted how in his letter of January 1913 hebegan to move away from an analysis of propositions as complexes united bya copula—an analysis which, if not actually Russell’s, was certainly Russellianin spirit and therefore wholly merited Wittgenstein’s description of it as ourtheory of symbolism. In the letter he adopted the Fregean idea that the unityis to be explained not by a pure copula but by one of the meaning-contributingcomponents of the proposition. However, the way in which he implementedthis Fregean idea was still Russellian. As a result, it is hardly a position thatcould have satisfied him for long.

!".!!".!!".! Form as nameAs we saw in chapter 8, Wittgenstein proposed in his January 1913 letter toanalyse ‘Socrates is human’ into two parts, of such different kinds that it isimpossible to reassemble them the wrong way round: the first is a name andthe second what we have been calling a form. I contrasted this proposal withWittgenstein’s earlier view that the proposition contains two names ‘Socrates’and ‘humanity’, combined by means of what he called a copula. What Idid not discuss then was what the kind of entity which is responsible for theunity of the proposition (whether it is a form or a copula) is supposed to be.What remained constant between the old theory and the new was preciselyhis conception of the sort of thing this entity is. In the old theory he analysedthe copula of a subject-predicate proposition as (

E

x,y)#1(x,y); or, in words, as‘Something is predicated of something’. In the new theory he analysed theform of ‘Socrates is human’ as (

E

x)!x; or, in words, ‘Something is human’.And similarly he now analysed the form of aRb as (

E

x,y)R(x,y).1

I suggested in §8.1 that Frege’s influence can be detected in Wittgenstein’smove from copula to form. The nature of the form was to be sufficient in it-self to explain why ‘Socrates is mortal’ makes sense but ‘Mortality is Socrates’does not. This was, I suggested, Wittgenstein’s attempt to implement Frege’s

1LW to BR, 16 Jan. 1913.

))* Forms

distinction between saturated and unsaturated expressions. But by maintain-ing his old analysis of the form, Wittgenstein failed to implement his Fregeaninsight convincingly. He was still struggling, it seems, to shoehorn the Fregeandistinction into what was still at base a Russellian conception of propositionsas complexes. Wittgenstein was no doubt right to insist in the letter that thecomponent of an atomic proposition which plays the role of the unsaturatedexpression must be simple, since if it is not, we shall then have to analyse itand a regress quickly threatens. But his proposal that this simple componentis ‘something is human’ is hard to take seriously, since this is so obviouslycomplex.Indeed, one might well wonder whether the idea was really Wittgenstein’s

at all. Since this part of the account—the idea that the verb of the propositionshould be analysed as existential—is present in both the old theory Wittgen-stein was rejecting and the new one he was now recommending, it evidentlypredates the letter. Moreover, although it is surely an odd idea, Wittgensteindoes not pause to explain or justify it, which suggests strongly that Russellwas already familiar with it. And indeed when Russell, writing his Theory ofKnowledge manuscript a few months later, needed something to play the roleof the form of a relation, he used just the same device.2 If it was part of whatWittgenstein was still inclined to call ‘our theory of symbolism’3—part, thatis to say, of a collaborative enterprise that he and Russell were conductingjointly—it is surely quite plausible that it was Russell’s idea in the first place.By the time Russell used the device in his Theory of Knowledgemanuscript, how-ever, Wittgenstein had probably already abandoned it. The passage in whichwe see him doing so occurs early in MS1 of the Birmingham Notes.

It is easy to suppose that only such symbols are complexes as contain names of objects,and that accordingly ‘(

E

x,!)!x’ or ‘(

E

x,R,y) xRy’ must be simple. It is then natural tocall the first of these the name of a form, the second the name of a relation. But in thatcase what is the meaning of (e.g.) ‘"( Ex,y) xRy’? Can we put ‘not’ before a name?4

The answer Wittgenstein intended us to give to this last rhetorical question isof course ‘no’. The inference he drew, therefore, was that ‘(

E

x,R,y) xRy’ is notsimple and hence not a component of ‘aRb’.This argument against attributing the unity of the proposition to a compo-

nent of existential form is surely convincing. It may therefore seem surprisingto discover him returning to the very same point in the wartime Notebooks.

I thought that the possibility of the truth of the proposition !a was tied up with the fact(

E

x,!) !x. But it is impossible to see why !a should only be possible if there is anotherproposition of the same form. !a surely does not need any precedent. (For supposethat there existed only the two elementary propositions ‘!a’ and ‘"a’; and that ‘!a’were false: Why should this proposition only make sense if ‘"a’ is true?)5

2ToK, 114. 3To BR, 26 Dec. 1912. 4B5. 5NB, 23 Oct. 1914.

Form as function )))

There is no need, though, to draw from this the conclusion that Wittgensteinhad become dissatisfied with his earlier argument against treating the formof a proposition as existential. It is one of the features of his way of workingthat he quite often returned some time later to issues that he had apparentlysettled. Sometimes this seems to have been because he had come to doubt hisearlier arguments; sometimes, perhaps, it was just because he had forgottenwhat his earlier arguments were; quite often, though, he knew what they werebut wanted to think the matter through again in any case. I mention this herebecause interpreters of Wittgenstein have sometimes used his later discussionsof an issue to argue that he cannot have been clear about the issue earlier.There will no doubt be occasions when that is so, but it is a harder point toprove against Wittgenstein than against many other philosophers, because hismethod of working seems so often to have involved him in thinking through aproblem again and again, reaching the same conclusion each time by a slightlydifferent route. He once enjoined himself, ‘Don’t worry about what you havealready written. Just keep on beginning to think afresh as if nothing at all hadhappened yet.’6 And the Notebooks provide ample evidence that he obeyed hisown injunction.

!"."!"."!"." Form as function

If we reject the idea that the third component of ‘aRb’ is ‘(

E

x,y) xRy’, that doesnot yet point us to what else it might be. The next step was for Wittgensteinto see that the answer must be radically different: the third component cannotjust be another name. Propositions, he said, ‘cannot consist of names alone;they cannot be classes of names’.7 Unusually, he repeated this point later inthe Birmingham Notes, this time adding helpfully, ‘This is easily shown.’8 Andindeed it is easily shown. For a class of names offers no prospect of supplyingthe resources to explain the unity of the proposition.Wittgenstein also thought this observation worth including in the Tractatus.9

When we read it in this context, though, it is apt to engender puzzlement. Ourimmediate response to being told that a proposition is not a class of names isto reply, ‘Yes, but whoever said it was?’ Not Frege, certainly: for him a prop-osition is a name, not a class of names. Russell’s position is a little more com-plicated, since by 1913 he held that propositions are incomplete symbols. Butthis does not make them classes of names; and before he abandoned proposi-tions, he had consistently held that any proposition must contain at least oneverb. The answer to this puzzle is surely that it was Wittgenstein himself whohad, albeit temporarily, held in effect that a proposition is a class of names.

6NB, 15 Nov. 1914. 7B1. 8B47. 93.142.

))! Forms

Abandoning this view led him to require that the third element in ‘aRb’, theelement he called its form, must be something radically different from a name.But if it is not a name, what is this third component of ‘aRb’? A casual

account of Frege’s view might be that the third component is the incompleteexpression ‘xRy’. But this is casual. The expression ‘xRy’ is plainly not acomponent of ‘aRb’ in any ordinary sense of the word ‘component’, if onlyfor the trivial reason that the letters ‘x’ and ‘y’ occur in the former but notin the latter. Frege’s rather less casual answer was to think of the incompleteelement in the expression of a proposition as what Geach has called a linguisticfunction, i.e. a function taking linguistic expressions as arguments and having alinguistic expression as its value. That, at any rate, is surely10 the natural wayto read Frege’s explanation of the notion of a function in the Begriffsschrift.

If in an expression . . . a simple or complex sign occurs in one or more places and if weregard that sign as replaceable in all or some of these occurrences by something else(but everywhere by the same thing), then we call the part that remains invariant in theexpression a function, and the replaceable part the argument of the function.11

Thus, for example, the sentence ‘Socrates is mortal’ is the result of applying to‘Socrates’ the function which for any name $ as argument has !$ is mortal"as its value; and ‘aRb’ is the result of applying to ‘a’ and ‘b’ the function whosevalue for the arguments $ and % is !$R%".12The difficulty with this, however, is that we may doubt whether it is any

longer something that can appropriately be called an analysis of the ‘structure’of the proposition. That an object is a value of a function for some argumentis not generally thought of as contributing to an analysis of the object. (Toobserve that 4 is the cube root of 64 does not contribute to an analysis ofthe number 4, for example.) This has led Dummett to claim that ‘to talkof expressions and their structure, we need the notions of part and whole,not those of function and value’.13 The problem with the function–argumentanalysis, that is to say, is that

given only the value of a function for some argument, it is not possible to recoverthe function or the argument. For this reason it is inappropriate to regard either theargument or the function as a constituent or part of the value, since we naturally supposethat anything is uniquely analysable into its ultimate constituents, and that the partsof a thing may be discerned by scrutiny of it.14

Scrutiny of the proposition ‘Socrates is mortal’ cannot reveal the functionwhose value for the argument ‘Socrates’ it is, because there is any numberof such functions. So if the purpose of analysis is to reveal the constituents of athing and the manner of their combination, Frege’s application of the notionof a function cannot be relevant to the analysis of the proposition.

10Contra Baker and Hacker, Logical Excavations, 172n. 11Frege, Bs, §9. 12See Sullivan, ‘Thefunctional model of sentential complexity’. 13‘An unsuccessful dig’, 396. 14Ibid.

The form of a fact ))#

!".#!".#!".# The form of a factWittgenstein’s breakthrough came when he saw that he could respect theessence of Frege’s point by saying that what symbolizes in ‘aRb’ is not thesign ‘R’ on its own but the fact that it occurs between the names ‘a’ and ‘b’.From this point on, therefore, he maintained that a proposition is a fact, nota complex. When it was applied at the level of symbolism, the distinction be-tween fact and complex thus allowed him to refine the Fregean insight thatwhat makes a proposition expressive is the structure it has. Frege, let us re-call, had conceived of the expressions for concepts and relations as what hecalled unsaturated, i.e. as containing argument places which have to be com-pleted with names of objects in order to form a complete sentence. Whatcorresponds in Wittgenstein’s conception to a saturated expression is what hein 1913 calls a ‘name’; what corresponds to an unsaturated expression is whathe calls a ‘form’. What is expressive is not the complex consisting of the namesand form but the fact that they are combined in a certain manner.Not: ‘The complex sign “aRb” ’ says that a stands in the relation R to b; but that ‘a’stands in a certain relation to ‘b’ says that aRb.15

Or, as he glossed it in the Cambridge Notes,In aRb it is not the complex that symbolises but the fact that the symbol a stands in acertain relation to the symbol b. Thus facts are symbolised by facts, or more correctly:that a certain thing is the case in the symbol says that a certain thing is the case in theworld.16

In a relational proposition involving the names ‘a’ and ‘b’, what expresses therelation is not one identifiable component of the proposition on its own; ratheris it the fact that the names ‘a’ and ‘b’ are related in some particular way. Inthe case of a proposition written ‘aRb’, therefore, the sign ‘R’ does not expressanything on its own; it does so only when combined with the names ‘a’ and ‘b’in a certain way. What is expressive is not the complex consisting of the threesigns, but a fact about this complex, namely that in it the sign ‘R’ occurs withsomething to the left of it and something else to the right of it. The sign ‘R’functions only as a label to distinguish this relationship between the names ‘a’and ‘b’ from other possible relationships (such as the one exemplified in ‘aLb’,for instance). Borrowing our terminology from Long,17 who in turn borrowedit from the Tractatus,18 let us call signs used as labels in this fashion indices. Thecorrelation between an index and what it labels is a matter of convention.Indeed, the fact that there is an index at all is conventional. What is requiredin order to express that aRb is only that the names ‘a’ and ‘b’ should standin a certain relationship. In the present case, of course, the relationship is, asit happens, that of having the index ‘R’ standing between them, but there is15B57; cf. 3.1432. 16C44. 17‘Are predicates and relational expressions incomplete?’, 93.185.02.

))$ Forms

nothing essential about this. It would be perfectly possible to have a languagein which some relations are expressed not with the aid of indices but rather byother devices such as spatial relationships between the signs. This last sort ofnotation is indeed sometimes used for functions in mathematics: what does thesymbolic work in the exponential expression xy, for instance, is not that ‘x’ and‘y’ have an index standing between them but that they are spatially related in acertain way. In cases where there is an index, of course, mathematicians adoptthe convention of using the index to refer to the function, and for just thatreason they find cases where there is no index, such as that of the exponentialfunction, awkward. They are reduced to the device of setting exp(x,y) = xy, sothat they have an index ‘exp’ which they can then use to refer to the function.Wittgenstein’s conception is that the form of a proposition is the symbol-

izing relationship between the names in the proposition which makes it thecase that the proposition says what it does. The point of this conception is ina sense the mirror image of the point of conceiving of the world as made upof facts, not complexes. In order to make judgments about the world, whatwe must perceive are facts, not complexes; and the symbols that express thosejudgments, likewise, must be facts, not complexes.

!".$!".$!".$ The unity of the proposition

This is Wittgenstein’s response to the problem of the unity of the proposi-tion. Russell’s theory had been that a proposition is a sort of complex entitycontaining (at least in some cases) the things it is about. But this conceptionfaced a difficulty when it came to explaining what it is that makes it the sort ofcomplex that can be true or false. The elements of a complex do not becomea proposition on their own, so we are naturally tempted to posit somethingelse—propositionally expressive glue, so to speak—which combines the ele-ments in the complex in such a way as to make them expressive. But this isonly to postpone the problem, not to solve it, since we are now owed a cor-responding explanation of the expressiveness of a new proposition, namelythe one which says that the original complex is expressive. This is knownas ‘Bradley’s regress’, and versions of it dogged Russell’s various attempts toexplain the expressive unity of the proposition.Frege, on the other hand, avoided the difficulty of Bradley’s regress by sup-

posing that what makes a proposition expressive is the unsaturatedness of thepredicate or relation symbol it contains. He analysed the sentence ‘aRb’, forexample, as formed by substituting the names ‘a’ and ‘b’ into the argumentplaces in the expression ‘xRy’. Writing the expression for a relation as ‘xRy’rather than just ‘R’ is a way of reminding us that it is unsaturated, requiringthe addition of two names if it is to form a complete expression (a sentence).

The unity of the proposition ))%

What the expression ‘xRy’ represents, therefore, is a certain sort of function—a function which with the names ‘a’ and ‘b’ as arguments yields the sentence‘aRb’ as its value. Frege did not need to posit a further element, the ‘copula’of traditional logic, gluing ‘a’, ‘b’, and ‘R’ together to form ‘aRb’, because therelational expression ‘xRy’ already contains, through its unsaturatedness, therequisite glue.But how does this explain the expressiveness of ‘aRb’? Frege’s view was that

the unsaturated relational expression ‘xRy’ has corresponding to it somethingwhich, although not an object, is nevertheless objective: he called it a relation.The relation possesses an unsaturatedness analogous to that of the relationsymbol. So if it is determined what relation ‘xRy’ expresses and what objects‘a’ and ‘b’ refer to, no further resources are needed to determine the thoughtexpressed by the sentence ‘aRb’.This cannot yet amount to an explanation of the unity of the proposition,

however, because unsaturatedness is a feature of function-signs such as ‘x+ y’just as much as of relational expressions such as ‘xRy’. Wittgenstein’s concep-tion of propositions as facts seeks to capture what is right about Frege’s con-ception of unsaturatedness, while also explaining Frege’s other insight, blurredby Frege’s faulty assimilation of sentences to names, as to the essentially ar-ticulate nature of propositions. Wittgenstein’s idea is not only that the verbin a sentence has an essentially different nature from the subject, but that thisdifferent nature is such that it on its own explains the unity of the proposition.The two kinds of components of a proposition are fundamentally different, asFrege’s two kinds—saturated and unsaturated expressions—had been. Theproposition ‘aRb’, for example, contains three components. Two of them arethe names ‘a’ and ‘b’. The third component is not the index ‘R’ but the formof the fact which constitutes the proposition. ‘There is no thing which is theform of a proposition.’19 In Fregean terms, the unsaturatedness of what ‘x ishuman’ expresses suffices to ensure that when it is combined with ‘Socrates’,what results is a proposition: no glue is needed. But the form of the proposi-tion is not a kind of function, because what results when names are insertedinto its argument-places is not a complex name, as it would be in the case of afunction, but a symbolizing fact.Russell himself was certainly aware of the problem of propositional unity

which Wittgenstein was addressing. He put the point succinctly in the noteshe made in October 1912 for a projected article, ‘What is logic?’

In a complex, there must be something, which we may call the form, which is not a con-stituent, but the way the constituents are put together. If we made this a constituent,it would have to be somehow related to the other constituents, and the way in whichit was related would really be the form; hence an endless regress. Thus the form is nota constituent.20

19B48. 20CP, VI, 55.

))" Forms

If the form is not a constituent, what is it? To say, as Russell had done, thatit is a component of the proposition but not a constituent, is so far only tolabel the problem, not to solve it. But if the idea that a proposition is nota complex but a fact is Wittgenstein’s solution to the problem, Russell didnot immediately recognize it as such. He soon abandoned ‘What is logic?’,telling Ottoline that he would leave this sort of thing to Wittgenstein; andin the Theory of Knowledge manuscript, as we have noted already, he indicatedthat he was vaguely aware of a distinction between complex and fact, butevidently did not think it terribly important. Presumably it took the Notes onLogic to make him recognize that the distinction might play a role in solvingBradley’s problem. ‘Chiefly through the work of an Austrian pupil of mine,’he told Bradley, ‘I seem now to see answers about unities; but the subject is sodifficult and fundamental that I still hesitate.’21

!".%!".%!".% The symbolic turn againThe idea that we can infer features of the structure of reality from the structureof the symbols we use to represent it is what I have been calling the symbolicturn. From this we obtain the idea that the structure of an elementary propo-sition corresponds precisely to the structure of what it expresses.

Every proposition which says something indefinable about a thing is a subject–predi-cate proposition; every proposition which says something indefinable about two thingsexpresses a dual relation between these things, and so on. Thus every propositionwhich contains only one name and one indefinable form is a subject-predicate propo-sition, and so on. An indefinable simple symbol can only be a name, and therefore wecan know, by the symbol of an atomic proposition, whether it is a subject-predicateproposition.22

Notice, though, that Wittgenstein claims this precise correspondence onlyfor the case of elementary propositions (propositions which ‘say somethingindefinable’). The reason is presumably that his conception of the relationshipbetween symbolism and the world allowed him to infer from the simplicity ofa name to the simplicity of the object it refers to, but not yet in the oppositedirection. If, for instance, there is just one thing on the table, ‘the thing on thetable’ is a complex name of something simple.For a similar reason, although on this conception whatever a simple name

refers to is an object, we need a further argument if we are to hold also thatno name, simple or complex, can refer to something unsaturated. For Witt-genstein, though, this is not something to be argued for so much as a guidingphilosophical principle: it relies on the harmony between the symbolism andwhat it expresses that I have called his symbolic turn. In this instance, though,

2130 Jan. 1914 (RA1.710.047583H). 22B77.

The symbolic turn again ))&

it is also something that Frege maintained: on his view, whatever refers tosomething unsaturated is itself unsaturated. Expressed in Russell’s language,this amounts to saying that noun and verb are intrinsically different gram-matical categories. The unsaturated expression ‘x is mortal’ and the saturatedexpression ‘mortality’ cannot, contrary to what Russell had claimed, refer tothe same thing. When Wittgenstein claims that there is ‘no name which is thename of a form’,23 that is just the analogue in his terminology of Frege’s claimthat no saturated expression can refer to something unsaturated.Wittgenstein’s claim that ‘no name is the name of a form’ is thus related

to Frege’s discussion of the concept horse. The phrase ‘the concept horse’ issaturated and therefore what it refers to is saturated. But concepts are unsatu-rated. The conclusion Frege drew was, notoriously, that despite appearanceswhat the phrase ‘the concept horse’ refers to is an object, not a concept. Fregewas not as troubled by this conclusion as he might have been. ‘Here,’ he wascontent to say, ‘we are confronted with an awkwardness of language’24 whichputs ‘a quite peculiar obstacle in the way of understanding’.By a kind of necessity of language, my expressions, taken literally, sometimes miss mythought; I mention an object, when what I intend is a concept. I fully realize that insuch cases I was relying upon a reader who would be ready to meet me halfway—whodoes not begrudge a pinch of salt.25

Wittgenstein—plainly a man to begrudge a pinch of philosophical salt when-ever he could—was not the sort of reader Frege could rely on. In the latterparts of the Birmingham Notes we see him gradually enlarging the range ofexpressions which he recognizes as missing their targets in this way. In theTractatus, of course, this is expanded into a central theme, but in the Notes thepoint is made piecemeal. What is lacking at this stage, that is to say, is any signthat he had yet begun to think through in any generality the consequences ofthe idea that there are attempts at expressing things which inevitably misfire.

23B48. 24‘Über Begriff und Gegenstand’, 196. 25Ibid. 204.

Chapter !#!#!#

Russell’s theory of judgment

In June 1913 Wittgenstein’s objection to Russell’s multiple relation theory ofjudgment led Russell to give up writing his book on Theory of knowledge. It isone of the most famous incidents in their interaction, and has sometimes beenpresented as a turning point in their relationship, when the roles of masterand pupil were reversed. The truth, perhaps, is more complicated. Russell’sattitude to Wittgenstein’s views had already become one of deference somemonths earlier: in February 1913, for instance, Pinsent noted that Russell‘acquiesced in what he said without a murmur’;1 and as early as June 1912Russell told Ottoline, ‘I feel he will do the work I should do, and do it better.He starts fresh at a point which I only reached when my intellectual spring wasnearly exhausted.’2 If one insists on identifying a moment when the son’s bat-tle to wrench psychological control from the father was won, it was probablynot in June 1913 but the previous October: for that was when Russell aban-doned his project of writing an article with the title ‘What is logic?’, tellingOttoline that Wittgenstein’s criticisms disturbed him profoundly3 and that hewould in future leave this sort of thing to Wittgenstein.4

Nor was Wittgenstein’s objection responsible on its own for Russell’s turnaway from philosophy: that, as we shall see later, was brought about by anaccumulation of factors, not one catalytic incident. But the objection haspiqued the interest of philosophers for reasons that are not simply biograph-ical. Offering an exegesis of Wittgenstein’s objection seems to have becomea sort of rite of passage for scholars of early analytic philosophy.5 Perhapsthis is because the clues that are tantalizingly scattered through the surviv-ing documents have been thought to stop just short of telling us quite whatWittgenstein’s objection was.

!#.!!#.!!#.! The original multiple relation theoryWe have seen that when he wrote the Principles Russell thought of proposi-tions as complexes which actually contain parts of the physical world. ‘A14 Feb. 1913. 21 June 1912. 313 Oct. 1912. 414 Oct. 1912. 5E.g. Griffin, ‘Wittgenstein’scriticism of Russell’s theory of judgment’; Wahl, ‘Bertrand Russell’s theory of knowledge’; Lan-dini, ‘A new interpretation of Russell’s multiple-relation theory of judgment’; Weiss, ‘On thedemise of Russell’s multiple relations theory of judgement’; Stevens, ‘Re-examining Russell’sparalysis’.

The original multiple relation theory ))(

proposition, unless it happens to be linguistic, does not itself contain words:it contains the entities indicated by words.’6 So the snow-covered peak itselfis a constituent of the proposition that Mont Blanc is the tallest mountain inEurope. In a case such as this, in which the proposition is true, it is hard tosee any difference between the proposition, as Russell conceived of it, and thefact. We have, in other words, an identity theory of truth according to whicha true proposition just is the fact that makes it true. But in that case, whatabout false propositions? The proposition that Charles I died in bed cannotbe identified with the fact of his doing so, because there is no such fact. So wemight try instead to posit a complex made up from Charles I, dying, and hisbed: this complex—let us call it a ‘fiction’—is like a fact except that it happensnot to exist. Truth and falsity are thus properties that objective propositionsmay have: true propositions are facts; false ones are fictions. ‘When a propo-sition happens to be true,’ Russell said in the Principles, ‘it has a further quality,over and above that which it shares with false propositions.’7

But what is this ‘further quality’? Or, to put it another way, what is thedifference between fact and fiction? In the Principles Russell says merely thatthis question ‘belongs no more to the principles of mathematics than to theprinciples of everything else’8 and can therefore be left to the logicians. Intruth, though, the question he was there so patently dodging was a fundamen-tal difficulty for his 1903 conception of propositions, as he gradually came torealize.The key insight came with ‘On denoting’, when Russell realized that the

apparent references to mysterious fictional entities that abound in ordinarylanguage are (sometimes, at least) a by-product of misleading grammaticalforms that can be eliminated on analysis. As a result of this analysis we arefreed from supposing that the King of France somehow has his being in ashadowy realm of non-existent entities. Applied to the present question, thatinsight amounts to this: it might be that the term apparently denoting theproposition vanishes from the correct analysis of statements attributing prop-ositional attitudes. If so, we shall be freed from assuming the subsistence ofobjective falsehoods just as we were freed by the earlier analysis from assumingthe subsistence of a non-existent king of France.Yet if this was the key insight, Russell was slow to realize it. Even after the

revelation that came to him in the summer of 1905, Russell did not straight-away apply the device of incomplete symbols to propositions so as to dismissfalse propositions from his ontology. In 1906 he was willing to concede onlythat ‘the view which denies objective falsehoods is, on the face of it, moreplausible’, and still thought that ‘the difficulties in its way are formidable’.9 Itwas not until 1910 that he rejected objective falsehoods unequivocally, on the

6Principles, §51. 7Ibid. §52. 8Ibid. 9‘On the nature of truth’, 249.

)!* Russell’s theory of judgment

ground that positing them ‘leaves the difference between truth and falsehoodquite inexplicable’.10

From now on Russell tried to analyse a propositional attitude ascription asrelating the bearer of the attitude not to the erstwhile proposition but to itscomponents.

Judgment is not a dual relation of the mind to a single Objective, but a multiple re-lation of the mind to the various other terms with which the judgment is concerned.Thus if I judge that A loves B, that is not a relation of me to ‘A’s love for B’, but arelation between me and A and love and B.11

Of course, Russell is not here proposing an analysis of the proposition ‘I judgethat A loves B’, for any such analysis would only push the problem back onestage. Rather is he proposing to analyse only the fact, if it is a fact, that I makethe judgment; the problem of false propositions does not arise for such a fact,because in the case in which I make no judgment there is no fact and hencenothing to analyse.The central feature of Russell’s theory is that the proposition that is judged

disappears from the analysis of judgment. The ascriptions of other proposi-tional attitudes (belief, disbelief, understanding, etc.) will be analysed similarlybut with different relations in place of judging. It does not follow, though,that propositions disappear entirely: for theoretical purposes it is possible toreconstruct them as long as we accept that they have a rather different formfrom what we had previously supposed. Russell could now offer as a surro-gate for the proposition aRb the fact that (

E

x,!)!(x,a,R,b): what it says is thatsomeone bears some propositional attitude relation to the terms a, R, and b.12

Propositions therefore become subjective in the sense that if no one bears anyattitude to them at all, they do not exist even in this extenuated sense. Russellbelieved that his theory solved the problem of false propositions: it explainedhow I can judge something even though what I judge is false, because in judg-ing that aRb I do not bear any relation to the proposition (which has a whollydifferent structure) but only to its constituents a, R, and b.

!#."!#."!#." A problem for the original theoryDuring the early summer of 1913 Russell was writing a manuscript on the The-ory of Knowledge in which the multiple relation theory of judgment was to play acentral role. Epistemology had for some time held a central place in Russell’sconception of philosophy. ‘There is one great question’, he had told Otto-line in 1911. ‘Can human beings know anything, and if so, what and how?’13

Nonetheless, the very fact that he was writing about epistemology at quite thispoint probably owes something to Wittgenstein’s presence at his shoulder. He

10CP, VI, 119. 11CP, VI, 122. 12Cf. ToK, 115. 1313 Dec. 1911.

A problem for the original theory )!)

had recently put aside two incomplete projects, the work on matter which wediscussed in chapter 3 and the essay on the nature of logic which he had de-cided to leave to Wittgenstein. He even delayed telling Wittgenstein aboutthe epistemology book until he was eighty pages into writing it,14 apparentlybecause he was afraid of his criticisms. When he did eventually tell Wittgen-stein, his fear proved well founded. Shortly before he got to the part of thebook which was to deal with the multiple relation theory, Wittgenstein offeredhim an objection to it.

Wittgenstein came to see me last night with a refutation of the theory of judgmentwhich I used to hold. He was right, but I think the correction required is not very seri-ous. I shall have to make up my mind within a week, as I shall soon reach judgment.15

And when Russell came to the section on judgment about five days later, hedid indeed reject the 1910 theory which I described in the last section (thetheory which he ‘used to hold’). The reason he gave was as follows.

In an actual complex, the general form is not presupposed; but when we are concernedwith a proposition which may be false, and where, therefore, the actual complex is notgiven, we have only, as it were, the ‘idea’ or ‘suggestion’ of the terms being unitedin such a complex; and this, evidently, requires that the general form of the merelysupposed complex should be given. More simply, in order to understand ‘A and B aresimilar’, we must know what is supposed to be done with A and B and similarity, i.e.what it is for two terms to have a relation; that is, we must understand the form of thecomplex which must exist if the proposition is true.16

It is natural to wonder whether Russell is here reproducing Wittgenstein’sobjection. The language he uses in reporting it is admittedly not Wittgenstein-ian: he expresses the point as an epistemological one concerning our under-standing of the complex rather than, as Wittgenstein would surely have done,as a grammatical point about the words with which the judgment is expressed.Perhaps, though, Russell’s mode of expression owes something to his presentpurpose, which is to direct us towards his new theory (which, as we shall seeshortly, Wittgenstein liked no better) rather than to convince us of the error ofhis old one.It is true, too, that Russell does not explicitly attribute the objection to

Wittgenstein, but this need not in itself be especially significant. In some of hislater work Russell was punctilious about crediting Wittgenstein whenever hereported his ideas in print, but only a few days after writing the passage justquoted we find him commenting to Ottoline on ‘the difficulty of not stealinghis ideas—there is really more merit in raising a good problem than in solv-ing it’.17 It is tempting to hear in this remark the voice of bad conscience, arecognition by Russell that he had just been guilty of stealing something andshould therefore resolve to avoid that danger in the future. The most striking

14To OM, 14 May 1913. 15BR to OM, 21 May 1913. 16ToK, 116. 171 June 1913.

)!! Russell’s theory of judgment

feature of Russell’s account, though, is its curiously hesitant tone. Having putthe point several times in different ways, Russell concludes lamely, ‘I do notknow how to make this point more evident, and I must therefore leave it to thereader’s inspection, in hopes that he will arrive at the same conclusion.’18 Thisinclines me to think that the objection Russell raises against his 1910 theory isat base Wittgenstein’s.Russell represents the objection as being one that applies only in the case

where the judgment is false—where, in other words, there is no complex con-sisting of the relata combined as the proposition says—but the problem isrecognizably derived from the one that in chapter 8 we saw Wittgenstein us-ing against the theory that aRb should be analysed as #2(a,R,b). In order tounderstand ‘A and B are similar’, we must know what is supposed to be donewith A, B, and similarity. We must know, Russell might have said, what arethe types of A, B, and similarity. In the case where A and B are indeed similar,Russell supposed that we can read these types off from the fact of their simi-larity when we perceive the complex; what he now conceded was that whenthere is no such fact, we cannot.Wittgenstein’s point was thus that the types of A, B, and similarity ought

somehow to be deducible from the fact that I judge that A and B are similar,whether or not they are similar. As he put it in the Notes on Logic, ‘Every righttheory of judgment must make it impossible for me to judge that this table pen-holders the book. Russell’s theory does not satisfy this requirement.’19 Rus-sell’s theory did not satisfy the requirement because in it all the componentsof what is judged, including the verb, occurred as terms. Russell therefore didnot have the resources to distinguish, in what is judged, between terms (suchas ‘similarity’) which derive from verbs and those (such as ‘penholders’) whichdo not.

!#.#!#.#!#.# Russell’s revised theoryWhat Russell did in the Theory of Knowledge manuscript was to amend his the-ory so that the judging relation has a further argument place in which theform of what is judged occurs. If A judges that Socrates is mortal, that is tobe analysed as J(A, #1,Socrates,mortality), where #1 is the subject-predicateform. If A judges that John loves Mary, that judgment is to be analysed asJ(A, #2, John,Mary, loving), where #2 is the form of relational judgments. Andso on.Now if the form of the proposition was to occur as a term in the judging

relation, Russell needed to make a specific proposal about what entity that

18ToK, 116. 19B33.

Russell’s revised theory )!#

is. The proposal he made was in essence the one that Wittgenstein had beenusing in January 1913. Wittgenstein had then taken the form of a subject-predicate proposition ‘Socrates is mortal’ to be ‘something is mortal’. SinceRussell was treating mortality as a second constituent, the same device ledhim to take the form as ‘someone has some property’. And in the same way,the form of a relational proposition such as ‘John loves Mary’ would now be‘someone bears some relation to someone’.I have presented Russell’s 1913 theory as a response to the objection Witt-

genstein presents in the Birmingham Notes. Since we cannot be sure whenWittgenstein formulated that objection, the question arises whether we can besure that Wittgenstein intended it to apply primarily to the 1910 theory ratherthan to the 1913 modification. One reason to think he did is that he includeda variant of the same objection in the Tractatus. ‘The correct explanation ofthe form of the proposition “A judges p” must show that it is impossible tojudge a nonsense. (Russell’s theory does not satisfy this condition.)’20 In thisformulation, at least, Wittgenstein must have intended the objection to applyprimarily to Russell’s 1910 theory. For Russell had given up his 1913 theorywithout publishing it, and Wittgenstein knew this. (Russell famously told himthat the objection had paralysed him.) It would be a little odd, even by hissomewhat idiosyncratic standards, if Wittgenstein had chosen to include inthe Tractatus an objection to a theory of Russell’s that no one except him hadseen and which at the time he was composing the Tractatus he believed Rus-sell to have long ago abandoned. The lectures on logic which Russell gave atHarvard in 1914 confirm this. Russell there presented his ‘old theory of judg-ment’, analysing the fact that S judges that x has R to y as a relation betweenS , x, R, and y, before commenting, in what is plainly an allusion to Wittgen-stein’s criticism, that ‘if R was a thing you could substitute another thing (z) forit, and if you do, the j[udgment] is meaningless’.21

If Wittgenstein’s criticism was directed at the version of the theory in whichthe form of what is judged does not occur as a term, it does not follow, ofcourse, that Russell’s original purpose in adding the form was to circumventthis difficulty. It may be, for instance, that Russell had already decided toinclude the form for reasons connected with the so-called ‘direction problem’and his correspondence with Stout. Nonetheless, he evidently thought at firstthat including the form would meet Wittgenstein’s objection. And it is fairlyeasy to see why he might have thought this: he hoped that the form of what isjudged could have encoded in it the typing information that is needed to ruleout nonsense; the form of all relational propositions, for instance, would in ef-fect stipulate that the middle term must be a verbal noun (such as ‘similarity’),not an ordinary noun (such as ‘penholders’).

205.5422. 21Eliot’s notes, 9 Apr. 1914.

)!$ Russell’s theory of judgment

!#.$!#.$!#.$ Wittgenstein’s further objectionA couple of days after Russell had written the section of his manuscript ex-pounding the revised theory, the following incident took place.

Wittgenstein came to see me—we were both cross from the heat—I showed him acrucial part of what I have been writing. He said it was all wrong, not realizing thedifficulties—that he had tried my view and knew it wouldn’t work. I couldn’t under-stand his objection—in fact he was very inarticulate—but I feel in my bones that hemust be right, and that he has seen something I have missed. If I could see it too Ishouldn’t mind, but as it is, it is worrying, and has rather destroyed the pleasure in mywriting.22

There can be little doubt that what Russell showed to Wittgenstein was some-how related to the new version of the theory of judgment. But what was wrongnow?The first issue that arises in answering this is whether it was the general pro-

posal of treating the form as a term in the judging relation that Wittgensteinobjected to or Russell’s specific proposal as to the identity of that form. Wehave seen already that the specific proposal was indeed one that Wittgensteinhad tried and rejected, because it would have the absurd consequence that wecan put a negation sign before a name. And perhaps some of his inarticulatecomments were indeed directed at explaining this difficulty. However, I thinkwe can rule out the idea that this was the whole of Wittgenstein’s objection.If it had been, after all, he could simply have suggested to Russell that he re-place his faulty conception of the form of the proposition with Wittgenstein’sown. It seems much more likely, therefore, that Wittgenstein’s objection wasto any proposal according to which the form of the proposition, however it isconceived, occurs as a term in the judgment relation.If that is right, then the objection will have been in essence the one he had

already offered, namely that a theory of judgment should make it impossibleto judge that the table penholders the book. For Wittgenstein had indeed triedand rejected something very like the view Russell now proposed.

If I analyse the proposition Socrates is mortal into Socrates, Mortality and (

E

x,y)#1(x,y)I want a theory of types to tell me that ‘mortality is Socrates’ is nonsensical, because ifI treat ‘Mortality’ as a proper name (as I did) there is nothing to prevent me to makethe substitution the wrong way round.23

Wittgenstein thought, that is to say, that the presence of the form as a furtherterm in the judging relation did not meet his objection: it still did not make itimpossible to judge nonsense unless we simply add a stipulation according towhich ‘mortality’ is of a different type from ‘Socrates’.Russell ends the chapter of the manuscript on the understanding of propo-

sitions by drawing a map showing various connections between the elements22BR to OM, 27 May 1913. 23To BR, 16 Jan. 1913.

Acquaintance )!%

of the judging relation. Perhaps it is of significance, therefore, that when Rus-sell tried some years later to recollect Wittgenstein’s objection, he expressed itby saying that

you cannot make what I should call a map-in-space of a belief. . . You cannot get inspace any occurrence which is logically of the same form as belief. . . The discovery ofthis fact is due to Mr Wittgenstein.24

The point is that the map at once shows the futility of supposing that we canadd the form of what is judged as a term in the relation. By drawing a map—aspatial complex of a certain sort—Russell showed that he conceived of the factthat I judge something as having the same structure as this spatial complex.But the entities that are related in such a complex are things, not forms. Anyattempt to include the form as a term in such a relation will inevitably fail: bymaking it a term of the relation we turn it into a name and therefore makeit refer, if it refers at all, not to the form of the putative proposition but tosomething else. Judgment cannot be what Wittgenstein calls ‘a relation inthe ordinary sense’, because a relation in the ordinary sense has objects as itsconstituents, whereas one of the constituents of a judgment is the verb of theproposition judged, which is not an object.

!#.%!#.%!#.% AcquaintanceIt is futile, then, to include the form as a term in the judgment if judging isa relation in the ordinary sense, i.e. one with the same structure as a map.But for all we have said so far, the possibility remains open that judging maybe a higher-order relation in which the form does genuinely occur as a form.Wittgenstein needed a further argument to eliminate this possibility.

There is no thing which is the form of a proposition, and no name which is the nameof a form. Accordingly we can also not say that a relation which in certain cases holdsbetween things holds sometimes between forms and things. This goes against Russell’stheory of judgment.25

The phrasing is at first sight puzzling, since in Russell’s theory judgment isnot a relation that sometimes holds between things and sometimes betweenforms and things. Which relation is he referring to, then? The most likelycandidate is the relation of acquaintance. Russell held that if I am to judgesomething, I must be acquainted with all the terms of the judging relation. Soin the case where I judge that Socrates is mortal, I must be acquainted withSocrates, with mortality, and with the subject-predicate form. If the latter isto occur in this relation as a form, it cannot be the same sort of acquaintance

24CP, VIII, 198–9. 25B48.

)!" Russell’s theory of judgment

as in the other cases. So Russell would have to accept that acquaintance is nota univocal notion.Perhaps this ‘goes against’ Russell’s theory, but it does not yet refute it. Why

should Russell not simply have granted that our acquaintance with forms is adifferent sort of relation from our acquaintance with objects? The difficultywith this is one that Russell himself pointed out, in a section of his manuscriptwhich he wrote only a couple of days after the incident reported in the lastsection.

How can an object be at once simple and a ‘fact’, in the sense in which a ‘fact’ isopposed to a simple particular and is the sort of object whose reality makes a proposi-tion true? Why, if pure forms are simple, is it so obviously inappropriate to give themsimple proper names, such as John and Peter?26

These are good questions, and Russell could not answer them. He thereforeresorted to a familiar kind of bluster. ‘These logical questions can no doubtbe answered,’ he wrote, ‘but for our purposes the epistemological questionsare more pressing.’ A few days later Russell told Ottoline that although ‘in alllikelihood they are just’, Wittgenstein’s criticisms ‘have to do with problemsI want to leave to him’.27 If he meant by this that the problems he had inmind belonged to logic rather than to epistemology, then, as we have justseen, the problem of acquaintance with forms was his own prime candidate.And if he used this as an excuse to carry on advancing a view whose difficultieshad already been pointed out to him, then perhaps his behaviour merited thedamning assessment that he himself offered a few weeks later: ‘It is the firsttime in my life that I have failed in honesty over work.’28

!#.&!#.&!#.& Another formulationFor the time being Russell soldiered on, even though Wittgenstein’s criticismshad ‘destroyed the pleasure’29 in his writing and made the work ‘a task ratherthan a joy’.30 A couple of weeks later, however, Wittgenstein finally providedRussell with the succinct formulation of his objection that had hitherto eludedhim.

I can now express my objection to your theory of judgement exactly: I believe it isobvious that, from the proposition ‘A judges that (say) a is in a relation R to b’, ifcorrectly analysed, the proposition ‘aRb v "aRb’ must follow directly without the use ofany other premiss. This condition is not fulfilled by your theory.31

To some extent, the difference in expression between this and the formula-tion in the Tractatus—that the correct explanation ‘must show that it is impos-sible to judge a nonsense’—is cosmetic: Wittgenstein is here choosing to put26ToK, 130. 2728 May 1913. 28To OM, 20 June 1913. 29To OM, 27 May 1913. 30To OM,1 June 1913. 31[June 1913] (CL, no. 14).

Another formulation )!&

the point in a somewhat more formal manner that gestures towards Russell’smodes of expression. In the Principles it is taken to be a characterization ofwhat it is for p to be a proposition that p ! p should be true, but in Principia vand ! are taken as primitive, so that p ! p is an abbreviation for p v !p. Torequire that from ‘A judges that p’ should follow p v !p is therefore to requirethat p should be a proposition, i.e. that it should not be nonsense.Wittgenstein’s reformulation of the point is curious nonetheless. If he was

simply making again the point that a theory of judgment must show that it isimpossible to judge nonsense, this is a very odd way to make it. It is plausible,therefore, that Wittgenstein is here responding not to the theory we have beenconsidering, but rather to some new variant Russell had proposed, probablyduring their argument a fortnight earlier, as a way out of his predicament.The explanationmay be contained in a set of notes32 Russell probably made

around this time. (The first page of these notes is written on the back of arejected sheet from the main manuscript which Russell wrote around the timeof Wittgenstein’s visit on 26 May, so the notes themselves may well date fromvery shortly after that.) In the notes Russell explores a variant multiple relationtheory which is evidently an attempt to get round Wittgenstein’s objection.Judgment, he says, will ‘still be a multiple relation, but its terms will not bethe same as in my old theory’.33 Russell suggests that what is involved injudgment is not the positive fact that aRb or the negative fact that !aRb, butrather what he calls the ‘neutral fact’ which obtains when either of these factsobtains. It is surely very natural to represent this neutral fact as aRb v !aRb.Russell conjectures that the neutral fact is a constituent of whichever of thepositive and the negative fact actually occurs. ‘There will only be a neutralfact,’ he observes, ‘when the objects are of the right types.’34 That is to say, theneutral fact that aRb v!aRb is equivalent to saying that aRbmakes sense. So ifWittgenstein’s requirement that it should be impossible to judge nonsense is tobe directed against this new idea of Russell’s, it is natural to reexpress it as therequirement that whenever A judges that aRb, the neutral fact, i.e. aRbv!aRb,must follow. Even in Russell’s notes, though, we see him recognizing that toregard judgment as a relation to neutral facts ‘entails the old difficulties; itseems not intimate enough. . . It can’t be quite right.’35 Wittgenstein’s letterwas an attempt to explain succinctly why.To explain the point more fully, let us write #n(a0, . . . , an) for the result of

assembling a0, . . . , an in accordance with the form #n (so that, for instance,#2(a,R,b) is just aRb). Wittgenstein insists that from J(A, #2,a,R,b) it shouldfollow that aRb is a proposition, i.e. that aRb v !aRb; or, in greater generality,from

J(A, #n, a0, . . . , an),

32CP, VIII, App. B.1. 33CP, VII, 197. 34CP, VII, 199. 35Ibid.

)!' Russell’s theory of judgment

it should follow that #n(a0, . . . , an) is a proposition, i.e. that

#n(a0, . . . , an) v !#n(a0, . . . , an).

Russell—the Russell of Principia at any rate—had a simple method for dealingwith technical lacunae of this sort: if in doubt, just add another primitiveproposition. Thus if in the present case we add the axiom

J(A, #n, a0, . . . , an) ! (#n(a0, . . . , an) v !#n(a0, . . . , an)),

the required conclusion will follow at once. But this could hardly have satisfiedWittgenstein, since adding this axiom does not show, but only asserts, that it isimpossible to judge nonsense. Including the form in the judging relation as anextra term #n does not help one jot with this.Perhaps that is why Wittgenstein places a stronger demand on the theory

in his letter than in the Notes. The conclusion that we cannot judge nonsense,he now insisted, ‘must follow directly without the use of any other premiss’. In thenotebook entry he included in the Birmingham Notes he had required onlythat the theory should ‘make it impossible’ to judge nonsense. What he wasnow demanding was that the theory should, as he put it in the Tractatus, show(and not merely assert) that it is impossible to judge nonsense.All told, then, the exact formulation of his objection that Wittgenstein of-

fered in his famous letter is probably not so much his central argument againstRussell as a sort of intermediate skirmish, a response to a somewhat desper-ate rearguard action by Russell to shore up the defences of his theory by hisfavoured manoeuvre of adding a primitive proposition to it.

!#.'!#.'!#.' The fate of the multiple relation theoryWhen he received the letter containing the succinct expression of Wittgen-stein’s objection, Russell had in fact already stopped work on the book, butonly temporarily: the day after receiving it, he still said that he intended tostart again in September. If Wittgenstein wanted to get Russell to give upcompletely, he would have to make one last attack. The principal purposeof Wittgenstein’s letter was to make arrangements for Russell to have lunchwith his mother, who was visiting London at the time. Perhaps Wittgensteinused this lunch as the opportunity to make his attack; perhaps this time hesucceeded. At any rate it is noticeable that before the lunch Russell speaksonly of feeling that Wittgenstein must be right but not being able to see why,whereas the following day we find him writing to Ottoline of a recent andprofound change in his understanding of the effects of Wittgenstein’s criti-cisms of his work. ‘It was very difficult to be honest about it,’ he told her, ‘as itmakes a large part of the book I meant to write impossible for years to come

Other accounts )!(

probably.’36 Now, at last, Russell acknowledged the strength of Wittgenstein’sobjection, which until then he had felt rather than understood. ‘Yesterday,’he rather over-dramatically added, ‘I felt ready for suicide.’By 1916 Russell had become convinced that Wittgenstein’s criticism had

been ‘an incident of first rate importance in my life’.37 At the time, though,he seems to have bounced back quickly enough. By the beginning of 1914,indeed, he could tell Ottoline: ‘I don’t think my brain has been more clear,or more willing to do all I ask of it, any time since 1900.’38 And just in caseshe felt any temptation to doubt his own assessment of his intellectual stature,Russell was in the happy position of having independent evidence he couldadduce for it.

Whitehead . . . has been telling me that my mind has improved greatly in the last twoor three years, that in fact it has risen to an altogether higher class. He says . . . ifmy present work develops as it promises, it will put me among the few great philoso-phers.39

‘Although this is so agreeable,’ Russell continued modestly, ‘I think it is true.’

!#.(!#.(!#.( Other accountsAs I mentioned at the beginning of this chapter, there have been many at-tempts by commentators to explain what Wittgenstein’s objection amountedto—too many for me to give them all detailed attention here. One which isparticularly worth noting, though, is Griffin’s.40 Wittgenstein’s objection turnsout, he says, ‘to be an argument of the most extraordinary subtlety. . . WhatWittgenstein was able to show was that Russell’s multiple relation theory ofjudgment was inconsistent with the theory of types, the lynchpin of Principia.’41

The difficulty, according to Griffin, is that to ensure that an elementary judg-ment such as aRb makes sense, Russell would have to appeal to further judg-ments (such as the judgment that a and b are suitable arguments for a first-order relation) which according to the theory of types are of higher order thanthe original judgment whose sensicality he was trying to establish.Griffin’s account depends on quite detailed features of Russell’s theory of

types. However, it is just this feature that makes it unlikely to be correct.I have already noted that it is striking how little engagement Wittgenstein’swritings from this period display with the formal parts of Principia. Wittgen-stein did address some of the particularities of the formal system of Principia,but some of the comments he made about them are inept in the extreme. Itseems intrinsically more likely that Wittgenstein’s objection would have had aphilosophical rather than a narrowly technical basis. When Griffin says that

3620 June 1913. 37To OM, 4 Mar. 1916. 388 Jan. 1914. 39To OM, 23 Feb. 1914. 40‘Russell’smultiple relation theory of judgment’. 41The Private Years, 461.

)#* Russell’s theory of judgment

the argument he attributes to Wittgenstein is ‘of the most extraordinary sub-tlety’, he could just as well have said ‘very elaborate’. And that tells againstit too. One of the features of Wittgenstein’s arguments is that they are rarelyelaborate; most of them, indeed, are simple—infuriatingly so, when one hasstruggled to work out what they are. Perhaps that is why I find it difficult toregard Griffin’s argument, ingenious though it is, as truly Wittgensteinian.If the account I have offered here is correct, on the other hand, Wittgen-

stein’s criticism was indeed fundamentally simple. If I say ‘A judges that p’, Ido not, of course, myself judge that p, but I do, in the course of saying it, haveto express what it is that A is judging (namely that p). If I am to do that, I mustuse the verb of p as a verb. Russell’s theory, in any of its variants, forces theverb of p to occur as a term in the judging relation, and in such a position itcannot function as a verb.Even if I am right about what Wittgenstein’s objection was, that does still

leave open the possibility that Griffin might be right about why the objec-tion paralysed Russell. The evidence does not support even this concession,however. The notes that have survived of Russell’s lectures on epistemologyat Harvard the following year quote him as expressing just what I have rep-resented as Wittgenstein’s objection. If I believe that Jones hates Smith, theoccurrence of ‘hate’ in this fact is, he says, ‘not substantive’. Although it is asingle fact, nonetheless it ‘contains two verbs’ and ‘both must occur as verbs’.This, he concludes, ‘constitutes [the] oddity of propositional thought’. Andwhen Russell lectured on the same matter four years later, his recollection ofWittgenstein’s criticism had not changed. ‘When A believes that B loves C,’he said,

you have to have a verb in the place where ‘loves’ occurs. You cannot put a substantivein its place. Therefore it is clear that the subordinate verb (i.e. the verb other thanbelieving) is functioning as a verb, and seems to be relating two terms, but as a matterof fact does not when a judgment happens to be false. That is what constitutes thepuzzle about the nature of belief.42

What first led Wittgenstein to realize that Russell’s theory was susceptibleto this difficulty? Perhaps it is not wholly coincidental that Moore spent sometime criticizing Russell’s theory in his Cambridge lectures during Wittgen-stein’s first year there. While granting that a false judgment could not be arelation of the person making the judgment to a single non-existent entity,Moore complained that it could not be a relation to the proposition’s compo-nents either.

It is totally impossible that belief should consist in having a relation to something whichsimply isn’t; but it seems equally impossible that it should consist solely in havingrelation to something which indubitably is.

42CP, VIII, 198.

Other accounts )#)

Moore makes it clear that he objects to Russell’s multiple relation theory asmuch as to the theory it replaced. ‘To both forms of solution which [Russell]suggests there is fundamental difficulty, that whole J[udgment] solely consistsin my having relation to object which undoubtedly is.’ What both theoriesfail to explain is that ‘To believe that I am in Cambridge, is same thing as tobelieve that prop[osition] that I am in Cambridge is true, or that there is a factcorresponding to prop[osition].’ In his lecture notes Moore never quite expressesthe objection satisfactorily: he struggles repeatedly, in fact, to bring it to aclear formulation. False belief, he observes at one point, consists ‘somehowin taking for a fact, something which isn’t a fact’. That, surely, is the essenceof Wittgenstein’s objection to Russell’s original multiple relation theory, andhence, as we have seen, of his later objections too.

Chapter !$!$!$

Meaning

Russell was troubled, as we have seen, by the difficulty of explaining falsepropositions. He had begun with an account which identified true proposi-tions with facts, but this left false (or ‘unasserted’) propositions as a puzzlingkind of sham, like facts except for their non-existence. The great advantageof the multiple relation theory was that according to it true judgments havefacts (or, as Russell was just as happy to say, complexes) corresponding tothem, whereas false judgments have nothing at all. Rejecting this theory leftWittgenstein with the problem that had originally motivated it. His elegantanswer was that what it is in the world that makes p true, if it is true, is thesame as what makes !p false; and conversely, what makes p false, if it is false,is just what in that case makes !p true. This allowed him to satisfy the desireto ensure that every proposition, true or false, should have some part of theactual world responsible for its truth or falsity. This part of the world he calledits meaning: the meaning of!p is the same as the meaning of p,1 but they pointat it, as it were, from opposite directions.If propositions are to have meaning at all, the advantage of saying that p

and !p have the same meaning is that it avoids the need to commit ourselvesto the fictional facts that had troubled Russell. But it comes at a price. Wemust give up any pretence of the delightful simplicity of Moore’s original iden-tity theory. Given that what makes p true if it is true is the same as what makes!p true if it is true, it plainly cannot more than half the time be the case thatwhat makes a proposition true is itself. It is something of a misnomer, there-fore, to call the meaning of a proposition its truth-maker: if the proposition istrue, its meaning is indeed what makes it true; but if it is false, its meaning iswhat makes it false. A proposition’s meaning might therefore more accurately(if less elegantly) be called its true-or-false-maker. And although the ‘meaning’is the worldly correlate of the proposition, it is not in any recognizable sensewhat the proposition means. So the word ‘meaning’ is a rather awkward oneto use in this context. Russell at any rate thought so: when he was explain-ing Wittgenstein’s theory, he preferred to call the true-or-false-making entitythe ‘objective reference’ of the proposition.2 Even Wittgenstein himself laterexpressed unease with his own terminology.3

1B37. 2CP, VIII, 297. 3NB, 2 Nov. 1914.

Russell’s lectures on logical atomism )##

!$.!!$.!!$.! Russell’s lectures on logical atomismBefore we discuss the details of Wittgenstein’s notion of meaning, somethingneeds to be said about Russell’s 1918 lectures on logical atomism, in order toexplain why they are an important source for the interpretation of the Notes onLogic. For this we need to continue the narrative some way beyond the pointwhen the Notes were compiled.Russell was due to spend the period from March to June 1914 at Harvard

delivering the Lowell lectures. After Wittgenstein had departed for Norway,therefore, Russell spent the autumn writing these lectures, later published asOur Knowledge of the External World. Although Russell mentioned Wittgensteinin the preface to this book, his influence on it is detectable much more in whatit does not discuss than in what it does. Only in chapter 2, which describesWittgenstein’s conception of atomic facts, and of logic as the study of the formsof facts, does Russell make significant use of his ‘vitally important discoveries’.4

The Lowell Lectures were intended for a general audience, but while he wasat Harvard Russell also gave two more specialized lecture courses, one onepistemology and one on logic (with a graduate student called Costello as histeaching assistant); it was in preparation for the latter course that he went tothe labour of translating and arranging Wittgenstein’s Notes in February 1914.Russell did not publish the logic lectures himself, but much of their content ispreserved for us in the notes of T. S. Eliot, a philosophy student at Harvardwho soon achieved fame for other reasons.During the Autumn of 1913 Russell and Wittgenstein exchanged letters

frequently, but shortly before Russell was due to leave for Harvard there wasa rift and philosophical discussions between them almost ceased. After he re-turned from Harvard, Russell read the notes Wittgenstein dictated to Moore,but he did not understand them and Moore, he claimed,5 could not help him.(Moore indignantly responded6 that he had not helped because Russell hadnot asked.) At this point the war intervened. Straightaway philosophical com-munication between Russell andWittgenstein became impossible (and, after ayear or so, non-philosophical communication too). From then on, moreover,Russell did very little by way of original philosophy. One reason for this wasno doubt the war: it energized him politically and led him to devote muchof his time to campaigning against it. Not only that, but his motivation towork in philosophy deserted him. The reasons for this, as is often the case insuch matters, were complex. The war certainly (and understandably) playedits part in making philosophy seem pointless to him. For example, his reactionto a reminder about a lecture he had agreed to give in Manchester early inthe war was to ask, ‘Are there really people in Manchester interested in phi-losophy still? I am not.’7 A year or so later, though, he felt more inclined toattribute his lack of motivation to Wittgenstein.

4OKEW, vii. 5To LW, 21 June 1919. 6Moore’s diary. 7To Samuel Alexander, 17 Oct. 1914.

)#$ Meaning

I had to produce lectures for America, but I took a metaphysical subject although Iwas and am convinced that all fundamental work in philosophy is logical. My reasonwas that Wittgenstein persuaded me that what wanted doing in logic was too difficultfor me. So there was no really vital satisfaction of my philosophical impulse in thatwork, and philosophy lost its hold on me. That was due to Wittgenstein more than tothe war.8

He now spoke of the war not as drawing him away from philosophy, but ratheras filling the gap his discouragement with it had left by supplying him with the‘new and less difficult ambition’ of political activism. And years later, in part ofhis autobiography believed to have been written around 1931, Russell had yetanother explanation: the effect of writing Principia, he now thought, had beento induce in him a long-term intellectual exhaustion which made it impossiblefor him to work in technical philosophy again.In 1917, however, Russell did return to philosophy after a break of three

years, giving two courses of popular lectures. Each course consisted of eightlectures at weekly intervals: the first course, on the philosophy of mathemat-ics, began in October 1917; the second, on logical atomism, in January 1918.Russell’s prosecution (for making statements ‘likely to prejudice His Majesty’srelations with the United States of America’) came to court while the sec-ond course was still in progress, and shortly afterwards he served a six-monthprison sentence (reduced to five for good behaviour), during which he con-verted the first lecture course into a book, his Introduction to Mathematical Philos-ophy. Meanwhile, the course on logical atomism was published in the form ofverbatim transcripts of the lectures.In the Introduction to Mathematical Philosophy Russell mentions Wittgenstein

only in a footnote9 (attributing to him a recognition of the importance of thenotion of tautology). Elsewhere in the book Wittgenstein’s influence is de-tectable only very sparingly. The logical atomism lectures, on the other hand,are according to Russell ‘very largely concerned with explaining certain ideaswhich I learnt from my friend and former pupil, Ludwig Wittgenstein’, who‘originally supplied many of the theories contained in them’.10 Even a casualinspection of the lectures confirms that Russell should here be taken at hisword. Large parts of them consist simply of exposition, in language suitablefor a non-technical audience, of ideas from the Notes. In some instances thereasons he offers for the claims he makes are plainly Wittgenstein’s. In othershe repeats things Wittgenstein says in the Notes, but seems unable to recol-lect Wittgenstein’s reasons for saying them and quickly changes the subject.Hardly anywhere does he directly contradict Wittgenstein’s views as he thenbelieved them to be.It might be thought odd that Russell agreed to publish the lectures if they

were very largely a report of Wittgenstein’s views. Two obvious reasons sug-8To OM, 4 Mar. 1916. 9p. 205. 10CP, VIII, 160.

Propositions are not names of their meanings )#%

gest themselves. One is that his lectureship at Trinity had not been renewed,and he needed the money. The other is that, as he noted in a preface tothe lectures, he did not at that point know whether Wittgenstein was alive ordead, and therefore could not judge how soon there might be any other op-portunity for Wittgenstein’s ideas to reach the wider audience he thought theydeserved. He had no way of knowing that the Tractatus was by then nearingits final form.Further evidence of the extent to which Russell had by now turned away

from philosophical logic is provided by the first paper he published on hisreturn to philosophy, ‘On propositions: what they are and how they mean’.In this paper, written in February 1919, Russell began by offering what wasin effect a summary of Wittgenstein’s views about propositions in the Noteson Logic, but then he devoted the remainder to considerations of psychology,which he had been reading up in prison. In 1913 he had told Ottoline thathe felt inclined to leave logic to Wittgenstein: here is evidence that this is justwhat he did.For our current purposes, however, Russell’s misfortunes are our fortune.

Because he had not been active in the field for so long, Russell’s views seemtruer to Wittgenstein’s than might otherwise be the case. In the lectures, es-pecially, he was more interested in recollecting what Wittgenstein’s views hadbeen than in developing his own. This is relevant to us here, because there hadbeen, as we have noted, very little philosophical contact between them sinceWittgenstein’s departure for Norway in October 1913. The views of Witt-genstein’s which Russell was recollecting are therefore overwhelmingly thoseof the Notes on Logic. For this reason Russell’s lectures represent a significantadditional source for the interpretation of these notes.

!$."!$."!$." Propositions are not names of their meanings‘Especially false,’ Wittgenstein said, is the statement “propositions are namesof complexes”.’11 Especially false, because it combines two errors, that ofconceiving of the meaning of the proposition as a complex rather than a fact,and that of conceiving of the relationship between the two on the model of therelationship between a name and its meaning.The main target at this point is Russell: we shall focus first on his view

that a proposition names its meaning, and postpone to the next section thequestion of whether the meaning is a complex or a fact. We need at thispoint to distinguish two questions. As we saw in §2.2, Russell had adoptedthe view that what look like names often are in fact disguised definite descrip-tions. Those names which resist this treatment and turn out on analysis tobe genuinely names Russell calls logically proper names. Russell would have

11B4.

)#" Meaning

needed no convincing that propositions are not simple names: if they were, wewould on his view have direct acquaintance with what they refer to, and wouldtherefore know whether they are true or false whenever we understood them.But there remains the question, not settled by this quick argument, whetherpropositions are names in the wider sense that includes disguised descriptions.Wittgenstein’s reason for saying that they are not is neatly expressed by Rus-sell in his 1918 lectures (lectures which, as we have just noted, consist largelyof exposition of Wittgenstein’s views).

It is perfectly evident as soon as you think of it, that a proposition is not a name for afact, from the mere circumstance that there are two propositions corresponding to eachfact. Suppose it is a fact that Socrates is dead. You have two propositions: ‘Socratesis dead’ and ‘Socrates is not dead’. And those two propositions corresponding to thesame fact, there is one fact in the world which makes one true and one false. Thatis not accidental, and illustrates how the relation of proposition to fact is a totallydifferent one from the relation of name to the thing named. For each fact there aretwo propositions, one true and one false, and there is nothing in the nature of thesymbol to show us which is the true one and which is the false one. If there were, youcould ascertain the truth about the world by examining propositions without lookinground you.12

The argument that Russell is expounding here depends on two of Wittgen-stein’s doctrines which we have met already. First there is the doctrine withwhich this chapter began, that the feature of the world that makes a propo-sition true or false is the same as the feature that makes its negation false ortrue. Then there is the doctrine, which we encountered in connection withthe argument for substance, that we can understand a proposition indepen-dent of knowing whether it is true or false. Expressed less epistemically, whatis relevant here is the observation—central to Russell and Moore’s 1898 re-volt against idealism—that when a feature of the world makes a propositiontrue or false, the proposition itself is not altered thereby. It follows from thisthat truth and falsity are not internal to a proposition but are different rela-tionships that may hold between a proposition and the relevant feature of theworld. But if a proposition were a name, there would be only one such rela-tionship, namely that of reference. Therefore propositions are not names ofthose features of the world that make them true or false.According toWittgenstein, then, propositions are distinguished from names

by having two possible relations to their meanings. He called this feature ofpropositions their bipolarity. This is a promising proposal for avoiding Russell’sproblem of fictional complexes, and we shall do what we can to develop thedetails of it in the coming pages. Notice, though, that neither Wittgensteinhimself nor Russell on his behalf offered an argument for it. What recommendsit, apart from whatever virtues the detailed theory will turn out to possess, is

12CP, VIII, 167–8.

Meanings as facts )#&

only the intuitive thought that what makes a proposition true if it is true isthe very same as what makes its negation false. If that thought should lose itsappeal, the proposal will be left without a priori support.

!$.#!$.#!$.# Meanings as facts

Wittgenstein’s idea, we have said, was that with each proposition there is as-sociated some part of the world which is its meaning. But which part? Wediscussed in §11.1 his view of facts as conceptually prior to (because closer tothe possibility of their representation than) complexes. It is to be expected,therefore, that he would conceive of the meaning of the proposition that Cae-sar died as being not the complex consisting of his death but the fact of it. Andso he did. The meaning of a true atomic proposition he called a positive fact.What makes an atomic proposition true is thus that something is the case, notthat something exists. Wittgenstein also had to suppose that there are negativefacts to act as the meanings of false atomic propositions. If an atomic prop-osition is false, there is no positive fact corresponding to it, but we are notrequired to suppose that something else magically springs into existence. Thenon-occurrence of the positive fact is itself already the negative fact.

If A loves B, it may be said, that is a good substantial fact; while if A does not love B,that merely expresses the absence of a fact composed of A and loving and B, and byno means involves the actual existence of a negative fact. But the absence of a factis itself a negative fact; it is the fact there is not such a fact as A loving B. Thus, wecannot escape from negative facts in this way.13

The logical book-keeping is thus done on this account not by a mysterious fea-ture of the world itself but simply by the grammar of factuality. Straightaway,though, we should recognize that Wittgenstein’s proposal about meaning hasembedded in it all the metaphysical problems that come with a commitmentto negative facts—problems which will be a recurring theme here.In the NotesWittgenstein is not at all explicit about what other facts, apart

from positive and negative atomic facts, his notion of propositional meaningrequires; nor does he give a worked out account of which facts are the mean-ings of which propositions. So let us try to fill in the gaps he has left. Somecases are easy enough. The meaning of not-p is the same as the meaning ofp. If ‘p and q’ is true, its meaning is just the sum of the meaning of p and themeaning of q. (We shall say a little more in the next chapter about what ‘sum’means here.) The problematic case is to say what the meaning is of ‘p or q’ ifit is true (or, dually, what the meaning is of ‘p and q’ if it is false).What Wittgenstein says about the meaning of ‘p or q’ is as follows:

13Russell, CP, VIII, 280.

)#' Meaning

Among the facts which make ‘p or q’ true, there are some which make ‘p and q’ true;but the class which makes ‘p or q’ true is different from the class which makes ‘p andq’ true; and only this is what matters. For we introduce this class, as it were, when weintroduce ab-functions.14

This remark does not yield up its meaning instantly. Wittgenstein says that‘the meaning is the fact’ (singular),15 yet he seems now to be suggesting thatwhat makes a disjunctive proposition true is not a single fact but a whole classof them. On the face of it, this makes no sense unless we understand him hereto be using ‘fact’ to mean ‘possible fact’. We might think, then, that when hespeaks of the class of facts which make a proposition true, he means not thosethat domake it true, but rather the class of possible facts any of which will, if it isindeed a fact, be a candidate to be the meaning of the proposition in question.On this reading, the meaning of a true proposition would in general be thesum of all those possible facts belonging to this class that are actually facts.But it unlikely that this can really be what Wittgenstein meant. The con-

text, let us recall, was Russell’s failed project of explaining the semantics ofpropositional thought without appealing to non-existent entities such as thedeath in bed of Charles I. And that context surely makes it improbable thatWittgenstein would really have intended to refer to what might nowadays becalled a ‘trans-world class’, i.e. a class some of whose members are merelypossible and not actual.A clue to whatWittgenstein intended is contained in the Cambridge version

of the Notes, where he remarks that in the explanation just given we have been‘talking of all p’s and all q’s’. What he intends to vary, in other words, seems tobe not the possible world referred to but the propositions involved. However,if this is what he means, what he says in the Birmingham Notes is obviouslywrong. For any fact whatever makes ‘p or q’ true for some values of p and q;and equally any fact whatever makes ‘p and q’ true for some p and q. So theclass of facts that make ‘p or q’ true for some values of p and q is exactly thesame as the class of facts that make ‘p and q’ true for some values of p and q.Perhaps this is why the version of the remark that Wittgenstein included in

the Cambridge Notes omits the passage about the two classes being different.But Wittgenstein has here got himself in a tangle quite unnecessarily. Whathe probably meant to say was that although for some values of p and q whatmakes ‘p or q’ true is the same as what makes ‘p and q’ true, for some othervalues what makes them true is different. In other words, he was merelyalluding to the obvious point that if ‘p or q’ is true, that may be because pis true, because q is true, or because p and q are both true. And the naturalthing to say, therefore, is that what makes it true, if it is true, is whichever ofthese is indeed the case. More precisely, if ‘p or q’ is true, its meaning is thesum of the meanings of whichever of p and q are true.

14B50. 15C39.

Meanings as facts )#(

We do not need to make any additional stipulation about the contrary casein which ‘p or q’ is false, since in that case the meaning is fixed by our earlierprescriptions: if ‘p or q’ is false, ‘not-p and not-q’ is true, so its meaning is thesum of the meanings of not-p and not-q, i.e. the sum of the meanings of p andof q; hence this is also the meaning of ‘p or q’. And dually we do not need tosay anything else about the meaning of ‘p and q’ when that is false, since fromwhat we have already said it will be the sum of the meanings of whichever ofp and q are false.We may summarize the account in the following table, in which we write

‘M(p)’ for the meaning of ‘p’ and ‘+’ for the operation of summing facts.

M(not p) = M(p)

p q M(p or q) M(p and q)T T M(p)+M(q) M(p)+M(q)T F M(p) M(q)F T M(q) M(p)F F M(p)+M(q) M(p)+M(q)

It is easy to extend this account to quantifiers. If (x f x is true, its meaning isthe sum of the meanings of all its instances; if it is false, its meaning is the sumof the meanings of all its false instances. Dually, if (

E

x) f x is true, its meaningis the sum of the meanings of all its true instances; if it is false, its meaning isthe sum of the meanings of all its instances.The theory just outlined does not always give especially intuitive results. It

is easy to check, for instance, that according to the theory M(p v !p) = M(p),whatever the truth-value of p. If M(p) &= M(q), therefore, M(p v !p) &= M(q v!q). This is at the very least uncomfortable because it has the consequencethat different logical truths will have different meanings: the meaning of ‘p ornot-p’ will generally be different from the meaning of ‘q or not-q’.Wittgenstein did not at this stage have anything very developed to say about

logical truths: that would come in Norway. But he does at least seem tohave realized that whatever motive we have for identifying something in theworld that makes a proposition true or false does not apply to logical truths,and therefore to have made them a sort of exception. But if it is clear thatWittgenstein already made logical truths an exception, it is less clear whatsort of exception. In the Cambridge Notes he appears to tell us that they donot have a meaning at all.16 But if we take this claim seriously, it causessevere difficulties for the theory, because logical truths may be componentsof propositions that do have meanings. Consider, for example, (p v !p) q:presumably we want it to have a meaning, even though on the view underconsideration its component p v !p has no meaning. The amendments we

16C36.

)$* Meaning

shall have to make to our theory to meet this difficulty quickly rob it of anyappealing simplicity it might have had.In fact, though, Wittgenstein probably did not intend to say that p v !p

has no meaning. The corresponding passage in the Birmingham Notes saysnot that p v !p is meaningless but that it is senseless.17 The conclusion to drawis probably that here we have no more than a difficulty of translation: whenWittgenstein said in English that a proposition ‘has meaning’, what he meant(confusingly) was that it has sense. Of course, this still leaves us to explain how(p v!p) q, for instance, can have sense even though part of it, p v!p, does not;but we shall postpone that problem until we discuss Wittgenstein’s notion ofsense in chapter 16. (We shall see there, in fact, that according to Wittgenstein(p v !p) q has the same sense as q.)Nonetheless, even if we do not read Wittgenstein as saying that logical

truths have no meaning, that is not to say that his theory is at all palatable,since it leads us out of one difficulty straight into another. Consider the casewhere p and q are true propositions expressing different atomic facts. Thenaccording to the theory just offered,

M((p v !p) q) = M(p v !p) + M(q) = M(p) + M(q) &= M(q).

So this is a case where two propositions have the same sense but differentmeanings. It is hard to believe that Wittgenstein would have regarded this assatisfactory, or even coherent, but it is equally hard to see how to avoid it.

!$.$!$.$!$.$ The demise of propositional meaningThis is perhaps the point at which to comment on the sketchiness of Witt-genstein’s theory of meaning. The reconstruction of that theory just offeredis hardly elaborate, but in the Notes Wittgenstein does not state it fully—theparts he does state he seems to bungle—and it is hard to tell in how muchdetail he ever bothered to work it out. This is an instance of a pattern inWittgenstein’s work which we have noted already. Unlike Russell, he neverlet technical considerations drive his philosophical views, and he filled in thedetails of technical proposals only reluctantly.A cynic might say, I suppose, that he did not work the theory out in detail

because he already had an inkling that it did not work, and there is probablysome truth in this. We shall later come across at least one other case in whichWittgenstein’s reluctance to work out the details of a theory, despite Russell’spestering, is plausibly a symptom that he subconsciously knew it did not work.(If so, this would not make him unusual, of course: many of us can reportcomparable experiences.)

17B40.

The demise of propositional meaning )$)

In this case, though, there is a further reason for the sketchiness of theaccount. Wittgenstein never spelt out the details, even in the Tractatus, becauseby the time he wrote it he had all but given up the idea that propositionshave meaning at all. The Tractatus does, it is true, contain the claim that ‘thepropositions “p” and “!p” have opposite senses, but to them corresponds oneand the same reality’.18 Since by this time Wittgenstein used the word ‘reality’to mean any sum of positive or negative facts,19 he was thus still assertingthat p and !p correspond to the same sum of positive and negative facts,i.e. that they have what in the Notes he called the same meaning. But theword ‘bipolarity’ does not occur in the Tractatus and the claim that p and!p correspond in different ways to the same fact is by now no more thanan idle wheel, one of a number of remarks from his early notebooks which heincluded in the finished work perhaps more for nostalgic reasons than becausethey were relevant to the exposition of his current views.What he seems by then to have abandoned is the idea that it is important

to identify just which bit of reality a proposition corresponds to. While propo-sitions still have a sense, in the Tractatus their meaning has faded into thebackground—not quite denied, but no longer doing any significant work inthe account. One way of putting this point stresses the overriding importanceof elementary propositions in Wittgenstein’s conception. The problems thatmake Wittgenstein’s theory of meaning collapse occur only when we considerthe meanings of molecular propositions, so nothing prevents us from contin-uing to say, if we wish, that elementary propositions and their negations havemeanings (which will always be positive or negative facts). Another way ofputting the reason for the demise of propositional meaning might be that bythe time of theTractatusWittgenstein no longer adhered to the correspondencetheory of truth which the notion of meaning entails. Once he had adopted thepicture theory, that is to say, it became less important to single out which partof the world it is that is responsible for making a proposition true or false.If we know the sense of a proposition, we know what it says. If it is true,things are as it says they are. Or, we might also say, what makes it true orfalse is how things stand. Once we have explained, through our account ofsense, how things have to stand for the proposition to be true, what furtherpoint is served by seeking to identify a particular part of the world as especiallyresponsible for bringing this about?

184.0621. 192.06.

Chapter !%!%!%

Metaphysics

!%.!!%.!!%.! Disjunctive factsIn order to make sense of Wittgenstein’s remarks we found it necessary in thelast chapter to appeal to what one might be tempted to think of as conjunctivefacts—the fact, for instance, that this rose is red but that one is not. In orderto provide meanings for generalizations, we had to extend this to allow forfacts made up of infinitely many component facts. At no point, though, didwe find ourselves having to appeal to disjunctive facts—the fact, for instance,that at least one of the roses in my garden is red. Of course the fact thatWittgenstein’s account did not appeal to disjunctive facts does not in itselfshow that he thought there are no such things. But if there were, they wouldmake the account significantly simpler: the natural thing to say would thenbe that the meaning of ‘p or q’, if it is true, is simply the disjunctive fact thatp or q. That he did not adopt this much simpler account is what shows thathe cannot have believed in them.What reason might he have had for this? One argument against disjunctive

facts that has become popular was advanced by Fine.1 Suppose that the factthat p or q is thought of as having p and q as constituents. In that case theoccurrence of the fact that p or q entails, it is argued, the occurrence of boththe constituent facts that p and that q. This is merely an instance of the generalprinciple that the existence of a compound entity entails the existence of itsconstituents.It is hard, though, to regard this argument as completely persuasive, be-

cause it depends on the premise that we should regard p and q as components,in the relevant sense, of the fact that p or q. It would presumably be open tothe defender of disjunctive facts to claim that there is a manner of composi-tion, of which they are an instance, for which the existence of the compounddoes not entail the existence of the components.Another, perhaps rather more Wittgensteinian, argument against disjunc-

tive facts is that even if there is a manner of combination such as has just beencountenanced, the fact that p or q will have to differ from the fact that p and q,and the difference can only be that one contains or where the other contains

1Cf. Fine, ‘Facts’, 67.

Negative facts )$#

and. But the existence of such logical ties as or and and stands in direct con-tradiction to Wittgenstein’s ‘fundamental thought’ that the logical constantsdo not represent. This kind of argument is limited, however: it shows at mostthat if we wish to countenance disjunctive facts, we need a radically differentconception of their structure, not that no such conception is possible. Afterall, exactly analogous arguments apply to negative facts, which, we have seen,Wittgenstein explicitly accepted in the Notes.Perhaps a more plausible explanation forWittgenstein’s reluctance to coun-

tenance disjunctive facts is offered by a combination of Occam’s razor and aconception of the world as determinate. It cannot be a fact that p or q unlessone or other of p and q is a fact. So whichever of these is a fact is part of theworld. What need is there to posit other, less definite facts? It is surely a rathernatural thought that devices such as disjunction are linguistic constructs de-signed to represent our ignorance about the ways of the world, not part of theworld itself.

!%."!%."!%." Negative factsWhat of negative facts? Russell later said that ‘the absence of a fact is itself anegative fact’.2 If this was supposed to be a report of Wittgenstein’s view onthe matter, then Russell did not get it quite right: what he should have said isthat the absence of a positive atomic fact is itself a negative fact. Wittgensteindid not think that the absence of a non-atomic fact is a fact. For, as we havejust seen, Wittgenstein rejected disjunctive facts. So the absence of the factthat p and q, for instance, cannot itself be a fact, since if it were, it would justbe the disjunctive fact that not-p or not-q.From now on, then, let us focus on the atomic case. Even if a positive

atomic fact represents the existence of something, it is not clear that the cor-responding negative fact has to be thought of as the existence of somethingelse. In the Notes, Wittgenstein suggests that the manner in which the absenceof something can determine reality is like the blind spot in the visual field.

The comparison of language and reality is like that of retinal image and visual image:to the blind spot nothing in the visual field seems to correspond, and thereby theboundaries of the blind spot determine the visual image—as true negations of atomicpropositions determine reality.3

Wittgenstein was presumably trying by the use of this analogy to make it plau-sible that reality might be characterizable as much by the absence of some-thing as by its presence.Let us now consider what can be said about the structure of a negative

fact. It cannot consist of the positive fact together with another element called2CP, VIII, 280. 3B11.

)$$ Metaphysics

negation, for reasons that parallel the ones offered earlier against disjunctivefacts: there is no such ‘thing’ as negation, and in the case in question the positivefact does not exist.A more plausible solution, perhaps, would be to roll up the negation into

the verb of the fact, so that if the positive fact is that aRb, the negative oneis that a!Rb. Of course, this is not a proposal we can use as an analysis ofcomplex facts: we cannot analyse the fact which makes ‘the present King ofFrance is bald’ false as being the same as the fact that the present King ofFrance is non-bald. But that is no objection in the present case, since hereit is only the facts corresponding to atomic propositions that are in question,and ‘the present King of France is bald’ is not atomic. Indeed a quick way tosee that it is not atomic is simply to note the difference in sense between ‘Thepresent King of France is non-bald’ and ‘It is not the case that the presentKing of France is bald’.The current proposal, that negation should be rolled up into the verb of

the proposition, was considered and rejected by William Demos in his articleobjecting to negative facts.4 However, his reason for rejecting it is barelymore than a restatement of his denial that there are negative facts. It is also,more significantly, the treatment of negative facts that Russell offers in thefirst section of his 1919 paper ‘On propositions’. In other respects this sectionof the paper consists largely of an exposition of ideas from the Notes on Logic:Russell notes that it ‘contains nothing essentially novel’, and claims, a littleinaccurately, that he has ‘defended its doctrines elsewhere’.5 This supportsthe hypothesis that his account of negative facts was Wittgenstein’s. If so,it is an account with which Wittgenstein himself later became uneasy. In theNotebooks he repeatedly mentions his concern with the structure of the negativefact and its relationship to the positive. ‘It is the dualism, positive and negativefacts, that gives me no peace. For there cannot be such a dualism. But how toescape from it?’6

Even when it is limited to the atomic case, the proposal is indeed far fromunproblematic. One of the things that makes facts more plausible truthmakersfor atomic propositions than complexes, I suggested in §11.1, is the implau-sibility of leaving the logical book-keeping to the world to sort out; but nowit seems as if we are reinstating that problem by supposing that the primitiveways in which objects can combine to form facts come in pairs: for every sim-ple way R in which objects can be related there is another way !R, with theconvenient property that in every case exactly one of the two relations holds.We might try to get out of this difficulty by viewing the convenient propertyjust mentioned as a criterion for the simplicity of R. In other words, the feature

4‘A discussion of a certain type of negative proposition’. 5CP, VIII, 278. 6NB, 25 Nov. 1914.

Summing facts )$%

we noted earlier in the case of ‘the present King of France is bald’—that it doesnot have a well-defined negation—is representative of something more gen-eral: what it is for a proposition to be atomic is for there to be two mutuallyexclusive possible facts, one positive and one negative, such that whicheverof them obtains is its meaning. Nonetheless, this hardly solves the difficulty.Once the forms of facts are aligned in contradictory pairs, R and !R, we canperhaps regard it as a criterion of atomicity that aRb and a!Rb should be mu-tually exclusive and exhaustive possibilities. But what reason is there to thinkthat forms can be paired off in this way at all? In his 1919 presentation of thetheory, Russell described the difference between R and !R as ‘ultimate andirreducible’,7 which was his favourite way of labelling a problem rather thansolving it.It may in any case be doubted whether the proposal now under discussion

drives a sufficiently large wedge between negative facts, which Wittgensteinadmitted, and disjunctive facts, which he rejected. For if the idea of rolling upthe logical tie into the verb is valid in the case of negation, it could in principlebe extended, albeit less tidily, to other logical constants such as disjunction.These difficulties suggest, perhaps, that a more radical approach is required. Ifwe really want to save negative facts, perhaps we should reject the temptationto conceive of their structure in such a way that the difficulty we have beendiscussing even arises.

!%.#!%.#!%.# Summing factsNo fact, Wittgenstein believed, can contain a logical tie. So even what I ear-lier called ‘conjunctive’ facts are not really conjunctive—do not combine theirconstituent facts by means of a logical tie of conjunction. In that case, though,we need an explanation of how there can be any complex facts; any facts, thatis to say, other than the positive and negative ones to which atomic prop-ositions correspond. The way I put the matter earlier was by appealing tofacts that are sums of other facts: the facts we are tempted to call ‘conjunctive’should more properly be thought of as something like mereological sums oftheir components. One might express the view, then, by saying that a mere-ological sum of facts is itself a fact. That, at any rate, is how Russell seemsto have conceived of non-atomic facts. In his Introduction to the Tractatushe used the language of part and whole, distinguishing8 between facts whichcontain parts which are facts and facts which contain no such parts.Mereological sums are sometimes held to be metaphysically innocent. Ac-

cording to this view, speaking of the sum of some things is just another way of

7CP, VIII, 280. 8CP, IX, 104.

)$" Metaphysics

speaking of those things. The most vigorous proponent of this view was DavidLewis.

To be sure, if we accept mereology, we are committed to the existence of all mannerof mereological fusions. But given a prior commitment to cats, say, a commitment tocat-fusions is not a further commitment. The fusion is nothing over and above the catsthat compose it. It just is them. They just are it. Take them together or take themseparately, the cats are the same portion of Reality either way. Commit yourself totheir existence all together or one at a time, it’s the same commitment either way. Ifyou draw up an inventory of Reality according to your scheme of things, it would bedouble counting to list the cats and then also list their fusion. In general, if you arealready committed to some things, you incur no further commitment when you affirmthe existence of their fusion. The new commitment is redundant, given the old one.9

In Russellian terms a mereological sum is an incomplete symbol whichcould be eliminated on analysis if our underlying logic was plural. ButWittgen-stein’s logic was not plural, and he displayed no sign of having been temptedin that direction. Perhaps, therefore, assuming the existence of complex factsis, strictly speaking, an inflation of his ontology beyond the positive and neg-ative facts that compose them; but if we take David Lewis’s line, we do notneed to think of it as an important inflation. This is presumably what Wittgen-stein meant when he noted that ‘whatever corresponds in reality to compoundpropositions must not be more than what corresponds to their several atomicpropositions’. What corresponds to p q if it is true, for instance, is the sum ofthe fact that p and the fact that q; but that is really no more (or at any rate notmuch more) than those two facts considered together.According to Wittgenstein, then, there are complex facts which are sums of

positive and negative fact. In the Birmingham Notes (but, for reasons we shallcome to shortly, not in the Cambridge ones) Wittgenstein made the furtherclaim that there is, in a certain sense, no more to the world than the positiveand negative facts. ‘If we formed all possible atomic propositions, the worldwould be completely described if we declared the truth or falsehood of each.’10

Anything we can say about a sum of facts can easily be reworded as a state-ment about the constituent facts. So to describe the world it is not necessaryto mention such composite facts at all.One point worth noting about this conception of the world is that it is in a

certain sense flat. Each positive or negative fact that there is has an internalstructure: it is composed of a form and a certain number of objects. But thereis no other structure to the world, except in the thin sense that atomic factsagglomerate into more complex facts as mereological sums. What there is notis any sort of structure to facts that would allow for the notion that they mighthave alternative atomic decompositions.

9Parts of Classes, 81–2. 10B36; cf. 4.26.

General facts )$&

!%.$!%.$!%.$ General factsMuch of the metaphysical picture outlined in the Notes Russell accepted in his1918 lectures without demur. The one point on which he disagreed, however,came when Wittgenstein insisted that the positive and negative facts are allthere is to the world. General facts, Russell believed, are a counterexampleto this claim. ‘You cannot,’ Russell said, ‘ever arrive at a general fact byinference from particular facts, however numerous.’11 There must, he insisted,be at least one fact over and above all the particular facts, namely the fact thatthey are all of them.

It is perfectly clear, I think, that when you have enumerated all the atomic facts in theworld, it is a further fact about the world that those are all the atomic facts about theworld, and this is just as much an objective fact about the world as any of them are.12

Russell was here repeating something he had said in the Lowell Lectures fouryears earlier.

It is easy to see that general propositions, such as ‘all men are mortal’, cannot be knownby inference from atomic facts alone. If we could know each individual man, and knowthat he was mortal, that would not enable us to know that all men are mortal, unlesswe knew that those were all the men there are, which is a general proposition.13

Now this looks superficially like, but is not the same as, the claim from theIntroduction to the Tractatus that ‘the world is fully described if all atomic factsare known, together with the fact that these are all of them’. The confusingsimilarity between the two claims is the result of a terminological shift: in theTractatus atomic facts (Sachverhalte) are always positive, whereas in the 1914 and1918 lectures Russell was following Wittgenstein’s prewar usage according towhich atomic facts may be positive or negative.Suppose for a moment that a, b, and c are the objects and f x and gx the

forms, so that there are six possible positive facts that may obtain, namely fa,f b, fc, ga, gb, gc. If I tell you that among these possible facts fa, f b, fc, andgb (say) do actually obtain, I do not describe the world completely: to do thatI would have to add the information that these are the only ones that obtain;or equivalently I would have to say that ga and gc do not obtain. The pointRussell is making in the Introduction is that a complete description of theworld cannot (except in the unusual case in which all the atomic facts obtain)be wholly positive; it will have to contain the negative information that noother of the atomic facts actually obtain.But in the 1918 lectures Russell’s point is different. Suppose now that in

addition to saying that fa, f b, fc, and gb obtain, I also tell you that ga andgc do not. By the lights of the Tractatus I have now described the world com-pletely. But Russell objects to this on the ground that the information supplied

11CP, VIII, 206. 12CP, VIII, 207. 13OKEW, 55–6.

)$' Metaphysics

does not license the inference to (x) f x. For that inference to be valid I have toknow that there are no objects other than a, b, and c, no forms other than f xand gx. The point, in other words, is not now about which of the possibilitiesare actualized, but about what the possibilities are.Russell’s insistence on general facts is striking partly because it is one of

the very few points in his 1918 lectures on logical atomism on which Russelldisagrees with the Notes on Logic at all, but also, more pointedly, because Russelldoes not give any indication that he is disagreeing with them. What is theexplanation for this? By the time of the Tractatus, of course, Wittgenstein hada developed view which allowed him to reject Russell’s point. To admit it as afurther fact about the world that just these atomic facts are possible would betantamount to granting that there could have been a different range of possibleatomic facts. But that would be to say that it is possible for the possibilitiesto be different from what they are, and hence to appeal to a second-ordernotion of possibility, which is precisely what Wittgenstein’s univocal accountof possibility14 denies. In the Notes, on the other hand, the notion of possibilityis not even discussed, and it is hard to see how, with the resources Wittgensteinthen had at his disposal, he could resist Russell’s argument. It seems likely,indeed, that the point simply had not occurred to him.This explains an otherwise puzzling difference between the Birmingham

and Cambridge versions of the Notes. In the Birmingham version Wittgen-stein insisted, ‘If we formed all possible atomic propositions, the world wouldbe completely described if we declared the truth or falsehood of each.’15 I aminclined to think that Wittgenstein repeated this claim to Russell in Cam-bridge, but that Russell then offered the objection just outlined. Withoutany response to this objection Wittgenstein was forced to withdraw the gen-eral claim. Instead he restricted himself to the more limited observation that‘whatever corresponds in reality to compound propositions must not be morethan what corresponds to their several atomic propositions’,16 stumblingly re-peated a little later: ‘Molecular propositions contain nothing beyond what iscontained in their atoms; they add no material information above that con-tained in their atoms.’17 By a ‘molecular proposition’ Wittgenstein presum-ably meant a truth-function of elementary propositions, so what Russell’s ob-jection now reduced him to asserting was no longer a substantial metaphysicalclaim, but rather a complete triviality.If in his conversations with Russell Wittgenstein withdrew the general claim

pending the considerations in the Tractatus which allowed him to reinstate it,that would be at least a partial explanation for Russell’s confident denial ofit in lectures in which he otherwise followed Wittgenstein’s line so slavishly.When he denied it in 1918, Russell was indeed still reporting what he still hadno reason to doubt was Wittgenstein’s view.

146.375. 15B36. 16C2. 17C10.

Logical data )$(

If my conjecture is right, it is also remarkable for what it implies aboutthe working relationship between Wittgenstein and Russell. If, as seems tobe the case, the question whether generalization introduces further facts hadsimply not been raised between the two men before October 1913, that canhardly fail to highlight their lack of genuine collaboration over the previousfew months.

!%.%!%.%!%.% Logical dataThe issue just discussed is important for epistemology, at least if we think,as Russell did, that ‘all empirical evidence is of particular truths’. For in thatcase ‘if there is any knowledge of general truths at all, there must be someknowledge of general truths which is independent of empirical evidence, i.e.does not depend upon the data of sense’.18 As Russell went on to note, thisresult ‘is important, since it affords a refutation of the older empiricists’.19 Heconcluded that ‘there is general knowledge not derived from sense, and thatsome of this knowledge is not obtained by inference but is primitive’. Hetherefore owed an account, which he did not supply, of how it is possibleto have primitive knowledge of facts not derived from sense. Russell calledthese facts ‘logical data’, but he might just as well (if more provocatively) havecalled them synthetic a priori truths. It was precisely this notion of logical datathat Wittgenstein wanted to resist. He may have been temporarily nonplussedby Russell’s apparent counterexample of the fact that the atomic facts are allthere are, but it remained his ambition to find an explanation which allowedhim to deny this. Part of his reason for this was epistemological: he wantedto deny that there could be ‘logical experience’, since such a thing wouldcontradict his fundamental insight that logic ‘must turn out to be a totallydifferent kind than any other science’.20 But the point is also metaphysical: ifthere are general facts not reducible to atomic facts, as Russell maintained andWittgenstein wished to find a principled reason to deny, then logical atomism,in the form in which Wittgenstein then understood it, is false. Wittgenstein’satomism, that is to say, amounted at the time of the Notes to the view that allthere is to the world is atomic facts; or, more precisely, all facts are just sums of(positive and negative) atomic facts. In particular, therefore, there is no roomin this conception for any such thing as a logical fact.In light of this, it is reasonable to ask in what sense Russell understood him-

self to be a logical atomist despite his willingness to allow that not all facts arereducible to the atomic facts. The phrase itself owes nothing to Wittgenstein:Russell described his own view as ‘un atomisme logique’21 during a discussionin Paris in March 1911 before the two men had met. But it was never as good

18OKEW, 56. 19Ibid. 20To BR, 22 June 1912. 21CP, VI, 412.

)%* Metaphysics

a description of Russell’s view as it was of Wittgenstein’s. The ontology ofthe Tractatus famously22 consists of facts, not of things. Wittgenstein no doubtmeant various things by this, but one of the things he meant was that a con-ception of the world will not count as truly atomistic unless it holds that allfacts are reducible to atomic facts. Russell was not an atomist in this sense: forhim atomism was primarily a doctrine about things, not about facts.Even if we accept that all facts—in particular, general facts—are reducible

to (i.e. nothing more than sums of) positive and negative atomic facts, thereremains the question whether there are any logical facts. Wittgenstein deniedthat there are (and in the Lowell Lectures,23 presumably under his influence,Russell agreed). But is there anything other than a prejudice against the syn-thetic a priori that counts as a reason against them? Wittgenstein’s atomismdoes at least entail that if there are any logical facts, the atomic facts that theyreduce to will have to be logical facts too. So the question reduces to whetheran atomic fact could be logical. Wittgenstein’s ‘fundamental thought’ alreadyrules out the possibility that there could be logical objects. So an atomic logicalfact would have to have non-logical objects as its constituents. What, then,would distinguish it from a non-logical fact? Nothing, presumably, except itsnecessity. But that—that it is necessarily so—would then be a further fact.In addition to the atomic fact that p, in other words, there would also be thefact that #p. But this leaves the entailment #p ! p unexplained, unless wesuppose that it is a further fact, thus inviting an obvious infinite regress. Theonly escape route consistent with Wittgenstein’s understanding of atomism,therefore, is to conclude that there are no logical facts at all.The concepts of necessity and possibility are conspicuously absent from the

Notes. The reason, I assume, is that they await the Tractarian conception ofa proposition as expressive of a possibility as to how things might be. Thatconception is quite absent from the Notes—it is probably one of the things hearrived at in Norway—but once it is in place, the notions of necessity and pos-sibility quickly follow. It is tempting, perhaps, to say that this makes modalityin the Tractatus a matter of language, not of the world: we express how thingsare by means of a contrast with how else they might have been, and hence wefashion our conception of possibility. But if we think this through, what wecome to realize is the flimsiness of the attempt to pin on the Tractatus a con-trast between language and world at all. For the possibilities of combinationare already present in the atomic facts; or, at any rate, in their being all thefacts. And there seems no reason to place that on the side of language ratherthan of the world.

221.1. 23OKEW, 53.

Chapter !&!&!&

Sense

A proposition has a meaning, which is the fact that makes it true or false.But it was central to Wittgenstein’s view that we can understand a propositionwithout knowing which of these two possibilities holds.1 The meaning is notsomething we come to know simply by virtue of understanding the proposi-tion, since it depends also on how things stand in the world. So there must bea second ingredient in what a proposition expresses, something we can graspin advance of finding out what its meaning is. This second ingredient Witt-genstein called the sense of the proposition. It consists in the conditions underwhich the proposition is true and the conditions under which it is false.What we know when we understand a proposition is this: We know what is the case ifthe proposition is true, and what is the case if it is false. But we do not know (neces-sarily) whether it is true or false.2

By saying that a proposition has both sense and meaning, Wittgenstein was ofcourse quite consciously mimicking Frege, but although he kept the same ter-minology, and some of the same motivation, there are important differencesin their usage. We have already seen that for Wittgenstein the meaning of theproposition is the fact that makes it true or false, not (as for Frege) its truth orfalsity itself. And although the sense is still (as it was for Frege) what I graspwhen I understand the proposition, Wittgenstein’s conception of it is based,as we shall see shortly, on a rather different understanding from Frege’s of thecontributions made to it by the proposition’s components.

!&.!!&.!!&.! Semantic valueA proposition can have various structures, subject-predicate, relational, etc.Wittgenstein focused solely on the relational case, not because he thought allpropositions have this form—he was explicit that they do not—but purely forexpositional simplicity. So let us follow him in this, and focus for convenienceon a relational proposition ‘aRb’. Here there are three components, the twonames ‘a’ and ‘b’, and the form ‘xRy’. The contribution an expression makes1C5; B42; cf. 4.024. 2C6. The version of this remark in the Birmingham Notes (B41) omits,presumably in error, the requirement that we should know what is the case if the proposition isfalse.

)%! Sense

to the sense of propositions in which it occurs is its semantic value. Wittgensteinheld that the sense of the proposition is determined by the semantic valuesof its components. To understand the sense of ‘aRb’, therefore, it sufficesto grasp the semantic values of its three components.3 But what are thesesemantic values?Giving the sense of a proposition consists, we have said, in determining the

conditions under which it is true and the conditions under which it is false.Wittgenstein starts by considering the idea that to understand the propositionp it is sufficient to be able to fill in the blank in the schema

‘p’ is true if and only if . . .

in such a way that what results is true. But, as he notes, this is too weak aconstraint.

It is incorrect to say: we understand the proposition p when we know that ‘ “p” is true’#p; for this would naturally always be the case if accidentally the propositions to rightand left of the symbol ‘#’ were both true or both false.4

It is accidentally the case, for instance, that

‘Snow is white’ is true if and only if grass is green.

We need a tighter constraint on the schema if we are to rule out cases likethis. In order to obtain such a constraint, Wittgenstein suggested, we needto introduce more structure into the T-schema, i.e. an extra variable, andinsist on a formal equivalence with respect to that variable. On the face ofit the most obvious such variable would be the possible world in which theproposition is to be true. Our criterion for understanding the proposition pwould then be that we could fill in the gap in the scheme

For every possible world W, ‘p’ is true-in-W if and only if in W . . .

in such a way as to make it true. The completion

For every possible worldW, ‘Snow is white’ is true-in-W if and only if inW grass is green

is now ruled out, because there are possible worlds in which snow is white but(because of a genetic mutation, perhaps) grass is red.I have already suggested in my discussion of Wittgenstein’s account of

meaning that he was reluctant to appeal to possible worlds in his semantics.His technique on that occasion was to resort to actual-world quantificationto achieve the restrictions he needed. It is therefore no great surprise that inhis account of sense he was not attracted by the appeal to possible worlds justsketched, but instead restricted the schema by an actual-world quantification.

3B41. 4B44.

The semantic value of a form )%#

What is needed, he says, ‘is bound up with the introduction of the form ofp’.5 And again, ‘What is wanted is the formal equivalence with respect tothe forms of the proposition, i.e. all the general indefinables involved.’6 Inthe case of our relational proposition ‘aRb’, for instance, quantifying over thename positions gives us

(x,y)(‘a’ means x ‘b’ means y ! ‘aRb’ is true " x has R to y),

where ‘a’ and ‘b’ are being used, like ‘R’, schematically.One thing we can deduce from this straightaway is what our earlier discus-

sion of Wittgenstein’s conception of simplicity already led us to expect, namelythat the semantic value of a name is its meaning. The object-level quantifica-tion in the schema ensures that different names which refer to the same objectmake the same contribution to the sense of propositions in which they occur,and so names do not have Fregean sense.

!&."!&."!&." The semantic value of a formWhat remains, though, is to work out the semantic value of the form ‘xRy’.Wittgenstein’s account of this is repeated in both the Birmingham and Cam-bridge versions of the Notes.7 What is striking about this account is the care hetook to proceed via the form, so as to reach the sense of the proposition onlyderivatively. He was not content, for instance, simply to say that ‘aRb’ is trueif a has R to b, and false otherwise. His reason for taking the more elaborateroute is that only by going via the form of the proposition could he ensurethe connection he wanted between the senses of ‘aRb’ and ‘cRd’: the relation-ship that ‘aRb’ expresses as holding between a and b must be the same as therelationship that ‘cRd’ expresses as holding between c and d. That is the con-straint the theory had to meet, and it is clear Wittgenstein understood it. Heemphasized, for instance, the need to explain ‘propositions such as (

E

x,y) xRyand similar ones’ in such a way as to show that they ‘obviously have in com-mon with aRb what cRd has in common with aRb’.8 As he could have learntfrom Russell, who had alluded to the point in the Principles,9 the dependencyof a proposition on the names occurring in it is not merely that of a functionon its arguments.Given his reluctance to countenance possible-world talk, it is no surprise to

find Wittgenstein insisting on an extensional understanding of forms.

Let us consider symbols of the form ‘xRy’; to these correspond primarily pairs of ob-jects, of which one has the name ‘x’, the other the name ‘y’. The x’s and y’s stand invarious relations to each other, among others the relation R holds between some, butnot between others.10

5B44. 6C40. 7B43; C38. 8B51. 9Principles, ch. 7. 10B43.

)%$ Sense

In other words, what it involves to specify the semantic value of the form ‘xRy’is primarily to determine which pairs of objects are related by the relation Rand which are not. But it is easy to see that this is not sufficient to determinewhich form we intend. To simplify the exposition, let us consider for a mo-ment the monadic case. The parallel proposal in this case would be that wecan fix the semantic value of a propositional function simply by determiningwhich objects fall under it and which do not. Consider a world with threeobjects a, b, and c, and two atomic forms f and g; suppose that the positivefacts that occur are fa, f b, and ga. I might try to determine a propositionalfunction by specifying that its extension is to be the class {a,b} (so that a andb fall under it but c does not). But this plainly does not specify the functionuniquely: it does not fix whether it is f x or f x v gx, for instance. The pointis, of course, that although it is true in the world just described that these twoforms are equivalent—i.e. the same objects satisfy them—the truth-conditionsof propositions involving them are not equivalent: the sense of fa is not thesame as the sense of fa v ga.So it is not possible to fix the semantic value of a form solely by determining

its extension: we need to make our specification more fine-grained. Wittgen-stein’s proposal for achieving this finer grain is that we have to correlate theproposition with facts. He therefore introduces the idea that facts may be gle-ichsinnig with, or entgegengesetzt to, a proposition. Russell translates this in theBirmingham Notes by talking of facts as ‘of like sense’ with, or ‘of oppositesense’ to, the proposition.11 In the Cambridge NotesWittgenstein talks insteadof the proposition as ‘true to the fact’ or ‘false to the fact’.12 (Notice, inciden-tally, that not all facts can be categorized as being of like or of opposite senseto a proposition. The fact that the cup is warmer than the saucer, for instance,is neither of like nor of opposite sense to the proposition that the cup is sittingon the saucer: they stand in quite different dimensions of variation.)Now that we have the notion of correlation with facts in place, we can re-

turn to the problem that led us to invoke it, namely that of determining thesemantic value of a form. This, we can now say, involves specifying whichfacts are of like sense with the form and which of contrary sense. In the ex-ample considered earlier, where the positive facts are fa, f b, and ga (so thenegative facts are ! f c, !gb, and !gc), we could fix the form f x by specifyingthat fa and f b are of like sense to it and ! f c is of opposite sense; to fix on theform f x v gx, we would specify that fa, f b, and ga are of like sense, ! f c, !gb,and !gc of opposite sense.Wittgenstein says elsewhere13 that a form is like a line dividing a plane.

Presumably the analogy he intends is that the form divides a certain group offacts (the plane) into two classes. In the case we are considering here, f x is likea line which has fa and f b on one side of it, ! f c on the other. Notice that

11Ibid. 12C38. 13B23.

The compass-needle analogy )%%

f x v ! f x now emerges as a special case because all the facts are of like sensewith it: it does not divide a plane. Similarly, f x ! f x does not divide a plane,since it is of opposite sense to all the relevant facts.

!&.#!&.#!&.# The compass-needle analogy

One of the challenges an account of sense faces is how to deal with the coun-terfactual element. It is worth pausing to note how Wittgenstein’s accountdeals with this challenge. Grasping a proposition is not simply a matter ofseeing the fact that actually makes it true or false, but rather of understandingwhat would make it true or false however things turn out to be. One wouldlike to say here, ‘which facts would make it true or false’. But we cannot saythat, because facts are actual: there are no such things as merely possible facts.Our first temptation, that is to say, is to explain the sense of the propositionas a categorization of possible facts as being of like or opposite sense to it, butwe are forced to recognize that that explanation is not available to us. Thisis yet again the problem of false propositions which had bedevilled Russell.Wittgenstein’s account of meaning had elegantly allowed him to provide ameaning for the false proposition that Charles I died in bed. The reason thatthe proposition is false is precisely that the fact that is its meaning is incompat-ible with its sense. So far, so good. However, it is also part of understandingthe proposition to know that if Charles I had died in bed, that—which wouldthen have been a fact—would have been of like sense with the proposition.But as things actually stand, it is not a fact; it is nothing at all. So instead Witt-genstein suggests that as part of our account of the sense of the proposition‘Charles I died in bed’ we point to the fact that Charles I did not die in bed,and specify that this fact is of contrary sense to the proposition.In an attempt to make this account plausible, Wittgenstein exploits another

meaning of the word ‘sense’, as applying to the direction of an arrow. ForFrege this would have been no more than an unintended pun; but for Witt-genstein it is the basis of an extended analogy. The sense of a proposition isakin to the direction of an arrow: p and !p point to the same fact, i.e. theyhave the same meaning; but they point to it in opposite directions, i.e. theyhave opposite senses. It is hard to avoid the thought that Wittgenstein has inmind some sort of analogy with electromagnetism. Perhaps the propositionis to be thought of as akin to the needle of a compass placed next to a coil ofwire. When current passes through the wire, the compass needle aligns itselfwith the magnetic field created; if the direction of the current is reversed, thecompass needle turns to point in the opposite direction. Wittgenstein’s use ofthe word ‘pole’ thus echoes its use to refer to the ends of a magnet, or theterminals of a battery. Reversing the direction of current flow by swapping

)%" Sense

the terminals corresponds to negation.Of course, variants of this analogy are possible too: I might hold the com-

pass needle fixed in my hand, in which case a magnetic field in the oppositesense to the needle will not make it rotate but will nonetheless generate a force,which I will be able to detect. As with any analogy, not every aspect of thestory is relevant, and there is room for doubt as to which aspects Wittgensteinwished especially to draw on. One aspect that surely attracted him, though, isthat the compass needle is disposed to react in a certain way to the presenceof a magnetic field whether or not the field is actually present. Wittgensteinpresumably hoped to draw on this analogy to make plausible the idea that aproposition provides the means to categorize facts as being of like or contrarysense even though some of the facts to be categorized are not actual. A prop-osition is to be thought of, that is to say, as carrying within it the disposition tobe of like or contrary sense to the facts, however the facts turn out to be.

!&.$!&.$!&.$ GrainIt follows fromWittgenstein’s account that two propositions can have differentsenses only if there is a difference in the circumstances that would make themtrue. But this does not straightaway determine how fine-grained the notionis. Russell, for instance, was quite happy with the idea that there are logicalfacts, and therefore could have maintained that different logical truths havedifferent senses. Wittgenstein reached the contrary view that all logical truthshave the same sense by way of his rejection of logical facts.Sense is therefore much less fine-grained for Wittgenstein than for Russell.

However, it is also much less fine-grained forWittgenstein than for Frege, whoin Grundgesetze held that logically equivalent propositions may differ in sense.According to Dummett,

this alone is enough to show that sense is not [for Wittgenstein] correlative to under-standing, since one may, in the ordinary meaning of ‘understand’, understand twosentences without realizing that they are equivalent.14

But this is to locate the difference between Frege and Wittgenstein at thewrong place. There may indeed be a meaning of ‘understand’ on which whatDummett says is right, but that is not Wittgenstein’s meaning. On his usage,we do not fully understand a proposition until we grasp what it says about theworld (or, in the case of a tautology, that it says nothing). And that is surely ausage that is to be found in ordinary language too. In the metaphor Wittgen-stein later introduced, we do not understand the proposition until we picturewhat it says. So on Wittgenstein’s use of the word ‘understanding’, sense iscorrelative to understanding.

14FPL, 680.

Grain )%&

For all we have said so far, though, this is no more than a difference ofterminology. To see why it is more than that, we must return to the contrastbetween Wittgenstein’s and Frege’s conceptions of logic. Both started fromthe idea that the sense of a proposition is what it says. We might be temptedto qualify this by adding that the sense is what the proposition says about theworld, but for Wittgenstein (though not for Frege) that is otiose, since thereare no logical objects and hence there is nothing else against which the worldcan be set—nothing else for the proposition to be about but the world. Nowlogical truths all say the same thing, which means, as just noted, that they saythe same thing about the world. But what they say about the world is nothing:they do not rule any atomic facts in or out, since if they did so, that wouldmake those atomic facts logical, contrary to Wittgenstein’s view that there areno such things as logical facts. It follows that if logic is assessed only withrespect to what it says, it is trivial. The only space left for Wittgenstein to findanything non-trivial in logic is not in what it says but in how it says it.This is where the true disagreement between Wittgenstein and Frege is

to be found. For Frege (and indeed for Russell) the fruitfulness of logic wasproof that it has content; for Wittgenstein the only explanation for logic’sspecial character was that it has none. This left Wittgenstein with an emergingproblem over what to say about the logicist project that Frege and Russell hadbeen conducting. Part of the reason for their conviction that logic is non-trivialwas their belief that logic entails substantial parts of mathematics: since theytook it as evident that mathematics is not trivial, it followed that they could notadmit logic to be trivial either. If Wittgenstein wished to maintain nonethelessthat logic is trivial, he had either to insist, implausibly, that mathematics istrivial too or to provide a quite different, non-logicist, explanation of its rolein reasoning.Perhaps it is no accident that just when Frege began to think his logicist pro-

gramme was a failure, he also came to accept that logically equivalent propo-sitions have the same sense.15 By the time Wittgenstein knew him, therefore,he would have been willing to grant that the sense of any logical truth is triv-ial, and hence that the explanation for the complexity of mathematics is to befound elsewhere.

15To Husserl, 9 Dec. 1906.

Chapter !'!'!'

Truth-functions

A truth-function is any function taking a finite list of truth-values as argumentsand producing a truth-value as its result. In the Notes, to Russell’s exaspera-tion, Wittgenstein chose to call them ‘ab-functions’. I shall mention his reasonsfor this later, but in the meantime I shall stick to the more standard terminol-ogy. In this chapter I want to describe the notations Wittgenstein used atvarious times for describing a truth-function. Since the eventual aim will be todiscuss the relationship between a logical connective and the correspondingtruth-function, it will be sensible to distinguish notationally between the two.I shall therefore, for the moment at least, use ‘not’, ‘or’, ‘and’, etc. for theconnectives, and reserve ‘!’, ‘v’, ‘.’, ‘!’, ‘"’, ‘|’ for the various truth-functions.

!'.!!'.!!'.! Using primitive signs

One sort of notation for expressing truth-functions is simply to make an ar-bitrary choice of sign to stand for each one. But the number of n-ary truth-functions is 22n , so even in the binary case there are sixteen that need to besymbolized. It turns out, though, that all the truth-functions can be obtainedby composition from a small subset of them. So all we need to do is to choosea symbol for each of these and then express others as compounded out ofthem. If such a notation is to be adequate to express all truth-functions, ofcourse, those for which we have primitive symbols must be chosen with care.The truth-functions ! and v are jointly adequate, for example; so are ! and!; but ! and " are not. This point is not mathematically sophisticated, andno doubt many of those who worked on aspects of the propositional calcu-lus in the nineteenth century were aware of it. Frege, for instance, presenteda formalization of the propositional calculus in the Begriffsschrift in which hechose ! and ! as the primitive functions, no doubt conscious that they aretruth-functionally adequate, whereas Russell chose ! and v. There are twotruth-functions each of which is expressively adequate on its own. They are thefunctions p | q and p † q defined as follows:

Using primitive signs )%(

p q p | qT T FT F TF T TF F T

p q p † qT T TT F TF T TF F F

These two truth-functions are known as ‘Sheffer stroke functions’ after theirdiscoverer. Confusingly, there is no agreement which of the two to denote bythe stroke sign ‘|’. In the table above I have followed Wittgenstein’s usageaccording to which p | q expresses ‘not-p or not-q’; the dual function, whichI have written here p † q, expresses the joint denial of p and q. The paperin which Sheffer showed that each of these functions on its own is adequateto express all truth-functions1 appeared late in 1913, but Russell was sent acopy in April and there is ample evidence in the Notes that Wittgenstein knewits content. He uses Sheffer’s notation, for example,2 and he states Sheffer’sresult that | is adequate to express all truth-functions.3

I mentioned in §8.3 the idea Wittgenstein got from Hertz that the indefin-ables in a system should be independent from one another. In the NotesWitt-genstein applied that idea to the truth-functions in three ways, none of themvery convincing. First, he claimed that ! and v are not independent: if bythis he meant that one can be defined in terms of the other, then the claim istrivially false; if not, it is wholly unclear to me what else he could have meant.Second, Wittgenstein thought that if a system can be presented using one inde-finable, then presenting it using two must contravene Hertz’s principle. This istrue for vectors in space, of course—different bases for the same space alwayshave the same number of vectors in them—but it is not true for axiom systems,as ! and v show. Wittgenstein’s attachment to this faulty reasoning seems tohave made him unduly impressed by the fact that Sheffer’s stroke function | ison its own adequate to express all the truth-functions. It may indeed be moreelegant to present logic using one primitive sign rather than two, but it doesnot have any ultimate theoretical privilege. Third, Wittgenstein thought thatthe existence of alternative sets of indefinables shows that ‘these are not theright indefinables’.4 It is hard to see why he might have believed this, sincenot even the corresponding claim about vectors is true—any non-trivial vectorspace has various different bases—nor is it true for his favoured Sheffer stroke|, since the dual stroke function † is also adequate. Russell reminded him ofthis in 1919, suspecting as a consequence that duality ‘persisted covertly inyour system’,5 but Wittgenstein brushed the point away with what looks verylike bluster. ‘This doesn’t matter! . . . All is said in my book about it and I feelunable to write it again. Try to understand it till we meet.’6

1‘A set of five independent postulates for Boolean algebras, with application to logical constants’.2B31. 3B30. 4B18. 513 Aug. 1919. 619 Aug. 1919.

)"* Truth-functions

!'."!'."!'." Truth-tablesOne obvious objection to the notations we have just been considering is thatbecause they are chosen more or less arbitrarily, they do not wear the truth-functions they stand for on their faces. An alternative approach is to devise asign for the truth-function which explicitly identifies the truth-table it corre-sponds to. Examples of truth-tables in Wittgenstein’s handwriting have beenpreserved on the back of one sheet of some of Russell’s notes for his work onmatter. So truth-tables were among the things they discussed at their meetings(most likely some time after Russell had received Sheffer’s paper). Whetherrepresenting a truth-function by means of a table was Russell’s idea or Witt-genstein’s is hard to say. (If I had to guess, I would opt for Russell.) But evenif these are the earliest known examples, they have no great significance inthemselves. The idea of notating a function on a finite domain by means of atable of arguments and values is an obvious one, and had been used frequentlyby nineteenth-century authors with whom Russell would have been familiar.All that he and Wittgenstein were doing was to apply this idea to the partic-ular case where the domain in question consists of the two truth-values. Thesignificant step for logic is rather that of identifying this case as an importantone to study.A truth-function, as we are using the term here, is a function in the math-

ematical sense: an n-ary truth-function takes a list of n truth-values as argu-ments and delivers a single truth-value as its value. Because the values deliv-ered by truth-functions always belong to the same set as the arguments, it ispossible to compose truth-functions with one another to form further truth-functions. The criterion of identity for truth-functions is the extensional onefamiliar from mathematics: functions count as identical if they deliver thesame value as output whenever they are given the same arguments as input.What this amounts to when expressed in terms of truth-tables is that all thatmatters in settling the identity of the truth-function a truth-table represents isthe main column, i.e. the one corresponding to the final result.A slightly more compact alternative to truth-tables, used in the Tractatus

itself,7 is to agree once and for all a convention as to the order in which thelines of the truth-table occur, so that a listing of the last column could thensuffice by itself to represent the whole truth-table. The truth-function p | q, forinstance, might then be written as (F T T T)(p, q).

!'.#!'.#!'.# Truth-diagramsAlthough, as we have just seen, Russell and Wittgenstein were both familiarwith the idea of truth-tables by October 1913, there are none in the Notes.

74.442.

Truth-diagrams )")

Instead Wittgenstein made use of another notation for representing truth-functions which, unlike the truth-table, is undoubtedly of his own devising.It works as follows. First write ‘T’ and ‘F’ (what Wittgenstein called the ‘innerpoles’) next to letters representing each argument place of the function. Thenwrite one ‘T’ and one ‘F’ on the diagram (what Wittgenstein called the ‘outerpoles’). Now add a series of lines connecting each outer pole with the setsof inner poles which, when used as arguments for the function, produce thatpole as value. (Wittgenstein used the letters ‘a’ and ‘b’ instead of ‘T’ and ‘F’,but his reasons for doing so need not detain us at the moment.)Here is an example.

T p F T q F

F

T

!""""""""""""""""#""""""""""""""""$ !"""""""""""#"""""""""""$

! " " " " " " " " " " " # " " " " " " " " " " " $! " " " " " " " " # " " " " " " " " $

....................

...................

......................

........................

..........................

!! ..................................

................................

...............................

..............................

............................

...........................

...........................

............................

The outer ‘T’ is connected to the pairing of the two inner ‘T’ poles; the outer‘F’ is connected to the other three pairs. So this diagram represents the samefunction as the following truth-table.

p qT T TT F FF T FF F F

In other words, it represents the conjunction function p q. In the same waythe following example represents the function which takes the value T just incase its arguments are both T or both F, i.e. the function p " q.

T p F T q F

T

F

! " " " " " " " " " " " " " " " " # " " " " " " " " " " " " " " " " $

!"""""""""""#"""""""""""$ !"""""""""""#"""""""""""$

! " " " " " " " " # " " " " " " " " $

""###

..............

..........

....

..................

.....................

................................

..........

.....

There is evidently no significance in the particular spatial arrangement chosenfor the diagram: all that matters is which outer poles are connected to whichsets of inner poles. For instance, we could write the diagram for!p as follows.

)"! Truth-functions

T p F

F

T

$$

$$

But it would amount to the same to write this linearly as F % T p F % T, orindeed as T% F p T% F. In the same way the diagram for p is just T p F, orindeed F p T. As Wittgenstein said in a letter to Russell, ‘The proposition phas two poles and it does not matter a hang where they stand.’8

It is also permitted, of course, for the same argument to occur more thanonce in a truth-function. The diagram for p" p, for instance, is obtained fromthe diagram for p " q above by substituting p for q.

T p F T p F

T

F

! " " " " " " " " " " " " " " " " # " " " " " " " " " " " " " " " " $

!"""""""""""#"""""""""""$ !"""""""""""#"""""""""""$

! " " " " " " " " # " " " " " " " " $

""###

..............

..........

....

..................

.....................

................................

...............

Now p " p is an example of what Wittgenstein would very soon after he hadwritten the Notes begin to call a tautologous function, i.e. a function which takesthe value T for all combinations of arguments. The fact that the functionis tautologous can be read off from the truth-diagram once we have learntto ignore impossible links—connections between an outer pole and oppositepoles of the same atomic proposition. In the diagram for p " p just given,for instance, there are two impossible links from the outer F-pole to both Tand F poles of p. If we ignore these impossible links, we observe that the onlyremaining links are to the outer T-pole, and so the function is tautologous.A recurring theme ofWittgenstein’s letters fromNorway is Russell’s evident

struggle to understand his truth-diagram method. Certainly what Wittgen-stein says in the Notes themselves could hardly count as a sufficient explanationon its own. And his response to Russell’s requests for clarification might bethought to lack a certain sympathy.

It distresses me that you did not understand the rule dealing with signs in my last letterbecause it bores me BEYOND WORDSBEYOND WORDSBEYOND WORDS to explain it. If you thought about it for a bit youcould discover it for yourself! . . . I beg you to think about these matters for yourself: itis INTOLERABLEINTOLERABLEINTOLERABLE for me to repeat a written explanation which even the first time I gaveonly with the utmost repugnance.9

(It was at about this time10 that he told Moore he thought Russell was past it.)Truth-diagrams can be composed in a manner that replicates the effect of

composing the truth-functions they represent. It cannot have helped Russell8LW to BR, [Nov.] 1913 (CL, no. 28). 9[Nov. or Dec. 1913] (CL, no. 32). 1019 Nov. 1913.

Comparison )"#

in his efforts to understand the method that Wittgenstein did not give him anyexamples of compound diagrams in the Notes. He would probably have foundit helpful to be told, for instance, that we can compose the diagrams for p qand p " q to obtain the following diagram for p " (p q).

T p F T q F

F

T

!""""""""""""""""#""""""""""""""""$ !"""""""""""#"""""""""""$

! " " " " " " " " " " " # " " " " " " " " " " " $! " " " " " " " " # " " " " " " " " $

....................

....................

.......................

.........................

............................

!! ...................................

.................................

.................................

.................................

.

...............................

.............................

............................

..........................FpT

! " " " " " " " " " "" " " " " " #

" " " " " " " " " "" " " " " " $

! " " " " " " " " " "" " # " " " " " " " " " "

" " $

!""""""""""""""""#

""""""""""""""""$

!""""""""""""#

""""""""""""$

T

F

.

..................................................

................................................

...............................................

.............................................

...........................................

. ..........................................................................

...................................................................

...............................

%%%%%%%

.......................

...................

......

.........................

..........................

We noted earlier that all that matters in settling the identity of a truth-function is the output for various possible inputs, and that this is represented inthe corresponding truth-table by means of the main column. When the truth-function is represented by means of one of Wittgenstein’s truth-diagrams, theanalogous point is that all that matters is which sets of inner poles are con-nected to which of the outer poles: intermediate poles drop out of the pictureas irrelevant. Or, as Wittgenstein put it, ‘In this notation all that matters is thecorrelation of the outside poles to the poles of the atomic propositions.’11 And,he ought to have added, we need as before to ignore impossible links. Whenwe do that, we see, for instance, that the diagram for p " (p q) just given isequivalent to the following diagram for p ! q.

T p F T q F

T

F

!"""""""""""#"""""""""""$ !""""""""#""""""""$

! " " " " " " " " " " " # " " " " " " " " " " " $! " " " " " " " " " " " " " " " # " " " " " " " " " " " " " " " $

.

...........................

..........................

........................

......................

....................

&&

As we shall see in chapter 20, Wittgenstein came to hope (wrongly) thatthe truth-diagram notation could be extended so as to represent quantifiersas well as truth-functions. But that idea emerges only at the very end of theCambridge version of the Notes; in the Birmingham Notes the method is usedonly for truth-functions.

!'.$!'.$!'.$ ComparisonSome time after the Notes (perhaps when he realized that it does not providea general method of the kind he sought for symbolizing quantified sentences)Wittgenstein put the diagram method aside. It makes no appearance in the

11C31.

)"$ Truth-functions

wartime notebooks or in the Prototractatus. But he did not forget it completely:an account of the method is one of the few things he added to the Tractatustypescript while he was a prisoner of war after the 1918 armistice. In theaccount he gives there12 he offers as an illustration the diagram for p!q above.He also works through an example to show how truth-diagrams can be usedto decide whether a proposition is a tautology. The case he discusses is theobviously tautologous proposition !(q !q), whose truth-diagram he piecestogether from the diagrams for conjunction and negation shown earlier.

T q F

F

T

$$

$$

T q F!"""""""""""#"""""""""""$

!"""""""""""""""""#"""""""""""""""""$

F

T

###

.

....................................

..................................

...............................

...............................

...............................

...............................

...............................

! """""""""""# """""""""""$

! " " " " " " " " " " " " " " " " # " " " " " " " " " " " " " " " " $

T

F

We can see from the diagram that!(q !q) is a tautology because the only linkbetween the outer F pole and the inner poles is impossible.In theTractatusWittgenstein bravely described this as an ‘intuitivemethod’13

for recognizing a tautology. It is worth comparing with the nowadays morefamiliar method of showing that !(q !q) is a tautology by means of the fol-lowing truth-table.

q !(q !q)T T FFF T FT

Theoretically, of course, there is nothing to choose between truth-diagramsand truth-tables, since the two methods are obviously equivalent. In prac-tice, on the other hand, truth-diagrams are at a significant disadvantage totruth-tables: even in simple cases they are hardly faster to write, and in morecomplicated cases the diagram tends quickly to become a tangled mess.

126.1203. 13Ibid.

Chapter !(!(!(

Truth-operations

From a single proposition pwe form its negation not-p. From two propositionsp and qwe form various further propositions, e.g. p or q, p and q, etc.; similarly(though less often) with three or more propositions. For each such method offorming propositions it is common to speak, as we frequently did in the lastchapter, of a certain truth-function as corresponding to it. We speak, for instance,of ‘the truth-table for negation’ or ‘the truth-table for the Sheffer stroke’. Onthe one hand, then, there is a function which takes senses of propositions asinputs and returns the sense of a proposition as output. (Later, most notably inthe Tractatus itself,1 Wittgenstein called such functions operations, and althoughthis is not the term he used in the Notes, it will be convenient to adopt it here.)On the other hand, there is a truth-function, i.e. a function taking some finitenumber of truth-values as arguments and giving a single truth-value as value,which is representable by a truth-table or truth-diagram. What I want todiscuss in this chapter is the relationship between the two.

!(.!!(.!!(.! The problemFor Frege, of course, this issue hardly arises, since he took sentences to benames of truth-values and could therefore simply combine them using signsfor the truth-functions. He could regard the expression ‘p v q’, for instance,as naming the output which results if we apply the truth-function v with thetruth-values named by ‘p’ and ‘q’ as inputs. On his account, it seems, thetruth-operation just is the truth-function.For Wittgenstein, though, this beguilingly simple account was not avail-

able, since he rejected the conception of sentences as names of truth-valueson which it depends. There is thus a mismatch in type between sentences andthe sorts of items that can fill the argument places of a truth-function. Andeven if that mismatch could be corrected by some procedure that convertedsentences into names, there would remain a further, more serious mismatchat the output level: the output of a truth-function is a truth-value, and thepoint of Wittgenstein’s objection to Frege’s account of sentences as names of

15.23.

)"" Truth-operations

truth-values was precisely that there is no general route back from there to theproposition expressing what it is that has that truth-value. Simply putting anassertion sign in front of it, or tagging ‘is true’ on the end, will not do the trick.In other words, if it is a mistake to think of ‘p’ as a name of a truth-value, itmust equally be a mistake to regard ‘!p’ as such.2

What is uncontroversial, though, is that some operations, those we mightcall ‘truth-operations’, give rise to truth-functions. Examples are the opera-tions of disjunction, conjunction, and negation. We have seen, on the otherhand, that such truth-operations cannot be regarded as simply being the sameas the corresponding truth-functions, or as obtained by composing them withsome suitable connecting devices. For all we have said so far, then, one mightentertain a doubt as to whether the truth-operation is obtainable from thetruth-function at all.One reason for doubt would be if the relationship between a truth-operation

and its corresponding truth-function were not one–one. If, for instance, weindividuated operations in such a way that p | q counted as different from!p v !q, then unless some reason could be given for privileging one of theseoperations over the other, neither of them would be obtainable from the cor-responding truth-function.That is not the point at issue here, however. Wittgenstein individuated

operations in just such a way as to ensure that the correspondence is one–one. Moreover, he evidently did think that it is possible to derive the truth-operation from the truth-function. He mentioned with evident approval, forexample, Frege’s truth-functional explanations of ‘not p’ and ‘if p then q’.3

!(."!(."!(." The solution

How, then, do we obtain the sense of the output of a truth-operation from thesense of the input propositions? Let us consider as an example the case of theconnective ‘or’. In order to make the connection between the truth-functionand the truth-operation, we first define

Val(p) =

%T if ‘p’ is trueF if ‘p’ is false

Our task is now to give a sense to expressions of the form ‘p or q’ in such away that we always have

Val(p or q) = Val(p) vVal(q).

2Cf. 4.431. 3B22.

The solution )"&

In order for this equation to hold, we must have

‘p or q’ is true " Val(p or q) = T

" Val(p) vVal(q) = T

" Val(p) = T or Val(q) = T

" ‘p’ is true or ‘q’ is true

We can use this equivalence to derive an account of the sense of any proposi-tion in which ‘or’ occurs. For instance,

‘aRb or cS d’ is true " ‘aRb’ is true or ‘cS d’ is true" the facts about the meanings of ‘a’ and ‘b’ ac-

cord with the sense of ‘xRy’ or the facts about themeanings of ‘c’ and ‘d’ accord with the sense of‘xS y’.

This account derives from the truth-function since it is this that tells us thecondition for propositions with the structure ‘aRb or cS d’ to be true.The key step in this account of disjunction is the requirement that

Val(p or q) = Val(p) vVal(q).

It is this that enables us to derive the equivalence

‘p or q’ is true " ‘p’ is true or ‘q’ is true,

and hence gives us a way of deriving the sense of ‘p or q’ from the senses of pand q. Can this be generalized to any truth-function? If f is an n-ary truth-function, can we introduce a corresponding n-ary truth-operation f ! by meansof the specification that for any propositions p1, p2, . . . , pn

Val( f !(p1, p2, . . . , pn)) = f (Val(p1),Val(p2), . . . ,Val(pn))?

What our discussion has shown is that simply writing down this equation isnot itself enough to secure what we want. What we need to show is that theequation suffices to give a sense to f !(p1, p2, . . . , pn) whenever p1, p2, . . . , pn

have senses. There is of course no great problem about doing this, but theprocedure has a feature that is worth noting. If we mimic the reasoning wegave earlier for the case of disjunction, we obtain easily enough

‘ f !(p1, p2, . . . , pn)’ is true " Val( f !(p1, p2, . . . , pn)) = T

" f (Val(p1),Val(p2), . . . ,Val(pn)) = T

" . . .

)"' Truth-operations

But what do we write next? We want to convert this last line into a conditiondepending on which of ‘p1’, ‘p2’, . . . , ‘pn’ are true. We could do that, of course,if we simply assumed the availability (‘in the metalanguage’, as we would saynowadays) of the truth-operation f ! whose sense we are trying to secure. Ifwe do not assume that, however, we have to inspect the function f in order todetermine just what the condition we want is. In other words, the method forconversion, although it depends uniformly on f , does not depend schemati-cally on it. This last point is of some significance since it shows that the waythat ‘p’ and ‘q’ occur in ‘p or q’ is fundamentally different from the way that‘or’ does. This difference is what makes it permissible to quantify over theformer positions in the proposition but not over the latter.

!(.#!(.#!(.# DualityOne further point is worth noting about the relationship between truth-func-tions and truth-operations. What we have been discussing is a means by whichwe can, for any truth-function f , find an n-ary truth-operation f ! such that

Val( f !(p1, p2, . . . , pn)) = f (Val(p1),Val(p2), . . . ,Val(pn)),

where Val is a function defined so that

Val(p) =

%T if ‘p’ is trueF if ‘p’ is false.

What we should note is that the identity of f ! depends directly on the choiceof the function Val. If, for instance, we used instead the function

Val"(p) =

%T if ‘p’ is falseF if ‘p’ is true

then we could certainly by an analogous procedure find an n-ary truth-oper-ation f !! such that

Val"( f !!(p1, p2, . . . , pn)) = f (Val"(p1),Val"(p2), . . . ,Val"(pn)),

but it would not be the same operation as f !. Since Val"(p) = Val(!p), itfollows that

‘! f !!(p1, p2, . . . , pn)’ is true " ‘ f !!(p1, p2, . . . , pn)’ is false

" Val"( f !!(p1, p2, . . . , pn)) = T

" f (Val"(p1), . . . ,Val"(pn)) = T

" f (Val(!p1), . . . ,Val(!pn)) = T

" ‘ f !(!p1,!p2, . . . ,!pn)’ is true.

Duality )"(

So the relationship between f ! and f !! is that ! f !!(p1, p2, . . . , pn) has the samesense as f !(!p1,!p2, . . . ,!pn). In other words, f !! is the same as the op-eration ( f ")! derived from the dual f " of the truth-function f (i.e. the truth-function whose truth-table is obtained from that of f by swapping all the oc-currences of T and F).The point to note is that whether we associate the truth-function f with

f ! or with f !! depends entirely on whether we decide to adopt Val or Val".Each of these functions establishes a link between the concept of truth and theobjects T and F, but it is a matter of convention which way round we establishthat link. Of course, to any English speaker our choice of the letters ‘T’ and‘F’ as names of the two objects indicates which convention we intend to adopt:once we have made that choice, to adopt the opposite convention would bewilfully confusing. Nonetheless, it is worth stressing that the other convention,although notationally unhelpful, would not be in any deep way mistaken. Ourchoice of name is not a property of the object itself, and no object has anyessential connection with truth (or, for that matter, with falsity).

Chapter !)!)!)

Molecular propositions

As we saw in chapter 9, it is for Wittgenstein not the complex that does thesymbolizing in a proposition, but rather a particular fact about the complex,namely that fact about it which ensures that the proposition expresses thesense that it does. It is this fact—the symbolizing fact—that we see in thecomplex when we read it as expressing the proposition. The question there-fore becomes urgent of trying to say what this symbolizing fact is. It is not theidentity of the sentences that is in question now. The sentence ‘p or q’, forinstance, is readily enough described as a complex consisting of the sentence‘p’ followed by the word ‘or’ and then the sentence ‘q’. What we want to donow is to identify what the fact is that we see exemplified in this complex whenwe read it as expressing a proposition.

!).!!).!!).! TerminologyIn the Cambridge Notes Wittgenstein did not explicitly define what he meantby a ‘molecular’ proposition, but it is clear from the text that he meant anyproposition obtainable from elementary propositions by the application of afinite number of truth-operations.1 Confusion is possible, however, becauseRussell’s own usage around this time was different. What Wittgenstein calledelementary propositions Russell preferred to call ‘atomic’; the propositionsWittgenstein called molecular Russell called ‘elementary’; and Russell used‘molecular’ to mean non-atomic.There are traces, both in the Cambridge Notes and in the correspondence

they conducted thereafter, of a running battle Wittgenstein and Russell werefighting over terminology. Russell did not object to new words for new con-cepts, but he did (reasonably enough, perhaps) put pressure onWittgenstein tokeep to old words for old concepts—to use established terminology where pos-sible. ‘What you call ab-functions are what the Principia calls “truth-functions”,’he observed at one point. ‘I don’t see why you shouldn’t stick to the name“truth-functions”.’2 And in the Cambridge Notes he briefly managed to getWittgenstein to call his elementary propositions by Russell’s term ‘atomic’.3

1C11; C14; C31. 2CL, no. 29. 3C31; C33.

Which fact? )&)

This terminological battle resulted in a certain amount of confusion, notjust for the modern reader but also, it seems, for Russell. In his translation ofthe Birmingham Notes4 he used the term ‘complex proposition’ for what Witt-genstein was in English calling a ‘molecular proposition’. (The word Russellwas translating was probably ‘zusammengesetzt’.) And at one point he seemsto have forgotten that he had encouraged Wittgenstein to call Elementarsätze‘atomic propositions’, translating the term instead as ‘simple propositions’.5

Wittgenstein’s letters to Russell during the autumn of 1913 show recurringevidence that Russell was still struggling to grasp his terminology: in one ofthem, for instance, we find Wittgenstein helpfully glossing ‘zusammengesetzteSätze’ by using Russell’s term ‘elementary propositions’.6

ThusWittgenstein’s claim that ‘every possible complex proposition is a sim-ple ab-function of simple propositions’7 might have been clearer if Russell hadtranslated it, ‘Every possible molecular proposition is a simple truth-functionof elementary propositions.’ And Wittgenstein was here only reminding usof what he meant by a molecular proposition. There is no trace of Wittgen-stein’s later claim in the Tractatus that all propositions are (possibly infinite)truth-functions of elementary propositions, and indeed various remarks in theNotes actively exclude it.

!)."!)."!)." Which fact?In chapter 18 I explained a uniform procedure whereby each truth-functionf gives rise to a corresponding truth-operation f !. If we have a notation fordescribing truth-functions, we can convert it into a notation for the corre-sponding truth-operations. So each of the methods of representing truth-functions that we described in chapter 17 corresponds to a method of repre-senting molecular propositions. I mentioned earlier Wittgenstein’s hope thathe might be able one day to find a ‘perfect notation’ in which there is a one–one correspondence between propositional signs and propositions. One of theattractions he saw in the truth-diagram notation is that although it is not quitesuch a notation, at least it is closer to being one than Russell’s or Frege’s is. Forif a proposition is written in truth-diagram notation, it is at any rate possibleby a sort of selective inattention to detail—by shutting one eye, so to speak—to see in the complex the relevant (i.e. symbolizing) fact. The diagram for!!p, for instance, is T–F–T p F–T–F. It is in this case a relatively easy matterto ignore the intermediate detail so as to see here the same symbolizing fact asin T pF, and to conclude from this that !!p = p.But even once we have devised an appropriate notation for expressing

molecular propositions, how, in general, are we to discern the symbolizing fact

4B31. 5Ibid. 6[Nov. or Dec. 1913] (CL, no. 32). 7B31.

)&! Molecular propositions

in it? Let us consider first the apparently trivial case where the truth-functionin question is the identity function. The truth-table method, for instance, givesus the rather laborious expression

pT TF F

for the proposition p. The abbreviated truth-table method gives us the rathermore manageable expression (T,F)(p). Wittgenstein’s truth-diagram methodgives us T p F, or, if we follow his notational eccentricity of using ‘a’ and ‘b’to indicate the two poles, a p b. But for each of these we can ask the samequestion: what is it in the expression that symbolizes?In the case of the last mentioned notation, Wittgenstein himself offered an

explanation. What symbolizes in the expression a p b, he said, is that ‘a ison the left of p and b on the right of p’. The puzzle that this explanationgenerates, however, is to explain the manner in which ‘p’ occurs in here. If‘p’ is a complex, e.g. the sentence expressing the proposition, then accordingto Wittgenstein’s own doctrines it does not yet symbolize anything, since whatsymbolizes in a proposition is a fact, not a complex. What does symbolize isthe fact that the pole letters ‘a’ and ‘b’ occur on either side of the complex.So the effect of Wittgenstein’s device of attaching ‘a’ and ‘b’ to p is just thesame as that of writing ‘ “p” is true’, i.e. it turns the complex into somethingexpressing a proposition. But of course Wittgenstein knew quite well that thisis incoherent. ‘The verb of a proposition is not “is true” or “is false”, but whatis true must already contain the verb.’8

The alternative is to say that ‘p’ occurs in ‘a p b’ not as a complex but as afact. But nowWittgenstein’s explanation of what symbolizes in ‘a p b’, namelythat ‘a’ is on the left of p and b on the right of p, makes no sense whatever. Afact is not the sort of thing that can stand in a spatial relation to anything else.If Wittgenstein was right to say that ‘what is true must already contain the

verb’, he would have to say that ‘p’ already symbolizes the proposition. Themost Wittgenstein could hope to achieve by writing ‘a p b’ would in that casebe to symbolize the same proposition in a rather more complicated manner, injust the same way as writing ‘It is true that p’ does. But that would be to render‘a’ and ‘b’ wholly redundant. The only alternative, it seems, would be to saythat ‘p’ on its own does not express a proposition but is some sort of incompletesymbol. This, of course, looks worryingly similar to Russell’s misbegottendistinction between asserted and unasserted occurrences of the proposition, adistinction which in §10.2 we presented Wittgenstein as rejecting.

8B10; cf. 4.063.

Poles )&#

!).#!).#!).# PolesOne thing which the three methods of representing truth-functions consid-ered in chapter 17 have in common is that as we have described them so far,they all use the letters ‘T’ and ‘F’ in a uniform manner. But what manner isthat? When we introduced them as devices for notating truth-functions, theanswer was clear: truth-functions are functions whose arguments and valuesare objects for which we use the two letters ‘T’ and ‘F’ as names, and the roleof these letters in the notation corresponds directly to their use as proxies forthese objects. But when the signs are intended to refer not to truth-functionsbut to truth-operations, that direct connection drops away. As we noted in§18.3, the connection of the objects we have called ‘T’ and ‘F’ with the con-cept of truth is one we have to establish by convention. The role of the letters‘T’ and ‘F’ is now only that of a mnemonic to remind us which of the twoconventions we have chosen.In chapter 17 I talked as if Wittgenstein used ‘T’ and ‘F’ (or ‘W’ and ‘F’ in

German) as signs for the two poles of a proposition. As noted in the last sec-tion, though, he did not: he used the letters ‘a’ and ‘b’ instead (and correspond-ingly called operations that can be represented by means of truth-diagrams‘ab-functions’). I omitted mention of this before in order to stress that how hechose to label the two poles is a distinct issue from the relative merits of truth-diagrams and truth-tables. Whatever point Wittgenstein wanted to make byusing the letters ‘a’ and ‘b’ he could have made just as well by substitutingthem for ‘T’ and ‘F’ in truth-tables (and, I suppose, calling them ‘poles’ ratherthan ‘truth-values’).This use of ‘a’ and ‘b’ for the two poles seems to have been a matter of

some importance to Wittgenstein. The text of the Notes bears witness to an-other battle he fought on this point with Russell, who found it hard enough towork out fromWittgenstein’s explanations how truth-diagrams were supposedto work without the additional difficulty of having to remember which letterstood for truth. In the dictation made in his presence Wittgenstein briefly (atRussell’s insistence, presumably) used ‘W’ and ‘F’ instead,9 but a couple ofparagraphs later he reverted to ‘a’ and ‘b’.10 Moreover, Wittgenstein main-tained the practice of using ‘a’ and ‘b’ in the notes he dictated to Moore inNorway and continued to call the truth-functions ab-functions in his first twowartime notebooks.11

Why, then, was Wittgenstein so intransigent? What was wrong with Rus-sell’s suggestion of using ‘W’ and ‘F’ instead of ‘a’ and ‘b’? One reason that hassometimes been suggested is that Wittgenstein wanted to point up the dualitybetween truth and falsity already noted: by swapping the roles of ‘a’ and ‘b’ we

9C31. 10C33. 11E.g. NB, 17 Dec. 1914.

)&$ Molecular propositions

transform a proposition into its dual. Now Wittgenstein was plainly aware ofthis duality—he discussed it explicitly in the Notes dictated to Moore—and thereis little doubt that he was intrigued by it: the bipolarity of the propositionis, after all, an attempt to express this symmetry between a proposition andits negation. However, the idea of swapping the roles of ‘a’ and ‘b’ over canhardly be the whole explanation for Wittgenstein’s insistence on using theseletters, since it is as straightforward to make this point using ‘W’ and ‘F’ as it isusing ‘a’ and ‘b’. More plausible, I think, is the concern that using ‘W’ and ‘F’might reify truth and falsity: it might encourage us to think that truth and fal-sity are logical objects.12 In the Notes dictated to MooreWittgenstein emphasizedthe idea that ‘a’ and ‘b’ are not names by describing them as ‘scratches’. WhatWittgenstein says in the Notes is that ‘a proposition has two poles, correspond-ing to the case of its truth and the case of its falsehood’.13 He seems to haveseen the role of ‘a’ and ‘b’ as being not to name truth and falsity, but somehowto make visible the fact-like nature of what it is that makes p symbolize. Theyare there only to serve as hooks against which p can stand in a fact-like rela-tion. Indeed, it is hard to avoid the thought that Wittgenstein intended ‘a p b’to remind us of ‘aRb’: he wanted to suggest that the symbol ‘p’ on its own isincomplete in something like the way that ‘R’ is. How he could possibly havemade this idea consistent with other things he believed can only be a matterof speculation, however.Wittgenstein had abandoned his use of the ab-notation by the time of the

Tractatus: even when he resurrected truth-diagrams there, he used the conven-tional labelling of the poles with ‘W’ and ‘F’. The reason is presumably that inthe Tractatus truth-diagrams are presented only as an ‘intuitive method’ whichenables us to ‘recognize a tautology as such’,14 not as a notation for expressingpropositions. By that time he had abandoned (for reasons to be explained in§20.3) the idea that a truth-diagram is somehow a transparent medium forconveying the proposition it expresses. Perhaps, too, he had realized the futil-ity of attempting to make p more fact-like by putting scratches on either sideof it.

!).$!).$!).$ The inputsRather similar considerations apply when we ask how propositions occur incompound propositions. Suppose first that the occurrences of ‘p’ in ‘!p’ andof p and q in ‘p ! q’ function as names: they do not name the propositions pand q themselves, perhaps, but one might suppose that they name complexesrelated in some way to p and q. Some such view as this is widespread inRussell’s writings of the period. In Principia, for example, he reads !p as

12Cf. 4.441. 13C13. 146.1203.

The inputs )&%

‘the negation of p’ and p ! q as ‘p implies q’—locutions which require theletters to be replaced by names rather than sentences if they are to make sense.Wittgenstein rejects the view. Indeed he takes it to be obvious (even15 to the‘plain man’) that the logical connectives do not express relations. He does notexpand on why this is obvious, however. One reason he might have had inmind is simply that what we put alongside the logical connectives are sentences,not names. If we think of !p as expressing a property of p, and p ! q arelation between p and q, then the substituands for the letters must be names,as Russell’s way of speaking assumes. So what Wittgenstein may have takento be obvious is that what is negated in !p is exactly what is expressed in p.‘In not-p, p is exactly the same as if it stands alone; this point is absolutelyfundamental.’16 The reason this is ‘obvious to the plain man’ is that it is whatordinary language already suggests.

Ordinary language would not contain the whole propositions if it did not need them:However, e.g., ‘not-p’ may be explained, there must always be a meaning given to thequestion ‘what is denied?’17

In other words, the inputs to the process, as well as the output, must be propo-sitions and hence (symbolizing) facts.Wittgenstein also tells us that ‘propositions, owing to sense, cannot have

properties or relations’.18 If we put these two claims together, we obtain theresult that ‘!’ is not a property and ‘!’ is not a relation. On the other hand,Wittgenstein’s argument for this conclusion is not all that persuasive, sinceit seems to depend rather too much on features of language that might turnout not to be essential. Ordinary language, after all, certainly contains manyfeatures it does not need.A fact, being a sum of component facts (positive or negative) has no internal

structure other than that of the components. Applied in the particular case ofa symbolizing fact (i.e. a proposition) this amounts to saying that a fully per-spicuous expression of such a fact, an expression which fully captures its basicstructure, cannot have any need of a notion of alternative decomposition. Butconsider now a sign for a molecular proposition such as ‘p ! q’. The sym-bolizing fact expressed by this sign is, like any fact, uniquely decomposableinto atomic parts. But the very same symbolizing fact is expressed by the sign‘!pvq’, or by ‘!(p !q)’. Since the same fact may be expressed in these variousdifferent ways, none of them can be a true expression of the structure of thesymbolizing fact they have in common.In the TractatusWittgenstein tried to express this last point by emphasizing

the significance of the fact that our notation for expressing molecular propo-sitions makes essential use of brackets.

15B18. 16C40. 17B21. 18C15.

)&" Molecular propositions

The apparently unimportant fact that the apparent relations like v and ! need brackets—unlike real relations—is of great importance.The use of brackets with these apparent primitive signs shows that these are not

the real primitive signs; and nobody of course would believe that the brackets havemeaning by themselves.19

We now know, of course, that Wittgenstein was wrong about the particu-lar point he was making here: Polish notation provides the same expressiveresources as the methods we discussed earlier, but does not involve any brack-ets. And even if we put that issue aside, there is plainly something fumblinglywrong about what Wittgenstein says. His thought was presumably that thegenuine primitives are in some sense independent of each other and hencedo not interact, but he mislocated this point by aligning it with the distinctissue of unique decomposition, which in the usual notation is what bracketsare designed to ensure.Yet this does not prevent there being something right in the general terri-

tory of Wittgenstein’s remark. There is a distinction to be drawn between aconception of complexity which leaves room for alternative decompositionsof the same complex and one that does not. Wittgenstein was wrong to thinkthat the use of brackets is the characteristic mark of the first of these kinds ofcomplexity, but he was right to stress the importance of the distinction.What is significant here is the contrast between language and the world.

As we noted in §15.3, Wittgenstein’s conception of the world was in a certainsense flat: it can be completely described simply by listing all the positive andnegative facts; neither brackets nor anything else of comparable complexity isrequired. The fact that language is in this respect different is due to the widerrange of possibilities—in particular, disjunctive possibilities—that it aims toexpress.This case exemplifies a more general feature of Wittgenstein’s way of think-

ing. There are several points in the Notes where he seems to have been con-fused about technical matters in logic, or where subsequent developmentshave shown suppositions he made to be wrong. It is curious how often thephilosophical views he was making are not refuted by exposing the technicalerror in the pieces of logic he used to make them. And this suggests ratherstrongly that the technicalities were never for him the real reasons for holdingthe views in question.

195.461.

Chapter "*"*"*

Generality

One of the most obvious gaps in the account of logic we have presented so faris that it says nothing about how quantification operates. This seems to be anissue Wittgenstein had only recently begun to address when he compiled theNotes: it is notable how little he says there about it, and some of what he doessay is contained only in the Cambridge Notes, not the Birmingham ones.Evidently the same steps have to be gone through as in the molecular case.

First there is the question of a perspicuous notation for expressing quantifiedpropositions. Then we need an account of how the sense of such propositionsdepends on the senses of their components. And finally there is the questionof what it is in the chosen notation that symbolizes.

"*.!"*.!"*.! Variables as classes of propositions

If in a proposition we replace all the occurrences of the name ‘a’ with a vari-able ‘x’, we obtain a propositional function. We must not think of ‘x’ hereas performing anything like the role of a variable name, however, for there isno such thing. Perhaps this is the reason why Wittgenstein preferred to callit not a variable but a prototype. But what is a propositional function? Wesaw in chapter 5 that Wittgenstein objected, for impeccable logical reasons,to the primitive idea of the assertion of a propositional function which White-head and Russell introduced in Principia, $9. But he could not simply banishpropositional functions from logic completely: they are too useful. What heneeded to do was to provide an explanation for those uses of them that cannotbe eliminated, most notably their use within quantified expressions such as(x) !(x) or (

E

x) !(x).One of the central ideas of the Tractatus is that we should break down our

explanation of how a quantified expression such as (x) !(x) or (

E

x) !(x) sym-bolizes into two stages: first, the propositional function !(x) determines a classof propositions; then an operator is applied to this class to express that all orsome of the propositions in the class are true. This idea is already visible inthe Notes. ‘What is essential in a correct apparent-variable notation is this: (1)it must mention a type of propositions; (2) it must show which components of

)&' Generality

a proposition of this type are constants.’1 By the time it reached the Tractatusthis remark had been refined into the observation that ‘that which is peculiarto the “symbolism of generality” is firstly, that it refers to a logical prototype,and secondly, that it makes constants prominent’.2

Wittgenstein thus recommended that the propositional function !(x) shouldbe thought of simply as a means of specifying a class of propositions, namely allthose propositions we obtain if we replace the variable x in the function withany legitimate substituand. ‘If we change a constituent a of a proposition !(a)into a variable, then there is a class p{( Ex) !(x) = p}.’3 In much the same way,we could alternatively keep the name ‘a’ constant and replace the remainderof the proposition with a variable. The resulting propositional function wouldspecify a different class, the class of all propositions about a.The conventional way of thinking about generality, repeatedly encouraged

by Russell, focuses on the variable ‘x’, which is thought of as having as its pos-sible substituands a class of objects. Wittgenstein focuses instead on the wholesymbol ‘!(x)’, which he takes as having as its range a class of propositions. Itwould therefore be natural, I suppose, to see a link between Wittgenstein’saccount of the variable and Frege’s context principle, that ‘only in the contextof a proposition do words mean anything’,4 since the context principle maybe thought of as motivating just such a shift of focus from objects to proposi-tions. But although the similarity is suggestive, I do not think it is especiallydeep. Wittgenstein’s treatment of the variable shares with the context prin-ciple the methodological tendency to treat the proposition as primary, but itdoes not contribute, as Frege intended the context principle to do, to the goalof avoiding the temptation ‘to take as the meanings of words mental picturesor acts of the individual mind’.5 Although Wittgenstein repeated the contextprinciple in the Tractatus,6 no doubt in conscious homage to Frege, it did notplay for him there the role it played for Frege in the Grundlagen. What is strik-ing, indeed, is how little the Grundlagen seems to have influenced Wittgensteincompared with Frege’s other writings. For instance, even when Wittgensteindenies the central claim (or, more accurately, presumption) of the Grundlagen,that numbers are self-subsistent objects, he never so much as gestures towardsFrege as a possible target.The deeper resonance of Wittgenstein’s explanation of propositional func-

tions is rather with Frege’s treatment of variables in the Begriffsschrift. ThereFrege takes the proposition as primary not in order to avoid psychologism butin order to avoid the temptation to think of variables as analogous to names.Russell, by contrast, although he had long shared Frege’s understanding ofpropositional functions as obtainable from a proposition by replacing a con-stant with a variable, frequently succumbed to the temptation to regard the

1C49. 25.522. 3B19. 4Gl, §62. 5Gl, Preface. 63.3.

Variables as classes of propositions )&(

variable as a kind of ambiguous name, and hence the propositional functionas ambiguously denoting the various propositions that are its instances.7

Wittgenstein’s idea in a sense reverses the conventional way of conceivingof variables: replacing a constant with a variable serves to make prominentnot what has been removed from the proposition but what remains. The ideathat the role of a propositional function is to determine a class of propositions ishinted at occasionally by Russell. In the Introduction to Principia, for instance,he said that ‘a function is what ambiguously denotes some one of a certain to-tality, namely the values of the function’.8 But Russell did not suggest that thetotality of values is to replace the function. He had tried various devices (mostnotably his substitutional theory of 1908) to eliminate propositional functionsfrom his ontology, but never this one. There is surely something characteristicabout how Wittgenstein uses the device: the idea itself is very simple, and onits own it is hardly deep; yet it brings about a radical reconfiguration of ourway of looking at the notion. Russell had in Principia conceived of the variableas specifying a class of things; now we are asked to conceive of it instead asspecifying a class of propositions.This reconceptualization of quantifier–variable notation has two effects.

One is that by regarding propositional functions only as notations for pickingout classes of propositions it removes any temptation to think of them as a kindof entity. Higher-order quantification is no longer in any sense quantification‘over’ propositional functions. On Wittgenstein’s understanding, a quantifiedexpression which apparently quantifies over propositional functions does notreally do so: a propositional function ‘can only occur in a proposition throughits values’.9 On Russell’s account the expression (!) !!a, for instance, hadbeen explained as asserting that all the predicative functions !!x apply to a;whereas on Wittgenstein’s it is now the logical product of all the propositionsof the form !!a.The second consequence Wittgenstein’s presentation has is that it removes

the possibility that the variable could somehow enable a general proposition toinclude within its range cases which no individual proposition expresses. Thevariable had been the last remnant of Russell’s notion of a denoting concept—a concept whose presence in a proposition was supposed to enable the prop-osition to be about entities with which we are not acquainted. Now, finally,this last remnant was discarded. One effect of thus discarding the variable wehave already seen: it renders incoherent Russell’s pre-1912 project of infer-ring the existence of objects of one kind from our acquaintance with objectsof another. For that project depended on our supposed acquaintance with avariable which ranges over objects with which we are not acquainted.However, this introduces a tension into Wittgenstein’s position. The nota-

tional device of quantification enables me to express a logical product without7E.g. PM, I, 39. 8Ibid. 9PM, I, xiv.

)'* Generality

having to write down all the instances, but are the instances contained in thesymbol? It seems that they must be, since if I do not grasp the instances, Ihave not fully grasped what the proposition says. To someone who recognizesthe symbol in the sign, the proposition (x) f x ! f amust be completely trivial;but it can be trivial, it seems, only to someone who recognizes ‘ fa’ as alreadyoccurring in the symbol ‘(x) f x’. And that, as Frege remarked, is wildly im-plausible.

If I utter a sentence with the grammatical subject ‘all men’, I do not wish to say some-thing about some Central African chief wholly unknown to me. It is thus utterly falsethat I am in any way designating this chief when I use the word ‘man’, or that thischief belongs in any way whatsoever to what the word ‘man’ means. It is likewiseequally false that in such a sentence many judgments are put together by means of thecommon name.10

As an explanation of our power to grasp generalizations Wittgenstein’s appealto the process by which we convert signs into symbols is thus every bit as mys-terious as Russell’s appeal to the variable. The consequences of this tension inWittgenstein’s position for his understanding of the theory of types will occupyus in the next chapter.

"*.""*.""*." NotationSuppose that we accept Wittgenstein’s idea of decomposing a notation forquantification into two stages: in the first, a symbol picks out a class of propo-sitions; in the second, an operator is applied to that class. Plainly this is sofar only programmatic: we need a more detailed account of both parts of theprocess. Since in the Notes Wittgenstein wanted the device of truth-diagramsto play the central role in his explanation of symbolism, what he needed to dowas to extend that method to quantified expressions. The remarks about thisissue at the end of the Summary are among the few things in the CambridgeNotes that are not simply reexpressions of, or glosses on, what is in the Birming-ham version. Perhaps, therefore, the ab-notation for generality is one of thethings he had been working on during his holiday in Norway with Pinsent,and was therefore not contained in the notebooks from which the Birming-ham Notes were drawn. If that is the case, however, one has to say that he hadnot made much progress with it. All he had to offer Russell on the subject11

was the following:

for (x)!x : a$(x)$a!xb$( Ex)$bfor (

E

x)!x : a$( Ex)$a!xb$(x)$b

10‘Kritische Beleuchtung einiger Punkte in E. Schröders Vorlesungen über die Algebra derLogik’, 454. 11C43.

Undecidability )')

This is plainly so half-baked that one suspects he had not really thought aboutthe matter at all and simply made something up on the spot when Russellasked him about it.In one of his letters from Norway Wittgenstein was willing to concede that

the rule for ab poles ‘applies first of all only for what you called elementarypropositions’, (i.e. what Wittgenstein called molecular propositions—truth-functions of atomic propositions). But he insisted that ‘it is easy to see that itmust also apply to all others’.12 Not unreasonably, however, Russell claimednot to understand, and in his next letter Wittgenstein was reduced to patentbluster. ‘I beg you to think about these matters for yourself: it is INTOLERABLEINTOLERABLEINTOLERABLE

for me to repeat a written explanation which even the first time I gave onlywith the utmost repugnance.’13

The fact that Wittgenstein hoped to extend the method of truth-diagramsso that it would encompass a notation for quantification explains, inciden-tally, his otherwise mystifying response to another of Russell’s observationsin the same letter. ‘What you call ab-functions are what the Principia calls“truth-functions”. I don’t see why you shouldn’t stick to the name “truth-functions”.’14 Wittgenstein’s reply to this was, ‘Whether ab-functions and yourtruth-functions are the same cannot yet be decided.’ The explanation for thereply can only be that Wittgenstein was now using ‘ab-functions’ to meanwhatever functions can be expressed using the method of truth-diagrams. Ifhe had succeeded in extending this method as he hoped, ab-functions, in thesense in which Wittgenstein was now using the term, would not have coin-cided with Russellian truth-functions but would have included those express-ible using quantifier-variable notation. It should be stressed, though, that thiswas a change of terminology on Wittgenstein’s part. In relation to the Notes onLogic, Russell’s observation is perfectly correct: Wittgenstein does indeed use‘ab-function’ there as a synonym for what the Principia calls ‘truth-functions’.

"*.#"*.#"*.# UndecidabilityThe desperation of Wittgenstein’s attempts to avoid the question Russell waspressing is patent. As is so often the case where instances of psychologicalavoidance are concerned, there was a good reason for Wittgenstein’s blus-ter: there cannot be a notation for quantification which in every case makesmanifest which proposition is being symbolized. There might, that is to say,have been some hope that Wittgenstein could extend the truth-diagram no-tation to cover some cases of generality, but it was forlorn for him to hopethat he could extend it to cover all such cases. The reason is the Church–Turing undecidability theorem, which tells us that once we have a notation to

12[Nov. 1913] (CL, no. 30). 13[Nov. or Dec. 1913] (CL, no. 32). 1425 Oct. 1913.

)'! Generality

express multiple generality, it is not mechanically decidable which sentencesare logically valid. It was precisely the purpose of the truth-diagram methodto make it immediately apparent which proposition a particular diagram ex-presses; and, in particular, to make it immediately apparent which diagramsexpress logically valid sentences and which do not. So any adequate notationfor generality will inevitably leave us some way short of this goal.Moreover, what the mechanical undecidability of first-order logic reveals is

something more than merely the failure of Wittgenstein’s favoured notation.It reveals also a curious feature of Wittgenstein’s conception of propositionsas symbols fully revealing what they express. It is perfectly possible to beconfronted with a logically valid sentence in some logical notation (Russell’s,for instance) whose validity we have not yet grasped. Until we do grasp this,we have not yet recognized the symbolizing fact that is being presented to us,and hence do not understand the proposition being expressed. I take it, forinstance, that most of us cannot instantly spot whether

(x)( f x ! gx) !!( Ex)( f x !gx) (1)

is logically valid. Yet if we were asked whether we understand it, we would beinclined to treat that as a question only about our grasp of Russell’s symbolism.On Wittgenstein’s account that is incorrect: in order to understand the prop-osition we have to understand what it says, and that includes understandingwhether it is logically valid or not.On this view the question I might set in a logic exam asking whether (1)

is logically valid is really a question about signs: what I am testing is whichproposition the sign (1) expresses, not whether that proposition is valid. WhatI want to find out is thus in a certain sense inherently linguistic. If I try toask the question within the language, what I am asking is trivial to anyone whounderstands the question. Of course, that is not to say that I could not achievethe aims of the logic exam by asking the question within the language: by an-swering it correctly the student could demonstrate precisely the understandingwhich I want to test. Nonetheless, it would remain the case that what I amtesting is not the same as what I am asking. (In just the same sense the exam-iner in a French oral asks questions to test not whether the candidates knowthe answers but whether they understand the questions.)The underlying point is that there is a difference in complexity between

two tasks, that of recognizing the sign and that of understanding the symbol.What the undecidability result shows is that in the case of polyadic quantifiedlogic that difference is stark: the first task is mechanically decidable, in thecase of any logically adequate formal language, whereas the second is not. Inthe simpler case of propositional logic, on the other hand, there remains a dif-ference of complexity, but it is only one of degree: both tasks are mechanically

Undecidability )'#

decidable, but the second is more complex than the first. The mechanical un-decidability of polyadic logic is what makes it the case that there cannot be awholly transparent notation which reveals in every case the logical structureof what it expresses: if there were, logic exams would be easier than they are.

Chapter "!"!"!

Resolving the paradoxes

Our discussion of Wittgenstein’s account of quantification in the last chapterleft unaddressed what sort of theory of types it commits us to. To answerthis question we need to look in more detail at the motivation for believingin logical types at all. That motivation derives from Russell’s paradox, theproblem which had originally attracted Wittgenstein’s notice back in 1909.

"!.!"!.!"!.! Russell’s theory of types

The paradox which Russell discovered in 1902 concerns classes: it is contra-dictory to suppose that there is a class of all those classes which do not belongto themselves. Russell’s solution to this paradox made use of the method ofincomplete symbols which he discovered in 1905. In his theory terms appar-ently referring to classes are incomplete symbols which disappear on analysis.The solution was to be that any sentence in which the term ‘the class of allclasses which do not belong to themselves’ occurs would resist rewriting ac-cording to the translation rules and would therefore turn out not to express aproposition at all.This solution does not just drop out all by itself, however. It is easy enough

to formulate rewriting rules for eliminating class terms (so that, for instance, aproposition that appears to be about the class of all men turns out really to beabout the property of manhood), but if that is all we do, we simply transfer thefocus of attention to the corresponding paradox for propositional functions,which involves the function which applies to just those propositional functionswhich do not apply to themselves. In order to avoid such paradoxes as this,Russell found it necessary to stratify propositional functions into types.The primary method of stratification is to categorize a propositional func-

tion according to the types of its real variables. If this is the only method ofstratification we adopt, the theory of types is said to be ‘simple’. The dis-tinctive feature of a simple theory of types is that since the only principle ofclassification is based on the types of the real variables, and propositions donot contain real variables, all propositions belong to a single type. As Russellwas aware, the simple theory of types suffices to eliminate his original para-dox. When we attempt to define the class of all classes that do not belong to

Russell’s theory of types )'%

themselves, we are stymied. But there are other paradoxes which the simpletheory does not solve. One of the best known is Grelling’s:1 if we define theadjective ‘heterological’ by the prescription that it applies to just those adjec-tives which do not apply to themselves, then the word ‘heterological’ itself isheterological if and only if it is not. So far, this is a paradox in ordinary lan-guage: the problem Russell had to address is that it can also be formalized ina simple theory of types. Let us write ‘Het x’ for

(

E

!)(x means !z !!x).

Now‘Het’ means Het z,

and so

Het(‘Het’)" (

E

!)(‘Het’ means !z !!(‘Het’))

"!Het(‘Het’),

since ‘Het’ cannot mean two different things.In order to solve paradoxes such as this, Russell adopted a ‘ramified’ theory

of types, so called because propositional functions (and hence propositions) arecategorized not only according to the types of the real variables they containbut also according to the apparent variables. The derivation of a contradictionfrom the definition of Het is stymied because the variable ! in this definitionis permitted to range only over propositional functions of a single order. Wecan, for instance, define a function by

(

E

!)(x means !!z !!!x),

where the variable ! ranges only over predicative functions, but the functionthus defined is not predicative, since it contains a quantification over pred-icative functions. So in the ramified theory, unlike the simple theory, thefunction does not fall in the range of the variable ! occurring in its definition.This solves Grelling’s paradox.The philosophical principle which Russell adduced in support of the the-

ory of types was what he called the vicious circle principle: whatever involves allof a collection cannot be one of the collection. But as a philosophical basisfor the theory this principle has two weaknesses. First, Russell does not re-ally explain why it should be true. In order for it to seem problematic thatsomething should belong to a collection which it involves we must presum-ably conceive of the notion of involvement as contributing in some way to theconstitution of the entity in question, so as to make the consequent circularityvicious. After all, no one thinks that the tallest man in the room is somehow

1Grelling and Nelson, ‘Bemerkungen zu den Paradoxien von Russell und Burali-Forti’.

)'" Resolving the paradoxes

compromised by the fact that he may be singled out in terms of a collectionto which he belongs. The manner in which we here pick him out, althoughimpredicative, is harmlessly so, because it is not an internal property of him.By formulating the vicious circle principle quite generally Russell divorces itfrom the surroundings in which it might most naturally be justified.Second, which theory of types the vicious circle principle justifies, simple

or ramified, depends on how we understand the notion of involvement inquestion. If every propositional function involves the real variables in it, thatgrounds classifying them into levels; if every propositional function also in-volves the apparent variables occurring in it, that justifies ramifying the hi-erarchy so as to distinguish propositional functions of different orders. Whyshould we say that the propositional function !x involves its instance !a, andnot conversely that !a involves !x? Russell says that this is ‘sufficiently obvi-ous, in any particular case’, because the proposition ‘Socrates is human’, forinstance, ‘can be perfectly apprehended without regarding it as a value of thefunction “x is human” ’. As Gödel later observed,2 Russell’s statements of thevicious circle principle waver between ‘involves’, ‘is definable only in termsof’, and ‘presupposes’. If we hoped that it would arbitrate between the simpleand ramified theories, we should first have to settle which kind of involvementis relevant. The fact that Russell was so vague about this central issue suggeststhat perhaps he did not really envisage it as playing so decisive a role. Whatwas driving his decision to adopt the ramified theory was really the need toavoid Grelling’s paradox, not the vicious circle principle.

"!.""!.""!." Wittgenstein’s vicious circle principleWittgenstein seems to have thought up several solutions to Russell’s paradoxat various times. We have already noted that he wrote to Jourdain with onewhile he was at Manchester in 1909. Then in 1912 Pinsent wrote in his diarythat Wittgenstein had explained to him

a new solution he has discovered to a problem {in the most fundamental Symboliclogic} which was puzzling him greatly in Iceland, and to which he made a somewhatmakeshift solution then. His latest is quite different and covers more ground, and ifsound should revolutionize lots of Symbolic logic: R[ussell], he says, thinks it sound,but says nobody will understand it: I think I comprehend it myself however (!). IfW[ittgenstein]’s solution works, he will be the first to solve a problem which has puz-zled R[ussell] and Frege for some years: it is the most masterly and convincing solutiontoo.3

If, as one assumes, the ‘problem which has puzzled Russell and Frege for someyears’ is that of finding a philosophically defensible solution to Russell’s para-dox, it is perhaps a shame that Pinsent did not record Wittgenstein’s ‘most

2CW, II, 121. 325 Oct. 1912.

Types as classes of propositions )'&

masterly and convincing’ solution for posterity. Whatever it was, though, per-haps it did not satisfy Wittgenstein for very long: the following summer wefind him reporting from Norway that he was once more ‘thinking about thebeastly theory of types’.4

The principle Wittgenstein eventually settled on as guiding the resolutionof the paradoxes was that ‘no proposition can say anything about itself, be-cause the symbol of the proposition cannot be contained in itself; this mustbe the basis of the theory of logical types’.5 If giving an account of the theoryof types was really Wittgenstein’s main goal while he was at Cambridge, thenthis principle is in one sense a poor showing for two years’ work. For Russell,as we have just seen, had already tried to justify his own solution to the para-doxes by appealing to the vicious circle principle, that whatever involves allof a collection cannot be one of the collection. What Wittgenstein was nowadvocating was not much more than a transposition of Russell’s principle intothe language of symbolism that he had by then adopted. It is, that is to say,what Russell’s vicious circle principle becomes when viewed through the lensof Wittgenstein’s symbolic turn. A proposition involves all those propositionswhich it says something about, and hence by the vicious circle principle can-not say anything about itself.On the other hand, although Wittgenstein’s vicious circle principle looks

very similar to Russell’s, the background it relies on is different. Wittgenstein’sphrasing makes his solution to the paradoxes dependent upon his theory ofsymbolism, since whether a putative proposition attempts what is impossible,namely to say something about itself, is to be settled by examining its sym-bol. This might at least be thought to overcome the first weakness which wenoted earlier in Russell’s vicious circle principle: Wittgenstein, by giving it asymbolic turn, could motivate it by appealing to the non-circularity of under-standing. On the other hand, Wittgenstein’s vicious circle principle on its ownis surely programmatic in just the same manner as Russell’s. On its own, thatis to say, Wittgenstein’s principle does not settle whether the theory of typesshould be simple or ramified. That will be settled only once we have a de-veloped account of when one proposition ‘says something about’ another; or,equivalently, an account of propositional symbols that allows us to determinewhen one contains another.

"!.#"!.#"!.# Types as classes of propositionsWe saw in the last chapter that Wittgenstein conceived of quantification intwo stages: first, the formation of a symbol to pick out a class of propositions;second, the application to that class of an appropriate operation. The onlymethod for achieving the first stage of the process that he countenanced in

4To BR, 5 Sep. 1913. 5B76; cf. 3.332.

)'' Resolving the paradoxes

the Notes was by means of a propositional function, conceived of as obtainedfrom a proposition by replacing some component with a variable. Supposenow that we continue the process of replacing parts of the proposition withvariables as far as it can be taken. We shall then obtain a symbol every com-ponent of which is a variable. This will pick out a class of propositions whichwill be maximal, in the sense that the classes picked out by the propositionalfunctions referred to earlier will be subclasses of it. Wittgenstein called thismaximal class of propositions a logical type.6 A logical type may thus be thoughtof as a kind of limiting case of a class of propositions picked out by a proposi-tional function: the class picked out by a propositional function will always bea subclass of some logical type.So to determine what sorts of quantification are possible we need to deter-

mine what the logical types of propositions are. Suppose, for instance, that‘Socrates is mortal’ is an atomic proposition. What is its logical type? If we re-place ‘is mortal’ with a variable, we obtain a class of propositions !(Socrates).But prima facie it seems as if there might be various such classes, dependingon the range of the variable !: there is the narrow range consisting only ofthose propositions ascribing qualities (i.e. simple properties) to Socrates; butthere might be broader ranges containing propositions that ascribe increas-ingly complex properties to Socrates, such as that of being mortal if Plato iswise or, more complex still, that of having all the qualities of a great philoso-pher. And, as Ramsey famously noted,7 the same ambiguity between narrowand broad ranges is present in the other part of the proposition. If we replace‘Socrates’ with a variable, we obtain a class of propositions ‘x is mortal’: thisis normally taken by logicians to be the narrow range consisting of the propo-sitions obtained by replacing ‘x’ with a simple proper name; but we shouldconsider the possibility that there may also be broader ranges which includepropositions where we replace it with words such as ‘someone’ or ‘anyone’.More generally, the logical type of ‘Socrates is mortal’ is the class of propo-

sitions we obtain if we replace both parts of the proposition with variables soas to obtain a class picked out by the symbol ‘!(x)’. To determine whichclass this is we need to settle the range of both these variables. If both rangesare taken as narrowly as possible, we obtain the elementary subject-predicatepropositions, so the logical type certainly contains all of these. The questionwe have to settle is whether it contains any other propositions. According toWittgenstein this ‘is not dependent upon any convention, but only upon thenature of the symbol “!(x)” ’.8 We saw earlier that Wittgenstein thought Rus-sell had attempted to superimpose type restrictions by stipulation on symbolswhose sense already determined their types; and here Wittgenstein explicitlyadvances the alternative conception.Moreover, Wittgenstein is specific about how this is to be settled. In the

6B19. 7FoM, 125. 8Ibid.

Types and molecular propositions )'(

Notes he says that his vicious circle principle ‘must be the basis of the theory oflogical types’;9 in the Tractatus he says, more explicitly, that it is the ‘whole’ ofthe theory of types.10 What this means, I take it, is that the question we arenow considering, of the logical type of the proposition that Socrates is mortal,is to be settled solely by reference toWittgenstein’s vicious circle principle: thisis the only constraint capable of restricting the range of values of a variable.Anything else could only be an artificial constraint—an illegitimate attemptto restrict by stipulation the range of applicability of a symbol. But to beexplicit about the means of solving a problem is not yet to solve it. WhetherWittgenstein’s vicious circle principle leads to a simple or a ramified theory oftypes still depends on how the complexity of a propositional symbol is to beunderstood.

"!.$"!.$"!.$ Types and molecular propositionsConsider first the case of a molecular proposition. We are agreed that ‘Soc-rates is mortal’ and ‘Plato is human’ are of the same logical type, but whatabout ‘Socrates is not mortal’ or ‘Socrates is mortal unless Plato is human’?To settle whether these are of the same type as ‘Socrates is mortal’, we need toask what symbols they contain. Does !p contain the symbol p? If it does, andif similarly!!p contains!p, then (assuming, as is plausible, that containmentis transitive) !!p contains p. But, as we have already noted, !!p is the sameproposition as p. So p contains itself, which directly contradicts Wittgenstein’sprinciple.The conclusion, then, is that the symbolizing fact in !p does not contain

the symbol p; in the same way the symbolizing fact in p v q does not containthe symbols p and q; and similarly for all the other truth-functions. So onWittgenstein’s conception of what is relevant to symbols the way is clear for theapplication of truth-functions not to alter the logical type of a proposition. Theanswer to the question we posed at the beginning of this section is thereforethat ‘Socrates is not mortal’ and ‘Socrates is mortal unless Plato is human’ areof the same logical type as ‘Socrates is mortal’. This is what led Wittgensteinto emphasize, for example, that

the function p | q is merely a mechanical instrument for constructing all possible symbolsof ab-functions. The symbols arising by repeated application of the symbol ‘|’ do notcontain the symbol ‘p | q’.11

The stroke ‘|’ contributes to the structure of signs for molecular propositionswithout contributing correspondingly to the structure of the symbols thesesigns express.

9B77. 103.332. 11B31.

)(* Resolving the paradoxes

However clear Wittgenstein may have been about this, however, it is intension with things he said elsewhere. For instance, he stressed the intuitionthat

ordinary language would not contain the whole propositions if it did not need them:However, e.g., ‘not-p’ may be explained, there must always be a meaning given to thequestion ‘what is denied?’12

This intuition surely encourages us to think that the symbolizing fact in !pdoes contain the symbolizing fact in p as a component. And if we need anyfurther encouragement, it is supplied in the Cambridge Notes. ‘In not-p, p isexactly the same as if it stands alone; this point is absolutely fundamental.’13

Wittgenstein’s conception of !!p as the same symbol as p thus conflictswith his intuition that in !p it is p that is being denied. If he was to resolvethis tension, he needed to explain our conviction that in !p it is p that isbeing denied in such a way that we are not required to see the latter symbolas contained in the former. Chapter 19 was our attempt to extract from theNotes whatever clues they contain towards the resolution of this tension. Ourlack of progress there may be taken to indicate how much work still remainedfor Wittgenstein to do on this question.

"!.%"!.%"!.% Types and generality

Let us take it as granted, however problematically, that truth-functions donot raise the type of a proposition. What about the corresponding questionfor quantification? The Birmingham Notes give no clue what Wittgenstein’sanswer to this was. Perhaps this was what he was contemplating as he satbeside a fjord in Norway thinking about the beastly theory of types: if so,it was presumably the product of this contemplation that he offered Russell,almost as an afterthought, at the very end of the Cambridge Notes.

Take (!) !!x. Then if we describe the kind of symbols, for which ‘!!x’ stands and which,by the above, is enough to determine the type, then automatically ‘(!) !!x’ cannot befitted by this description, because it CONTAINSCONTAINSCONTAINS ‘!!x’ and the description is to describeALLALLALL that symbolises in symbols of the !! kind.14

What this explanation makes clear is that Wittgenstein intended his ac-count to justify the ramified theory of types of Principia, not the simple theory.He here considers the propositional function (!) !!x, in which the variable !

ranges over all the predicative functions of x. Could this function itself bepredicative? Could it, that is to say, be among the functions over which thevariable ranges? Wittgenstein’s answer is that it could not, because otherwiseany explanation of how it symbolizes would inevitably be circular.

12B21. 13C40. 14C51.

Uniting generality and truth-functions )()

Thus Wittgenstein conceived of a variable as containing the symbols for allthe propositions which are instances of it. The broadest of the ranges we can-vassed earlier for the variable !(Socrates) is therefore barred to us. The prop-osition ‘Socrates has all the qualities of a great philosopher’ is not of the sametype as ‘Socrates is mortal’, because mortality is one of the qualities whichgreat philosophers possess: the symbol of the former proposition contains thesymbol of the latter.The difficulty with this, though, is that it seems to go against Wittgenstein’s

decision to regard the role of a proposition as being to express a certain sense.That decision, let us recall, led him to regard two symbols as the same if theymake the same contribution to determining the sense of the proposition. Witt-genstein had not yet worked out in detail his account of the sense of proposi-tions involving quantification, but what he had already said about molecularpropositions (that p = not-not-p, for instance) shows that he regarded logicallyequivalent propositions as having the same sense and hence as being the samesymbol. The problem this causes is that it is possible for two propositions ofdifferent types to be logically equivalent. But the same symbol cannot belongto two different types. So Wittgenstein’s account of sense drove him towards asimple theory of types, while his understanding of a propositional variable asinvolving the symbols for the propositions in its range directed him towards aramified theory.

"!.&"!.&"!.& Uniting generality and truth-functionsWhat we have seen is that at the time of the Notes there must have been a fun-damental difference in Wittgenstein’s understanding of how molecular andgeneralized propositions depend on their components: a molecular proposi-tion does not contain symbols for its components, whereas a generalized prop-osition does. This was the reason why he thought that molecular propositionsbelong to the same type as their components but generalized propositions donot.Later, though, Wittgenstein united the two methods of forming proposi-

tions by adopting a generalization of Sheffer’s stroke. In the Tractatus he intro-duced a single operation N which could be applied to any class of propositionsso as to express the joint denial of all the propositions in the class.15 So if (& )is a symbol picking out a class of propositions, N(& ) expresses the propositionwhich is true if and only if all the propositions in the class are false. If (& ) picksout the class expressed by the propositional function f x, for instance, then

N(& ) = (x) ! f x.

But Wittgenstein now widened his conception of the kind of symbols thatcould be used to pick out a class of propositions: in addition to propositional

155.502.

)(! Resolving the paradoxes

functions, he allowed finite lists of propositions. This enabled him to treatmolecular propositions as particular cases of the same method. Thus

N(p) = !p

N(p,q) = !p !q.

Now that the symbols for molecular propositions were the result of the samegeneral process of symbolization as general propositions, it became even lessplausible than before to think that the theory of types should discriminatebetween them. To think that, we would have to hold that the symbol ‘ f x’picks out the class of propositions { fa, f b, fc, . . . } in such a way as to involvethe symbols for all the instances fa, f b, etc., but that the symbol ‘{p, q}’ doesnot in the same manner involve the propositions p and q. And why would onethink that?But even if we accepted this as a reason to hold that molecular and general

propositions are in the same logical boat, and hence that Russell’s ramifiedhierarchy is untenable, that would not in itself determine which direction togo: towards a simple theory of types in which neither the formation of molec-ular nor of general propositions raises the type of a proposition; or towardsan ultra-ramified theory in which both do. To settle that, we would have todecide whether the symbol for a class of propositions involves the symbols forthe propositions in the class, in whatever sense of ‘involves’ is relevant to thevicious circle principle. But the vicious circle principle is, as we noted earlier,only programmatic. So more would still need to be said about the structure ofsymbols.

"!.'"!.'"!.' The general form of propositionJust when Wittgenstein invented the N-operator is a question on which thewritten record is curiously silent: the device is absent from the Notes dictatedto Moore as well as from the surviving wartime notebooks, but it occurs onthe first page of the Prototractatus. Whenever it was that Wittgenstein adoptedthe N-operator, however, the tension implicit in his adherence to a ramifiedtheory of types, already severe, surely now became intolerable; yet it tookWittgenstein a remarkably long time to resolve this tension. When he beganto compile the Prototractatus in late 1915 or early 1916 he still seems to haveadhered to a ramified theory.16 Matters were then made still worse by the cen-tral role that came to be played in Wittgenstein’s conception by the generalform of proposition. The culmination of the Tractatus, as Wittgenstein origi-nally conceived it, was to be a statement of the general form of proposition.Propositions, he said, form the range of the propositional variable

[p0, & ,N(& )].16Cf. PTLP, 4.432.

Unsayability )(#

What matters for current purposes is not the particular form of this variablebut merely its existence. If there is a general form of proposition (whether thisor anything else), then an expression for this general form is a variable whichhas as its range all propositions, in which case we are forced to have a simpletheory of types. AndWittgenstein’s philosophical instincts pulled him stronglytowards a belief in such a variable. This had its roots in Wittgenstein’s muchearlier insight, present already in his letter to Russell of June 1912, that therecannot be any such thing as a logical experience.

If the most general form of proposition could not be given, then there would have tocome a moment where we suddenly had a new experience, so to speak a logical one.That is, of course, impossible.17

Thus he insisted in November 1916 that

it must be possible to erect the general form of proposition, because the possible formsof proposition must be a priori. Because the possible forms of proposition are a priori,the general form of proposition exists.18

And yet he evidently had not yet seen his way through to the simple theory,for in the next sentence he seems still to admit the ramified theory at least asan open possibility, but to encourage us to ignore it for the purposes of thepresent discussion.

In this connection it does not matter at all whether the given fundamental operations,through which all propositions are supposed to arise, change the logical level of propo-sitions, or whether they remain on the same logical level.19

And as late as January 1917, at the very end of the surviving Notebooks, Witt-genstein continued to feel an unresolved tension between these two competingconceptions.

In the sense in which there is a hierarchy of propositions there is, of course, also ahierarchy of truths and of negations, etc.But in the sense in which there are, in the most general sense, such things as propo-

sitions, there is only one truth and one negation.

In one sense, he was saying, a ramified theory is correct; but in another sensea simple theory is.

"!.("!.("!.( UnsayabilityWittgenstein did eventually resolve the tension, and he did so in favour of thesimple theory of types: in the published version of the Tractatus almost all theremaining traces of the ramified theory are at last removed, and propositionsare no longer presented as belonging to a hierarchy of types. (The principalexception is Wittgenstein’s lingering mistrust of the ancestral,20 which is hard

17NB, 9 July 1916. 18NB, 21 Nov. 1916. 19NB, 21 Nov. 1916. 204.1273.

)($ Resolving the paradoxes

to motivate in the context of a simple theory of types.) But the final deci-sion to opt for the simple theory must have occurred rather late, since by thebeginning of 1917, when he still thought there was in a sense a hierarchy ofpropositions, almost seventy pages of the Prototractatus were already filled in.Why did it take him so long? As we have seen, his philosophical instincts—the unity of the self, the emptiness of logic, and his conception of propositionalsense—pushed him unequivocally towards a simple theory. Only one thingpushed him in the other direction—the need to solve the paradoxes. He real-ized, no doubt, that a simple theory suffices to solve Russell’s original paradoxof the class of all classes which do not belong to themselves. But Russell hadbeen driven towards a ramified theory by the need also to solve paradoxessuch as Grelling’s. If Wittgenstein was to adopt a simple theory, he thereforeneeded some reason other than ramification to prevent the formation of thepropositional function

(

E

!)(x means !z !!x),

which is responsible for generating Grelling’s paradox.The reason Wittgenstein settled on was his doctrine of unsayability, which

disallows the use within his system of semantic concepts such as meaning. Themeaning of a symbol is shown by its use in propositions, but we cannot say inlanguage what the symbol means. In particular, therefore, we cannot definea propositional function Het (and even if we could, we could not say that‘Het’ means Het). This suffices to block Grelling’s paradox (and others like it)without the need to ramify types.Now it is true that this solution was not really available to him at the time

of the Notes, since only the earliest traces of the distinction between saying andshowing are visible there. But unsayability—in particular, the unsayability ofsemantic relations such as that of meaning—is something Wittgenstein madea great deal of in the notes he dictated to Moore in Norway the following year.So he did have by then all the components he needed to resolve the tensionwe have been discussing. Yet it seems to have taken him three years fromthen to see how the resolution should go. At the very least this shows rathervividly that his reasons for adopting the doctrine of unsayability must havebeen very distant from the technical considerations we have been discussinghere. Its ability to contribute to a resolution of the paradoxes can only havebeen a welcome consequence of the doctrine, not its original motivation.Even so, the slowness of Wittgenstein’s adoption of a simple theory of types

remains one of the central puzzles in our understanding of the genesis of theTractatus. What it demonstrates above all is once again the extent to whichWittgenstein’s logical insights were independent of formal considerations. Hehad a conception of the sort of thing a perspicuous notation would be, and

Unsayability )(%

then he set about finding it. And in the same way he had a conception of whatthe general form of proposition amounts to before he saw how this translatedinto the formal system. It was philosophical, not technical, considerations thatmotivated these conceptions.Although Wittgenstein’s discussion of types inevitably suggests that Principia

is his intended target, we have noted already how little trace of that book thereis in his discussion. It is not merely that the Notes do not discuss the details ofthe theory of types that is presented there: they bear remarkably little signthat their author had even got beyond the Introduction. Wittgenstein maderoutine use of the logical signs Whitehead and Russell employed in Principia,but of course he would not have had to read the book in order to acquire hisknowledge of them. Only once, at the very end of the Cambridge Notes,21 didhe make use of a notation (a variable ranging over predicative propositionalfunctions) that is distinctively type-theoretic. And if, as is curiously plausible,he delayed until the summer of 1913 any serious consideration of the parts ofPrincipia which deal with quantification, there is no sign that his engagementwith them became deeper during the war years. (By 1922 his recollection ofPrincipia was so shaky that he could no longer remember22 whether the signused there for the existential quantifier was ‘

E

’ or ‘E’.) Although it was a desireto solve the paradoxes in the philosophy of mathematics that put him on hiscourse towards the Tractatus, his method of achieving that goal was, to the last,strikingly indirect.

21C51. 22To Ogden, 10 May 1922.

Chapter """"""

Typical ambiguity

In §5.4 we saw how Wittgenstein thought Russell had been led into errorby his practice of using formulae containing real variables. To correct thismistake, it would be necessary to rewrite Principia so that any use of such for-mulae is either eliminated or explained. In the elementary parts of the book,where it is simply a treatment of first-order predicate calculus that is in ques-tion, eliminating the problematic occurrences is feasible (although, given theeccentricities of the treatment, not trivial). Very soon after that, however,Whitehead and Russell use formulae containing real variables in a differentmanner that aroused new suspicions in Wittgenstein, namely as a device forexpressing various propositions simultaneously at different levels in the type-theoretic hierarchy.

"".!"".!"".! Typical ambiguity

The device of typical ambiguity is a response to a difficulty which arises oncewe accept that the paradoxes prevent us from quantifying over absolutely ev-erything. We find ourselves nonetheless wanting to make assertions that applyto each level in the type-theoretic hierarchy. We might wish to say, for in-stance, that everything, at whatever level in the hierarchy, is self-identical. Ifwe say that, however, we quantify over all the levels in the hierarchy, which isprecisely what the theory of types renders illegitimate.The solution Whitehead and Russell adopted was to make use of the device

of typical ambiguity. In order, for instance, to regard ‘# !(x)’ as expressingsomething determinate, we need to know what sort of variable x is; we needto know, that is to say, which type it is supposed to range over. Until thatis determined, what we have written remains ambiguous as to type. Theirconvention was that by writing ‘# !(x)’ we should be taken as ambiguouslyasserting any such determination of the formula.Nowadays we would formulate a typically ambiguous generalization met-

alinguistically by means of a schema which only becomes a sentence express-ing a determinate proposition once the schematic letters in the schema are

Typical ambiguity )(&

replaced by specific symbols of the object language. It is striking, therefore,that in the Notes Wittgenstein uses the word ‘scheme’ in what is recognizablythis modern sense.

Those symbols which are called propositions in which ‘variables occur’ are in realitynot propositions at all, but only schemes of propositions, which only become propo-sitions when we replace the variables by constants. There is no proposition which isexpressed by ‘x = x’, for ‘x’ has no signification; but there is a proposition ‘(x) x = x’and propositions such as ‘Socrates = Socrates’ etc.1

One central case in which Whitehead and Russell made use of the deviceof typical ambiguity was the axiom of reducibility. Wittgenstein complainedabout this in a letter to Russell probably dating from the summer of 1913.

Your axiom of reducibility is

! (

E

f ) !x #x f !x;

now is this not all nonsense as this proposition has only then a meaning if we can turnthe ! into an apparent variable. For if we cannot do so no general laws can ever followfrom your axiom. The whole axiom seems to me at present a mere juggling trick. Dolet me know if there is more in it. The axiom as you have put it is only a schema andthe real P[rimitive] p[roposition] ought to be

! (!) (E

f ) !x #x f !x,

and where would be the use of that?2

But Wittgenstein had mislocated the source of the problem. Whether avariable is typically ambiguous is nothing to do with whether it is real or ap-parent. What makes this hard for the reader of Principia to see is that theirnotation omits any visible indication that a variable has been disambiguated.Modern treatments of the theory of types standardly indicate which type avariable is supposed to range over by means of a subscript. If the theory oftypes is simple, so that there is only one dimension of variation, the subscriptneed only be a single numeral; if the theory is ramified, as that in Principia is,we need a more complicated notation, but the principle is the same. What itamounts to to disambiguate a schema of Principia is to decorate all the vari-ables that occur in it with imaginary type-subscripts indicating the types theyrange over. But when we do this, we must decorate the real variables as wellas the apparent ones. So in Principia ‘(x) x = x’ is typically ambiguous just as‘x= x’ is: in both cases the letter ‘x’, which looks as if it is expressing a variable,does not really do so until it is decorated with a subscript determining the log-ical type it ranges over. If we adopted a notation which insisted on attaching

1B13. 2CL, no. 20.

)(' Typical ambiguity

type-subscripts to all variables, the point would emerge more clearly. The lawof self-identity, for example, would in such a language be written

(x' ) x' = x' ,

where ' is schematic; if we replaced ' with a specific type-subscript in theobject language, the resulting proposition would express the self-identity of allthe entities in the particular type over which the corresponding disambiguatedvariable ranged.What Wittgenstein had failed to see, then, is that this point applies to ap-

parent variables just as much as to real ones. So what makes the axiom ofreducibility ‘only a schema’ is not really, as Wittgenstein evidently supposed,the unbound occurrence of ‘!’. The formula which Wittgenstein offers as the‘real primitive proposition’ is just as much a schema as Russell’s version. Itmay indeed be reasonable to ask (as Wittgenstein does) what the use of the ax-iom of reducibility is, but that question has nothing to do with whether we canturn the ! in it into an apparent variable. The relevance of real variables tothe point Wittgenstein was fumbling for was only that Russell’s willingness touse formulae containing real variables to assert ambiguously all the instancesof the formula in a single type made him less critical than he should have beenof extending this device so as to assert ambiguously instances belonging todifferent types. The first kind of ambiguous assertion, although strictly incor-rect, is innocuous, because by prefixing the formula with a universal quantifierwe can turn the real variable into an apparent one. The second kind is notinnocuous, because no corresponding manoeuvre is available to us.There is also another sense in which Wittgenstein’s criticism of the axiom

of reducibility is inept. Wittgenstein’s objection is that the ‘axiom’ is really justa schema awaiting determination of the types in order to obtain a propositionusable in inference. But in that case it is curious that he had not spotted theproblem already in $9, where we find primitive propositions such as ‘# !x !(

E

z) !z’; this is a schema in which ! awaits the determination of its type, justlike the axiom of reducibility.It was not until he was in Norway that Wittgenstein was able to give a

more plausible reason to think that reducibility is a ‘mere juggling trick’. Theobjection was not that it was a schema, but that even if it is true, it is only ac-cidentally true.3 And to be in a position to formulate this objection he neededto be clear about the distinction between truth and tautology, and that, aboveall, is what he achieved in Norway. But even if it is understandable that hecould not formulate his objection to the axiom of reducibility until later, it issurely notable nonetheless that the letter quoted earlier gives the impressionof being the first time he had thought much, if at all, about it.

3LW to BR, [Nov. or Dec. 1913] (CL, no. 32).

Independent indefinables )((

""."""."""." Independent indefinablesI mentioned in §8.3 Wittgenstein’s tendency to see Russell’s theory of typesas an attempt to impose artificial restrictions on symbols whose grammar hasalready been fixed by the meanings we have assigned to them. We see thistendency at work in many of Wittgenstein’s comments about Principia.Every proposition belongs to a logical type. Something Wittgenstein was

repeatedly concerned to stress, however, was that types should not be thoughtof as containers into which propositions can be sorted. To see the point Witt-genstein was trying to make we need to take a step back and consider his viewson the issue of how language gets its meaning. Once more, the starting pointis Fregean. One of the central concerns of his Grundgesetze had been to ensurethat what he set up there was not a purely formal language. In order that logiccan be applied, it is necessary that we invest the primitive terms occurring init with meaning.Wittgenstein called the primitive terms indefinables. These, he said ‘must be

independent of each other’. The reason was that he wanted—to exploit theanalogy frommechanics which wementioned in chapter 8—a system in whichthe number of indefinables is equal to the number of degrees of freedom.

If an indefinable is introduced, it must be introduced in all combinations in which itcan occur. . . In short for the introduction of indefinable symbols and combinations ofsymbols the same holds, mutatis mutandis, that Frege has said for the introduction ofsymbols by definition.4

This is why Wittgenstein held that words such as ‘individual’ or ‘particular’must not be taken to be primitive ideas in logic. ‘It is easy to suppose,’ heobserved, ‘that “individual”, “particular”, “complex” etc. are primitive ideasof logic. Russell e.g. says “individual” and “matrix” are “primitive ideas”.’5

Wittgenstein went on straightaway to offer a diagnosis of Russell’s error. It is,he said,

to be explained by the fact that, by employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have been deprived of allproperties except thing-hood, and with propositions deprived of all properties exceptcomplexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occurunder the generality-sign, never outside it.6

The fact that Russell added the German ‘Urbilder von Zeichen’ suggests thathe was uncertain about the translation, and indeed it might more happily betranslated here as ‘prototypes of signs’. What Wittgenstein is referring to issimply Russell’s variables and pure forms. Wittgenstein takes Russell’s deviceof ambiguous assertion to have misled him into thinking that logic has a sub-ject matter, namely the abstract forms of propositions and their constituents.

4B51. 5B71. 6B71.

!** Typical ambiguity

Part of the point here is that the type of a proposition is intrinsic to it: ifit belonged to a different type, it would be a different proposition. But this isonly part of the point. It is intrinsic to the table that it is made of wood: ifit were not, it would be a different table. In this case, however, it is perfectlyintelligible, even if false, to say that the table is made of metal. In the case ofthe proposition, however, it does not even make sense to say that it belongs toa certain type.To see why, suppose for a moment that it does make sense to say that a

proposition belongs to a certain type. Then it cannot also make sense to saythe same of a proposition of some other type. This is because if one proposi-tion can be substituted for another salva sensu then they are of the same type.(That, after all, is just the point of categorizing propositions according to type.)But if it does not make sense to say of a proposition that it has any type otherthan the one it in fact has, it cannot make sense to say that it does not havethat type (since that is tantamount to saying that it has some other type). Andin that case it does not even make sense to say that the proposition does havethe type it in fact has, since if it did make sense, the negation of that claimwould make sense too. The conclusion we must draw from this is that logicaltypes are not really properties of propositions at all.7

This conclusion, that types are not properties, is significant partly because itpoints to a fundamental difficulty with the theory of types, namely that it seemsimpossible to state it without violating its own constraints. But it is significantalso because it is one of the first signs in Wittgenstein’s work (perhaps, indeed,the very first) of the theme of unsayability that would play such a large rolein the Tractatus. We cannot say which type a proposition belongs to, and yet,we feel, the proposition surely does belong to it. We seem to be gesturingfrustratedly towards a truth that is just out of our expressive reach.

"".#"".#"".# WhiteheadTypical ambiguity does pose a genuine problem, even if Wittgenstein did notaccurately pinpoint its source. The problem is that the device of typical am-biguity creates an illusion of unity where, if we take the theory seriously, thereis none. If we assert a typically ambiguous formula !(x), we are to be takenas asserting an infinite number of formulae of this form, in each of which thevariable x is disambiguated to range over a particular type. But each of theseinfinitely many formulae obtained by disambiguating the typically ambiguousformula !(x) is distinct: there is nothing except an accident of notation to unitethem. A remark early in the Notes, although perhaps not originally directed atthe problem of typical ambiguity, is certainly applicable to it.

7B16.

Whitehead !*)

It can never express the common characteristic of two objects that we designate themby the same name but by two different ways of designation, for, since names are ar-bitrary, we might also choose different names, and where then would be the commonelement in the designations? Nevertheless one is always tempted, in a difficulty, totake refuge in different ways of designation.8

Wittgenstein was not the first person to have been suspicious of the device oftypical ambiguity, however. Whitehead himself expressed the point to Russellin January 1911.

I vaguely conjecture that some careless sentence of mine has led you to believe that Iobject to limiting the type of a variable. This is a gross libel. So far from that, my viewis that our symbols remain mere unmeaning forms until the types of all the variablesare determined.9

A couple of days later he put the point even more clearly. ‘According to meuntil all ambiguities are definitely settled there is simply a sequence of mean-ingless shapes.’10 In the same letter Whitehead also appealed to a distinc-tion between what we now call the object language and the metalanguage—between what we prove within a formal theory and what we say about it fromwithout.

We do talk in summaries, etc. of all types. Certainly we do,—and the words have avery complicated meaning, as do most notions of common thought. . . In the case ofaddition it has been worth while defining the complication in the S[ymbolic] T[heory];in the case of ‘all types’ it is not, chiefly because the complication is much greater.Hence any essential use in the S[ymbolic] T[heory] of ‘all types’ is forbidden to us.11

According to the theory of types it is illegitimate to quantify within our formallanguage over all the levels in the type-theoretic hierarchy. Nonetheless, wemay do so informally in ‘summaries’ (i.e. the pieces of explanatory text placedat the beginning of each section of Principia).There is an evident difficulty with this idea. What forced Whitehead and

Russell to introduce the complexities of the theory of types was the paradoxes,which showed that there is something profoundly mistaken about the attemptto quantify over everything. If that is so, then it remains mistaken whether theattempt is made formally or informally. A paradox is a paradox whether it iswritten down in a formal language or not. What Whitehead was advocatingthus seems to be once again an instance of the sort of chickening out from theconsequences of a theory which we saw with Frege and the concept horse, andwhich Wittgenstein so obviously disapproved of.Whitehead did at least recognize that he and Russell needed to say more

about this issue than they had in the text of Principia so far, even if he perhapsdid not know what more that was. ‘This correspondence has convinced me

8B3; cf. 3.322. 927 Jan. [1911] (RA1 710.057409). 1029 Jan. 1911 (RA1 710.057414). 11Ibid.

!*! Typical ambiguity

that we must sketch somewhere the talk the sort of thing ‘all types’ (in sum-maries) means.’12 Whitehead also recognized that if the typical ambiguitiesin schemas were to be removed, there had to be a procedure for determiningtheir types, and this procedure would itself have to be typically ambiguous,thus leading to the danger of an infinite regress.

We have . . . rules for the assigning of meaning to the symbols so as to produce aprop[osition]. There are really a succession of sets of rules one set for each type asit is defined. All the sets have a formal identity of procedure in relation to the varioussymbols.To say that a prop[osition] holds for any type is to say,—If after any acts of defini-

tion of types, the corresponding sets of formal rules of interpretation be applied to asequence of symbols, we can then rightly assert the proposition or propositional func-tion which is then obtained. Thus the P[rimitive] I[dea] ‘act of definition of a type’is used—but not in the S[ymbolic] T[heory]. In the S[ymbolic] T[heory] we prove forthe one or two types we do define, and outside the S[ymbolic] T[heory] we say thatwhoever works away at defining more types will find the corresponding prop[osition]sstill true.. . . What I want to see (and we get no further forward till we have it) is a full discus-

sion by you of the legitimacy of ambiguity of type under the assert[ion] sign.13

Russell’s reply (now lost) was evidently to the effect that we can escape thedifficulty by explaining that when we assert a formula containing an ambigu-ous variable, this is to be considered true if it holds however the variable isdisambiguated. Whitehead was not reassured:

What do you mean by ‘considered true’? This is the point of the whole question—the lawof excluded middle cries aloud for vengeance. The truth is that under this phrase youhide the fact that you are really dealing with ‘all types’.14

The end result of this correspondence was that Whitehead wrote a ‘prefa-tory note’ which appears at the beginning of Volume II of Principia. (Theremarks in this note do not apply just to Volume II, incidentally: it was boundin there simply because by this time Volume I had already appeared.)If typical ambiguity was one of the issues Wittgenstein was thinking about

during the summer of 1913, it is to be expected that he should have wantedto talk about it to Whitehead, who evidently grasped the issue more clearlythan Russell. Wittgenstein at any rate took considerable trouble to go andvisit Whitehead in August 1913,15 and then again before leaving for Norwayin October:16 one suspects that these were not merely courtesy calls.The issue of typical ambiguity is a prime candidate for the application of

the distinction between sign and symbol which we shall be discussing in chap-ter 24, and indeed Whitehead’s way of expressing his concerns about it in his

12Ibid. 13Ibid. 14To BR, 31 Jan. 1911 (RA1 710.057458). 15McGuinness, Young Ludwig, 179.16LW to BR, 17 [Oct.] 1913.

Whitehead !*#

letters to Russell is strongly suggestive of Wittgenstein’s later way of conceiv-ing of that distinction. A ‘proposition’ of Principia which contains a typicallyambiguous variable is, strictly speaking, merely a sign: it becomes a symbolonly when the ambiguity is resolved. But there is nothing in the sentence onthe page that tells us how to do that. We resolve the ambiguity by reading thesentence as saying something, i.e. by disambiguating all the variables that oc-cur in it, and when we do that, the resulting proposition says something aboutthe entities in some particular type.More generally, then, a sign is, as Whitehead said in the particular case of a

typically ambiguous formula, ‘simply a sequence of meaningless shapes’ untilit is read as a particular symbol. What Whitehead’s discussion makes salientbut is quite absent from the Notes, however, is the distinction between whatcan be said in the symbolic theory and what can only be said outside it—or,as we would now say, between object language and metalanguage. Once weattend to that distinction, however, it is then a short step to recognizing thedanger that the paradoxes which the theory of types was designed to solve willsimply recur in the metalanguage.

Chapter "#"#"#

Identity

Whatever its virtues, the account of logic that Wittgenstein offers us in theNotes is, as he plainly knew himself, incomplete. One of the issues that he knewstill required further thought was how to represent the relation of identity.

"#.!"#.!"#.! Russell’s definitionIt was evidently Wittgenstein’s ambition at the time when he compiled theNotes to extend his method of truth-diagrams so that it would provide a wayof symbolizing all the logical notations of Principia. But on the face of it therewas no need for him to address the question of identity, because in Principia itis not one of the primitive notations. In Principia identity is a defined relation:

x = y =Df (!) !!x " !!y.

Two things are thus said to be equal, according to this definition, just in casethey have all their predicative properties in common. By October 1913, how-ever, we know that Wittgenstein had formulated an objection to the definition:he reported the objection to Moore,1 but for some reason did not include it inthe Notes. Nor, unfortunately, did Moore record it, so we are left to guess justwhat it was.We have noted that by about this time Wittgenstein had begun to suspect

that the axiom of reducibility is no more than a ‘juggling trick’. It would benatural in that case to suspect Russell’s definition of identity in turn. For Rus-sell needed reducibility in order to obtain the indiscernibility of identicals infull generality. Without this axiom, he could not rule out the possibility thatthere are objects which are equal according to the definition (because theyshare their predicative properties) but differ as to their higher-order prop-erties. But although Wittgenstein could perhaps have objected to Russell’sdefinition on this ground, I doubt if he did. It is more likely that his objectionis the one that appears in the Tractatus. The objection there is not that thedefinition does not entail the indiscernibility of identicals, but rather that itdepends unjustifiably on the identity of indiscernibles.

1Moore’s diary, 3 Oct. 1913.

Russell’s definition !*%

Russell’s definition of ‘=’ won’t do; because according to it one cannot say that twoobjects have all their properties in common. (Even if this proposition is never true, itis nevertheless significant.)2

To see just what Wittgenstein’s point was, let us start by noting Russell’sobservation in 1913 that

it is erroneous to regard [the identity of a sense-datum] as constituted by its predicates,because a precisely similar sense-datum may exist in another place. (By ‘preciselysimilar’ I mean ‘having the same predicates’.)3

But even if two sense-data cannot be distinguished by means of their predi-cates, Russell did not think they would be indistinguishable: he thought theycould be distinguished by means of their relational properties. For if a and bdiffer in relation to other entities with respect to the relation R, for instance,so that there is an object c such that aRc &" bRc, then by applying Russell’sdefinition of identity with the propositional function xRc as ! we obtain a &= b.Wittgenstein was not disagreeing with this. His point was that even if as a

matter of fact a and b are distinguishable, they might not be. We can under-stand what it means to say of two distinct objects a and b that they have alltheir predicative properties in common: the sentence (!) !!a " !!b is signifi-cant even if it is false. The conclusion Wittgenstein drew from this was thatthe relation Russell had defined was not identity.How, though, could Wittgenstein be so sure that it is significant to say that

two objects have all their properties in common? In the Tractatus, this followsfrom the doctrine that elementary propositions are logically independent ofone another, and that doctrine in turn depends on Wittgenstein’s hostility tothe Kantian synthetic a priori, a hostility which may well be of early date. Onthe other hand, Wittgenstein need not have formulated the logical indepen-dence of elementary propositions in full generality in order to believe that itis significant to say two objects have all their properties in common. He hadample motivation for that view independent of his developing views aboutpossibility. For otherwise the fact that they do not have all their properties incommon would be a further fact about them not reducible to the atomic facts.And we have noted already his conception of atomism as involving the claimthat there can be no such fact.One of the oddities of this story, of course, is that there is no record of Witt-

genstein reporting his objection against Russell’s definition to Russell himself.Perhaps he knew what Russell’s reaction would be. After all, Russell thoughtthat there are general facts which are not reducible to the atomic facts—thereis, for instance, the fact that there are no other atomic facts—and we notedearlier the evidence in the Cambridge Notes to suggest that when Russell putthis point to him, Wittgenstein had nothing to say in reply. Why, in that case,

25.5302. 3CP, VI, 94.

!*" Identity

should there not also be further facts not reducible to atomic facts, such asthe fact that two objects do not ever have all their properties in common?Whatever other of Russell’s philosophical views Wittgenstein had succeededin persuading him to give up, Russell did not waver in his belief that there isan a priori structure to which the atomic facts are subject.After the war, when Russell eventually learned of Wittgenstein’s objection

to his definition from the typescript of the Tractatus, he was initially persuaded:it was, he said, ‘a destructive criticism from which there seems no escape’.4

Quite soon, though, he came to doubt it. However, his doubts were, as sooften with Russell, largely regressive. Without the definition of identity, hethought, mathematics would be lamed. This was hardly the sort of objec-tion to impress Wittgenstein: so much the worse, he would have said, formathematics. Russell’s only non-regressive defence of his definition was epis-temological. ‘I should maintain,’ he said, ‘that to say that a and b are twomeans that they have different properties.’5 By now, though, Russell was notengaging with Wittgenstein’s system as a whole, but assessing the plausibil-ity of his claims piecemeal. If I have recognized a and b as distinct entities,Russell insisted, they must ipso facto have different properties. Quite so, butWittgenstein did not deny this. His point was not that there are two differentobjects which share all their properties, but only that it is intelligible to say thatthere are. The relation Russell had defined did indeed coincide with identityin the actual world, but Wittgenstein thought that in other worlds it need not.If something is possible, it does not follow that it is possible for it to be actual.

"#.""#.""#." Eliminating identityEven if the objection Wittgenstein reported to Moore before departing forNorway was the one just described, the most that it can achieve is to showthat Russell’s definition is incorrect, because the relation it defines might notbe identity. What it cannot do in itself is to show that there is no such relationas identity at all. It leaves open, for instance, the possibility that identity issimply a primitive relation which may hold or fail to hold between objects.What rules this out is rather Wittgenstein’s conception of the metaphysics

of facts. For each proposition, let us recall, there is on his account a fact whichis its meaning. Suppose that identity is a primitive relation, so that sentencesof the form a= b are atomic. So in the case when a= b, the fact that a= b is apositive fact—the very same, indeed, as the fact that a = a. Now if a positivefact does not occur, there is a negative fact which occurs in its stead. So in thecase when a &= b, the negative fact in question ought to be the fact that a &= a.But there cannot be any such fact.

4CP, IX, 107. 5CP, X, 108.

The notational problem !*&

To put the point in other words, the bipolarity of the proposition requiresthat the negation of any elementary proposition should represent a possiblesituation. But if simple things are identical, then they are just the same thing,so to represent that they are not identical would be to represent that a singlething is not itself. But there is no room in Wittgenstein’s conception for such afact, and there is therefore no relation of identity. But this has a consequencefor his conception of simple objects. It was part of his conception of simplicitythat it is unintelligible to say of two distinct simple objects that they are iden-tical. His explanation of any informative identity statement would thereforealways be that the things which are asserted in it to be equal are complex.When exactly Wittgenstein arrived at this conception of objects as the con-

stant elements around which possibilities turn, and hence of identity as notproperly a relation, is not clear: what little he says about identity in the Notesdictated to Moore does not show that he had fully embraced such a conceptionby the time he dictated them in April 1914. Yet he must by then have beentantalizingly close to just this conception, since it seems to be firmly in placeright at the beginning of the wartime notebooks, where Wittgenstein observesthat to say of two things that they are identical means nothing.6

One of the characteristic methods we have seen Wittgenstein applying totechnical problems that got in his way is to dissolve them rather than to solvethem—to reconfigure the territory in such a way that the alleged problemdisappears. Something like this is what Wittgenstein did in the case of iden-tity, when he arrived at the conclusion that there is, properly speaking, nosuch relation. What Wittgenstein did in order to reach this conclusion was toadopt a fundamentally modal conception of sense according to which whatvaries between possible worlds is how the objects are configured, not whichobjects there are. Only now, perhaps, might we describe Wittgenstein’s viewas genuinely logical atomism.

"#.#"#.#"#.# The notational problemIf we agree that there is no genuine relation of identity, we might hope thatthere should be no need for a notation for identity (since in that case there isnothing in the world for such a notation to express). And in its occurrences tolink proper names we can indeed eliminate the need for the sign of identity bythe device of adopting a language in which one object never has two differentnames. But that does not absolve us completely from the need to have a no-tation for identity. For Wittgenstein’s conception depends on Russell’s theoryof descriptions, according to which f (the g) is to be analysed as

(

E

x) f x (y) gy " x = y,6NB, 5 Sep. 1914.

!*' Identity

which still makes use of the sign of identity. Wittgenstein therefore needed anaccount of the notation of identity as it occurs linking variables in sentencessuch as this.Wittgenstein’s letters throughout the autumn of 1913 document his contin-

uing failure to make progress with this problem.Identity is the very Devil and immensely important; very much more so than I thought. Ithangs—like everything else—directly together with the most fundamental questions,especially with the questions concerning the occurrence of the same argument in dif-ferent places of a function. I have all sorts of ideas for a solution of the problem butcould not yet arrive at anything definite. However, I don’t lose courage and go onthinking.7

Shortly afterwards, Wittgenstein believed that ‘the ab-notation’ (by which hemeant his method of truth-diagrams) provided a general test of whether aproposition was a logical truth. But in the case of identity he had to admitthat this was still no more than a hope. ‘The ab-Notation for identity is not yetclear enough to show this clearly but it is obvious that such a Notation can bemade up.’8 A little while later, he repeated his belief that logical propositionsare those whose truth or falsehood is, in a perspicuous notation, visible in thesign itself. Nonetheless, he had to admit that identity still defeated him.I have not yet succeeded in finding a notation for identity that satisfies this condition;but I have NONONO doubt that it must be possible to find such a notation. . . I find identity, asI say, still far from clear. So I will deal with that another time.9

Wittgenstein did eventually solve the problem: in the Tractatus he adopted10

a notational convention according to which the ranges of nested variables areto be understood exclusively. For example, the sentence ‘(

E

x,y) xRy’, whichaccording to Russell says that there exist things related by R, on Wittgenstein’sreading says that there exist distinct things related by R. Similarly, ‘(x,y) xRy’is now to be read as saying that anything is related by R to anything else. Ifwe adopt this way of reading the quantifiers, the Russellian analysis of f (the g)can be expressed without any use of the identity sign as

(

E

x) f x gx !( Ex,y) gx gy.

It seems to have taken Wittgenstein some time to hit on this notation. Notuntil November 1914 did he mention it in his notebook (although it is hard totell from the entry whether he had just invented the device or was remindinghimself of something he had devised earlier).11 Yet the core idea is surelyalready present in the Notes, where he observes that it is ‘useless to replace!(a,a) by !(a,b) a = b’.12

7To BR, 29 Oct. 1913. 8To BR, [Nov. 1913] (CL, no. 32). 9[Nov. or Dec. 1913] (CL, no 32).105.532. 11NB, 29 Nov. 1914. 12B29.

Chapter "$"$"$

Sign and symbol

One of the themes of this book has been Wittgenstein’s understanding of thenotion of a symbol. This notion was placed at the very centre of his concep-tion of logic as soon as he took the symbolic turn described in chapter 6. Bycharacterizing a simple object as what is expressed by a simple symbol, hecommitted himself to a conception of symbols quite different from the signsthat routinely confront us on the page. In this he was to some extent follow-ing Russell, whose conception of simplicity committed him to the view thathardly anything which counts grammatically as a proper name is logicallysuch. In other respects, however, Wittgenstein’s conception of symbols owesmuch more to Frege’s notion of sense. In this chapter we shall explore thisconception, both in the form that is implicit in the Notes and in Wittgenstein’sdevelopment of it in the Tractatus.

"$.!"$.!"$.! Seeing through to the symbol‘In regard to notation,’ Wittgenstein reminded us, ‘it is important to note thatnot every feature of a symbol symbolizes.’1 The way I put it in the discussionof Wittgenstein’s symbolic turn was that recognizing the symbol is a matter ofseeing through the irrelevant details to the feature of the complex that doesthe symbolic work. At that point in the discussion this was an approximatestipulation, since it left open quite what the work is that we are trying to do.Recent chapters have made the issue more determinate. The sense of a prop-osition is what we have to grasp in order to be said to understand it. So thesymbolic work the proposition has to do is just that of expressing this sense.The symbolizing fact in the proposition is therefore the fact which ensures thatit expresses the sense it does.The symbolizing fact is what we see exemplified in the complex when we

understand the proposition. So it must determine the sense: we could nothave two different senses symbolized by the same fact, because there wouldbe nothing else to determine which sense was meant. On the other hand, thesymbolizing fact does no more than to determine the sense. So propositions

1C14.

!)* Sign and symbol

that express the same sense must have the same symbolizing fact, and hencemust be the same. In summary, propositions are the same if and only if theyexpress the same sense.That, at least, is what Wittgenstein evidently held. It is why he says, for in-

stance, that ‘ “not-not-p” is the same as “p” ’.2 It is a doctrine which gives riseto some difficulties, however. To explore these difficulties it will be convenientto make use of the distinction Wittgenstein draws in the Tractatus3 between apropositional sign and a proposition. Both are facts, but the identity criteriaare drawn differently in the two cases. A propositional sign is what I shall calla signifying fact: it is the fact that various syntactic units are arranged in thesentence in a certain manner. In the sentence ‘John met Mary’, for instance,the propositional sign is the fact that the word ‘John’ is followed by the word‘met’, which is followed in turn by the word ‘Mary’. The proposition, on theother hand, is a symbolizing fact: it is what the signifying fact becomes whenwe read it as saying something, namely that John met Mary. This distinctionbetween propositional sign and proposition, between signifying and symbol-izing fact, is in the Tractatus an instance of a more general distinction betweensign and symbol. A symbol is what a sign becomes when we read it (or asentence in which it occurs) as saying something.Wittgenstein did not himself draw this terminological distinction between

‘sign’ and ‘symbol’ until he was composing the early part of the Prototractatusaround the end of 1915. Although both these words (as well as their cognates,‘signify’ and ‘symbolize’) occur in the Notes, there is no reason to think thatthis amounts to any more than stylistic variation. This naturally invites thequestion whether the terminological distinctions are relevant to a discussionof the Notes at all, given the anachronism that Wittgenstein had not yet for-mulated it when he wrote them. I hope, though, that enough has been saidin earlier chapters about Wittgenstein’s concern for symbols to establish thatthe anachronism is only terminological: if he had not settled on his later termi-nology, this was not because he had not yet drawn the distinction, but ratherbecause he had hardly any need at that stage to talk about signs at all.

"$.""$.""$." Same sign, different symbolSymbols are what signs become when we invest them with meaning. Whenwe read the words on the page, we turn them into the living expression ofa situation; mere signs, on the other hand, do not yet say anything aboutthe world. One might be tempted to think, perhaps, that what Wittgensteinmeant by a sign was a mere pattern of words on the page. But in that case mytalk of the ‘signifying fact’ would seem puzzling: a pattern is a complex, not a

2C14. 33.12.

Same sign, different symbol !))

fact. Yet my usage here follows Wittgenstein’s own: in the Tractatus he statesexplicitly (three times) that a propositional sign is a fact.4

The question that arises, therefore, is which fact. If we focus on the signrather than the symbol, which features of the pattern of words before us onthe page should we attend to? I have forgotten the Greek I learnt at schoolso comprehensively that I can no longer even recognize which are the nounsand which the verbs; all that has remained is the residual ability to read thecharacters. So when I am shown a piece of classical Greek and read the wordson the page without the least understanding of their grammar or their sense,am I identifying the signifying facts or not?According to the Tractatus, ‘The propositional sign consists in the fact that

its elements, the words, are combined in it in a definite way.’5 Moreover, it isclear from the examples Wittgenstein offers that instances of different gram-matical categories can, sometimes at least, count as the same word. At anyrate, this will have to be the case if the sign–symbol distinction is to be of anyuse in explaining the device of typical ambiguity discussed in chapter 22. Forthe point there was that variables of different types are genuinely of differentgrammatical categories: to think of them as forming a single category wouldjust be to collapse the distinctions on which the theory of types depends.Wittgenstein’s answer therefore appears to be that the grammatical cate-

gory of a word belongs not to the sign but only to the symbol, and hence thatmy almost total ignorance of Greek is no bar to identifying its signs. Oncewe take this view, though, it is far from clear what stable notion of signify-ing fact is available. The point of regarding propositional signs as facts, notcomplexes, was to give them a grammar that put them on course to have arepresentational function, but the grammar of signs now seems too plastic todo any such representational work.When Wittgenstein introduced the terminological distinction between

propositional sign and proposition, it was, superficially at least, because hewanted to talk about cases where the same propositional sign is used to ex-press two different propositions or, more generally, where two different sym-bols can be read into the same sign. This is supposed to occur most notablyin the case of puns. The issue is not quite as clear as it seems on the surface,however. In ordinary language there are cases in which the same sentence isused to express two different propositions (i.e. the same complex exemplifiestwo different symbolizing facts), but it is worth noting that there are two waysin which this can happen.In some cases the right explanation for an ambiguity is that there are two

different propositional signs exemplified in the same sentence. For instance,we might plausibly explain the ambiguity of the complex ‘not p or q’ by de-scribing two different facts: first, that ‘not’ occurs in front of ‘p or q’; second,

43.14, 3.143. 53.14.

!)! Sign and symbol

that ‘or’ occurs between ‘not p’ and ‘q’. It seems right to describe both of theseas facts at the level of signs rather than symbols because they are straightfor-wardly syntactic. There are, in other words, two different propositional signsexemplified in the complex ‘not p or q’. In order to avoid this ambiguity,of course, all that is required is to add appropriate brackets to the sentencein order to disambiguate it. Other examples require other devices (such asconventions to disambiguate variable clashes in predicate logic). We mightcall a notation ‘uniquely parsable’ if no sentence exemplifies more than onesignifying fact.On the other hand, a slightly different case arises when an ambiguity can-

not be explained by a difference of compositional fact residing in the samecomplex; in this other sort of case the explanation of the ambiguity is thatone sign is used to name two different objects, e.g. two different people called‘John’ either of whom we might be referring to in the sentence ‘John metMary’. To regard this as a difference of propositional sign we need to treatthe two uses of the word ‘John’ as different signs, not just different symbols.Let us call a language ‘uniquely readable’ if no component of it is ambiguousas to the contribution it makes to determining the sense of a sentence in whichit occurs.What is clear, though, is that if we are to avoid linguistic confusion, we need

to eliminate both sorts of ambiguity. Let us therefore call a language ‘logicallyadequate’ if it is both uniquely parsable and uniquely readable. In that caseno sentence in the language will exemplify more than one signifying fact, andno signifying fact will allow more than one symbolizing fact to be read in it.I mentioned earlier that Wittgenstein wavered in the Notes over whether thefeatures of ordinary language should be taken as reliable indicators of logicalstructure. In the TractatusWittgenstein famously said that ‘all propositions ofour colloquial language are actually, just as they are, logically completely inorder’,6 but presumably he was by now placing considerable weight on his useof ‘propositions’ rather than ‘propositional signs’. He was suggesting not thatcolloquial language is logically adequate in the sense just adumbrated, butrather that if we do succeed by the use of colloquial language in expressinga proposition, then that proposition is logically completely in order. Makingsense is not, on this view, something we can do by halves.

"$.#"$.#"$.# Same symbol, different signPerhaps it is debatable whether Wittgenstein should have insisted that a dif-ference of symbol entails a difference of sign. What is uncontroversial, how-ever, is that two different propositional signs may be used to express the sameproposition, even in a logically adequate language. In Russell’s notation, for

65.5563.

Same symbol, different sign !)#

instance, !q, !!!q, !!!!!q, etc. are all signs for the same proposition.Even specifying all the signs in this notation which express the same propo-sition as !q is a non-trivial task. It is not just q preceded by an odd numberof instances of ‘!’, since we have to include other signs such as ‘!q v !q’, orindeed ‘!q (p v !p)’. The difficulty of specifying the ‘definite rule’ for gener-ating all such signs is one to which Ramsey drew attention in his critical noticeof the Tractatus. ‘It may,’ he observed, ‘be doubted whether it is possible to for-mulate this rule as it seems to presuppose the whole of symbolic logic.’7 AndRamsey was (as so often) right: to settle whether pv!p is the same propositionas p, for instance, is just to settle whether p is a tautology, and so the formerquestion cannot be any easier to solve than the latter.I mentioned in chapter 6 Frege’s quest for a ‘logically perfect language’.

With the Wittgensteinian terminology now available to us we can express log-ical perfection as the demand that there should be a one–one correspondencebetween signs and symbols, so that we never have two different propositionalsigns expressing the same proposition, as happens with!q and!!!q in Rus-sell’s notation. It was plainly Wittgenstein’s hope at the time of the Notes tobe able to find such a notation. It is not only ordinary language that is notperfect; neither are the tidier languages of the Begriffsschrift and Principia. Thisis something Wittgenstein warned us to be conscious of, since otherwise thereis a danger that we shall be misled by inessential features of our symbolism.

If p=not-not-p etc., this shows that the traditional method of symbolism is wrong,since it allows a plurality of symbols with the same sense; and thence it follows that, inanalysing such propositions, we must not be guided by Russell’s method of symboliz-ing.8

If we can find a notation in which not-not-p is represented by the same symbolas p, Wittgenstein believed, then that must be the right notation and Russell’smust be wrong.

The very possibility of Frege’s explanations of ‘not-p’ and ‘if p then q’, from whichit follows that ‘not-not-p’ denotes the same as p, makes it probable that there is somemethod of designation in which ‘not-not-p’ corresponds to the same symbol as ‘p’. Butif this method of designation suffices for logic, it must be the right one.9

On the other hand, if a perfect notation is impossible, as Frege eventuallycame close to realizing, then of course these pronouncements will have to besoftened somewhat. The content of ‘wrong’ in the first quotation will haveto be understood as no more than that the traditional method of symbolismis misleading if taken as a guide to the underlying complexity of what it ex-presses. A symbolism will be more accurate as a guide to this the more nearlyit approaches the goal of perfection.

7FoM, 278. 8B26. 9B22.

!)$ Sign and symbol

As we saw in chapter 20, a symbolism capable of expressing polyadic quan-tified logic transparently is impossible. Eventually, Wittgenstein himself seemsto have realized this, or at any rate to have given up on the ambition. But itwas plainly his goal at the time of the Notes. There is therefore some justice inRussell’s later tetchiness on the subject.At one time, Wittgenstein agreed with me in thinking that a logical language wouldbe useful in philosophy, and I attributed this view to him in the introduction whichI wrote to his Tractatus Logico-Philosophicus. Unfortunately, by this time, he had notonly abandoned the view, but had apparently forgotten that he ever held it. WhatI said about it therefore appeared to him as a misrepresentation. His followers eversince have vehemently rejected the suggestion that a logical language could possiblybe useful.10

"$.$"$.$"$.$ Symbol in terms of signThere is a natural temptation to explain the symbolizing fact in terms of thevarious signifying facts that can be used to express it. The account Wittgen-stein offered in the Tractatus is as follows.That which denies in ‘"p’ is . . . not ‘"’, but that which all signs of this notation, whichdeny p, have in common.Hence the common rule according to which ‘"p’, ‘"""p’, ‘"p v "p’, ‘"p "p’,

etc. etc. (to infinity) are constructed. And this which is common to them all mirrorsdenial.11

It is worth noting carefully what Wittgenstein does not say in this passage.He does not say that the symbolizing fact in an expression for the negationof p is that it is ‘!p’ or ‘!!!p’ or ‘!p v !p’ or ‘!p !p’, or any of theothers in a range of signs which can be generated according to a certain rule.If he continued to believe at the time of the Tractatus, as he had when hewrote the Notes, that there are no disjunctive facts, it would evidently havebeen necessary for him to avoid such an account, since according to it thesymbolizing fact would be disjunctive.Nor does Wittgenstein say that what symbolizes is the rule according to

which all the various signs for the negation of p are constructed. At the levelof phenomenology, at any rate, this would not be a plausible view to hold.As Ramsey noted, even in the case of Russell’s notation it may be doubtedwhether it is possible to formulate the rule at all; and it seems wildly implausi-ble to ascribe to me even an implicit grasp of such a rule merely on the basis ofmy ability to read signs such as ‘!p’ and ‘!!!p’ reliably. Wittgenstein’s useof the word ‘hence’ in referring to the putative ‘common rule’ suggests that hewished to avoid such an ascription. For each proposition there is a rule codify-ing what is common to all the signs expressing it. Such rules ‘are equivalent to

10MPD, 165. 115.512.

Symbol in terms of sign !)%

the symbols and in them their sense is mirrored’,12 but it does not follow fromthis (or at any rate not directly) that anyone who has the capacity to recognizethe symbol in any sign constructed according to the rule must have an implicitgrasp of the rule.Wittgenstein apparently wanted us to think of symbolizing facts not as dis-

junctions of signifying facts or as rules for generating such facts but as oc-curring at an altogether different level—in, to borrow Frege’s later usage, adifferent realm. Symbols are not worldly, but nor are they of anything likethe same nature as signs. One way to think of this might be by analogy withlooking at a distant object through a patterned window: when we focus onthe distant object, the pattern on the window becomes invisible. In much thesame way, recognizing the symbolizing fact is a matter of adjusting our logicalfocal length so that the sign by means of which the proposition is expressedbecomes invisible. Something very like this is indeed a familiar experience inreading. It is the difference, perhaps, between the experience I have of readingmy native language and that of reading one in which I am not fluent: in theformer, especially if the content of what I am reading interests me, I hardlynotice which words are actually being used; in the latter I am conscious ofeach sentence as I laboriously translate it word by word.In my earlier sketch of their respective theories I followed the orthodoxy

of describing Russell’s as a one-step, Frege’s as a two-step semantic theory.Which, then, was Wittgenstein’s? As I remarked in §7.1, he did not needthe notion of the sense of a proper name for the role which Frege originallyclaimed for it of explaining the different contributions two names of the sameobject may make to the thoughts expressed by sentences in which they occur.Even in a non-perfect language in which there are two signs for the same ob-ject, Wittgenstein held that how they symbolize will be identical: the semanticvalue of a name is the object it refers to, and so there will be only one sym-bol to be discerned in the two signs. If he had succeeded in the project ofdevising a notation that at least approximated to logical perfection, he mightthen have felt able to limit the sign–symbol distinction to the criterial role ofdefining what it is for a language to be logically perfect. It would then havebeen possible to say that, for logically perfect languages at least, his semantictheory counted as one-step in Russell’s sense.But, as we have seen,Wittgenstein eventually gave up the attempt to provide

an explicit language in which the symbols are transparently mirrored in thesigns used to express them. This left the symbol with the role of mediatingthe transition from sign to object, not, as in Frege’s case, to make good aconception of objects as multifaceted, but nonetheless for a purpose that isstill recognizably that of Frege’s senses, namely that of enabling language torepresent a world whose complexity it cannot hope to match. Frege and Witt-

125.514.

!)" Sign and symbol

genstein both bridge the gap between language and world in two steps, withthoughts as the intermediate point, but Wittgenstein’s picture theory, by in-sisting on the transparency of the second step, makes the first, from languageto thought, that much more mysterious.Frege’s third realm of sense has seemed mysterious to many of his readers:

twentieth-century philosophy is littered with attempts to eliminate or demys-tify it. Perhaps the Notes are one of the first such attempts. If so, this is a respectin which the Tractatus is not a continuation of the project sketched in the Notes.The Tractatus not only does not eliminate the mediating role of symbols butit in a certain sense identifies that role with the mystical. This is of course alarge theme, and one that is quite absent from the Notes. What is significantto note here is only that Wittgenstein’s abandonment of the goal of a logicallyperfect language, although not a prerequisite for there to be the mystical, un-doubtedly contributes to its mysteriousness. Recognizing the symbol in thesign is a process which the doctrine of unsayability tells us cannot be put intowords: the impossibility of a logically perfect language gives us a clue to itsimpenetrability.

"$.%"$.%"$.% The symbol vanishesThe job of a proposition, according to the Notes, is to divide facts into twoclasses, those that are of like sense, and those of opposite sense. Wittgensteininvoked an analogy with a line dividing a plane, but tautologies and contradic-tions were to be treated as special cases: the proposition p v !p, for instance,is of like sense with the facts, whatever they are; and in the same way p !pis of contrary sense to the facts. Wittgenstein’s response to this was to say thathere the analogy with a line dividing a plane breaks down: for any line in theplane, there are points on each side of it. Wittgenstein concluded that p v !pis senseless.13

Whether we say this, or instead say that pv!p has a trivial sense, is surely initself just a matter of terminology. But Wittgenstein evidently intended whathe said to have a consequence for our understanding of the proposition. Bysaying that p v !p has no sense, he intended to suggest that there is nothingto be expressed, and hence no symbolizing fact to be discerned in the prop-ositional sign. Later, the analogy he drew between propositions and pictureswould provide him with a reason for holding this view. If a proposition is apicture of how things would have to be for it to be true, then a logical truth isnot a proposition because there is nothing to picture. Saying that this rose iseither red or not red puts no constraint on the rose or anything else. To pic-ture how the world has to be for this to be the case is not to picture anything.

13Cf. §14.3.

The symbol vanishes !)&

If we try to assemble pictorial elements so as to picture a situation in which therose is either red or not red, we stumble: what results is not a trivial picture,but no picture at all. What is notable here, though, is howWittgenstein seems,as on other occasions, to have adopted a view before he had much reason tohold it. This is characteristic of Wittgenstein’s method of theory formation,which put a much greater premium on suggestive analogies than on reasons.Until we have formulated the picturing analogy, it is not really clear why weshould say that p v !p is senseless, rather than that its sense is trivial. Andeven when that analogy is in place, it is still only an analogy: it does not initself dictate what the senselessness of a tautology amounts to.In these remarks in the Notes about the vacuity of the propositions of logic

Wittgenstein was plainly only a small step away from the characterization ofthem as tautologies which he developed in Norway shortly afterwards. Buthe was also an equally small step away from an obvious problem for the cor-responding conception of the symbolism. If in the case of the sentence ‘Thisrose is either red or not red’ there is no symbol to be discerned, then the sameis true of the sentence ‘This rose is both red and not red’. Both, according toWittgenstein, say nothing. Yet they somehow contrive to say it in diametri-cally opposite ways. The duality between positive and negative facts, whichgave him no peace,14 here has its echo in the case where no facts are in ques-tion.

14NB, 25 Nov. 1914.

Chapter "%"%"%

Wittgenstein’s theory of judgment

In chapter 13, I described how Wittgenstein tore Russell’s multiple relationtheory of judgment to shreds. That discussion left one salient piece of unfin-ished business, however. The nub of Wittgenstein’s complaint against Rus-sell’s analysis, let us recall, was that in it the verb of the proposition believeddid not occur as a verb. If I say that A believes that p, I must express what itis that A believes, even though I do not myself assert it. So p must occur inthe analysis as a fact, not as a complex. But even if we are now clear what waswrong with Russell’s theory, it still remains to be determined what should beput in its place.

"%.!"%.!"%.! Russell’s later viewsAlthough it does not belong to the principal thread of our narrative, it is per-haps worth noting first that Wittgenstein’s criticisms did not lead Russell al-together to give up hope of resurrecting some version of the multiple relationtheory. When he reprinted in 1917 one of the articles in which he mentionedit, he added in a footnote, ‘I have been persuaded by Mr Wittgenstein thatthis theory is somewhat unduly simple, but the modification which I believe itto require does not affect the above argument.’1

Unfortunately Russell did not go on to explain the modification he had inmind. Some time later, though, he was a little more explicit. ‘Belief will re-ally have to have different logical forms according to the nature of what isbelieved.’2 What he had in mind was presumably that he could escape fromthe difficulty by positing different kinds of judging—a judging relation J1 ap-propriate for judging subject-predicate propositions, another judging relationJ2 appropriate for judging relational propositions, and so on. As he noted,‘The apparent sameness of believing in different cases is more or less illusory.’And of course he had been quite happy to take the analogous course with theconcept of truth as a way of resolving the liar paradox: in Principia he heldthat there is not just one concept of truth but a whole hierarchy of them. So itwould not have seemed to him to be too large a step to say similarly that there

1CP, VI, 154. 2CP, VIII, 199.

The theory of judgment in the Notes !)(

is a whole hierarchy of judging relations. (At any rate, it is surely a lot moreplausible to say this about judgment than about truth.)A moment’s thought shows, however, that the hierarchy of judging rela-

tions will have to be of considerable complexity. For eventually Russell wouldhave to deal with the cases where we judge not only atomic propositions butmore complex ones built up from them. For instance, he was in danger ofneeding a special relation just for judging ‘p or q’ where p is a subject-predicateproposition and q is relational. One of his students, Dorothy Wrinch, pub-lished an attempt to extend the multiple relation theory to molecular prop-ositions in 1919, but it was (as she herself admitted in her article) both verytentative and very complicated.3 Russell himself made no attempt to developthe idea, and it seems very likely that he regarded this as one of the matters hewould leave to Wittgenstein.

"%.""%.""%." The theory of judgment in the NotesIn the Notes, as we have seen, Wittgenstein was in the grip of the idea thatwhat makes the proposition p expressive is its relationship to its two poles aand b. So when he wanted a proposition to occur as a fact, not a complex,one way he liked to indicate this was to draw in its poles explicitly. His way ofrepresenting the idea that in ‘A believes that p’ the proposition p is expressedwas therefore to say that it consists in a relationship not between A and theproposition p but between A and the poles of p.‘This is obviously not,’ Wittgenstein says with some understatement, ‘a re-

lation in the ordinary sense.’4 Indeed not, because the poles a and b are notobjects in the ordinary sense. In fact, they are not objects at all. So it is puz-zling what relationship I am supposed to enter into when I believe something.If I conceive of it as a relationship between me and the proposition p, it seemsthat p will inevitably occur in the relationship as a complex, not as a fact. Onthe other hand, it cannot in any useful sense be thought of as a relationshipbetween me and the poles of the proposition, because there are no such things.So it is hard to avoid the thought that the diagram

Aa p b

which Wittgenstein offered in the Notes to represent his analysis of ‘A believesthat p’ is in its way just as desperate as the map of belief Russell had drawn inhis Theory of Knowledge manuscript.Wittgenstein was plainly struggling. All he could offer was the rather vague

suggestion that ‘the epistemological questions concerning the nature of judg-ment and belief cannot be solved without a correct apprehension of the form

3‘On the nature of judgement’. 4C41.

!!* Wittgenstein’s theory of judgment

of the proposition’.5 As it turned out, though, he was looking in the wrongplace. His error lay not in the bottom half of the diagram (the understandingof the proposition p), but the top half (the occurrence of the subject A as aterm in the relation). As long as A occurs in our analysis of the judgment, wewill have no alternative but to conceive of the judgment as consisting in a rela-tion between A and what is judged; yet we cannot relate A to the symbolizingfact p, precisely because it is a fact. The only entity in the vicinity for A to berelated to is a complex, not a fact, and hence not what we want.Wittgenstein’s error was thus that he had placed incompatible demands on

his theory. The proposition had to occur in the diagram as a propositionin order that the judgment should express what is judged; but the subject Acould not occur in the judgment as a further term outside what is judged andrelated to it, because ‘propositions, owing to sense, cannot have predicatesand relations’.6

"%.#"%.#"%.# Wittgenstein’s later theory of judgment

We can trace out a route (perhaps even Wittgenstein’s own route) to his even-tual theory of judgment if we start again from the theory of Russell’s that herejected. Let us consider, for simplicity’s sake, a relational proposition ‘aRb’.As we saw in chapter 13, Russell wanted to analyse this as #2(a,R,b), whereasWittgenstein had rejected this idea. But the question of how far into the prop-osition the form reaches is not what is at issue at the moment. So let us heresimply side with Wittgenstein and represent the symbolizing fact in ‘aRb’ ashaving two constituents ‘a’ and ‘b’ combined together according to a certainform. And let us, borrowing a notational device from Geach,7 refer to thisform as §R. In that case Russell’s theory, adapted to recognize Wittgenstein’sanalysis of the proposition, would have analysed A’s judgment that aRb as

J(A,§R,a,b),

i.e. it would have included as a term the form of what is judged. But that, aswe have seen, is precisely what Wittgenstein objected to—that §R occurredas a term and not as a form. So let us respond to this objection by removingthe form. We cannot let it disappear entirely, though: judging that a has R tob is not the same as judging that a has S to b, and that difference has to berepresented in the analysis. So let us indicate the difference by subscriptingthe judgment relation correspondingly, thus obtaining

J§R(A,a,b).

5B55. 6C15. 7Mental Acts, 52.

Wittgenstein’s later theory of judgment !!)

Thus far, we have not gone beyond anything available to Wittgenstein atthe time of the Notes. But now there is one further step to be taken, and it re-lates to the term in the relation that has not so far been the object of criticism,namely the subject A. What we saw in the last section was that judging can-not consist in a relation between A and the proposition, because propositionscannot be related to anything. But in the putative analysis just considered, Ais related not to a proposition but to its constituents. So although the previousargument is no doubt suggestive enough to give us reason to experiment withdropping A, it does not in itself prove that we must. It does not, that is to say,show that there is no more reason to think A is simple than that p is.But recall that ‘a’, §R, and ‘b’ combine to make the symbolizing fact

§R(a,b). If we are to respect the bareness of the notion of a logical picture,we have to think of this as a mere notational variant on aRb. In that case itis hard to see how there can be any place for A. If ‘aRb’ is complete, there isno relating left to do. (This is another way of putting the point made in thelast section that our analysis of ‘A believes that p’ cannot both express p andrelate it to A.)We conclude, then, that when the judgment relation is fully analysed, the

subject A will not occur as a term in it: so let us remove it. What this givesus is J§R(a,b). Of course, we would expect something similar to apply to otherattitudes we might take to the proposition. So ‘A doubts that aRb’ might beD§R(a,b), and ‘A believes that aRb’ might be B§R(a,b). What strikes us immedi-ately is that none of these is very different from §R(a,b). In other words, whatit is for A to judge, doubt, or believe that a has R to b is very like what it issimply for the symbolizing fact §R(a,b) to occur.Once he had given up the idea that the subject A occurs as a term in judg-

ment, the way was clear for Wittgenstein to reach the theory of judgmentoffered in the Tractatus. According to this theory, all that has to be addedto the occurrence of the symbolizing fact in order for it to count as a case ofsomeone’s believing something is that its constituents should occur ‘connectedin his mind and accompanied by a feeling of belief’.8 Similarly, a case wheresomeone doubts something will be analysed as the constituents of the proposi-tion connected in the person’s mind accompanied by a feeling of doubt. Andso on for other propositional attitudes.The occurrence of the constituents connected in the mind in this manner

expresses the proposition, and so the analysis fulfils Wittgenstein’s require-ment that it should express what it is that is judged. As he put it in the Trac-tatus, ‘ “A believes that p”, “A thinks p”, “A says p”, are of the form “ ‘p’ saysp”.’9 This account ‘shows that it is impossible to judge a nonsense’ because itinvolves us in expressing the proposition that is judged. If there were a judg-

8Ramsey, FoM, 145. 95.542.

!!! Wittgenstein’s theory of judgment

ment that the table penholders the book, for instance, it would have the sameform as the expression,

‘The table penholders the book’ says that the table penholders the book.

But this is nonsense. I can look at the string of words, ‘The table penholdersthe book’, but I cannot read it as saying anything. That is to say, I cannot dis-cern in it a symbolizing fact. So I cannot configure any of the contents of mymind so as to represent such a symbolizing fact, since there is no such symbol-izing fact to represent. The decisive step, then, is the one Wittgenstein took inNorway of realizing that judgment cannot be any sort of relation between thejudging subject and either the proposition or its components, and hence thatthe subject has no place in the analysis of judgment at all.

"%.$"%.$"%.$ RamseyWhether or not it quite represents Wittgenstein’s own route to his Tractar-ian theory of judgment, the narrative just sketched has one significant virtue,which is to bring out the similarity between that theory and Russell’s multiplerelation theory. This similarity is especially worth bearing in mind when weare assessing alternative accounts of Wittgenstein’s objection to Russell’s the-ory. It is true that Wittgenstein’s objection was general enough to apply to allthe variants of that theory that Russell came up with: that is to say, it appliesto any analysis of ‘A believes that p’ in which the judging subject A is multiplyrelated to the components of p. But any account which represents Wittgen-stein as offering an objection to any multiple relation theory whatever is veryunlikely to be correct, for the straightforward reason that Wittgenstein’s isalso a multiple relation theory. The difference is that in Wittgenstein’s theorywhat are multiply related are just tokens of the components of p: the judgerA disappears on analysis. This is important because it permits these tokens tooccur in the analysis of the judgment with their ordinary grammatical values.What happens when Othello believes that Cassio loves Desdemona is thatthere occurs as a belief in Othello’s mind a token of the thought that Cassioloves Desdemona. Or, more precisely, there occur in his mind tokens of thecomponents of this thought, combined just as these components are combinedin the thought. Among them is a token of the verb ‘loves’; and it occurs in thebelief just as the verb occurs in the thought, i.e. as a verb, not as a noun. Thusthe flawWittgenstein had identified in all of Russell’s various multiple relationtheories is not present in his own.What the Tractatus offers may thus with some justice be termed a multiple

relation theory of judgment. What connects the components in Wittgenstein’sanalysis is not what he called a ‘relation in the ordinary sense’—such as a spa-tial relation, for instance—because one of the components is a form; but it

Ramsey !!#

is at any rate more similar to Russell’s multiple relation theory than it is toany of the other theories of judgment that had been proposed. Surprisinglyfew commentators have remarked on this similarity. One who did was Ram-sey, whose account of Wittgenstein’s theory in ‘Facts and propositions’ clearlypresents it as a multiple relation theory.Ramsey’s treatment of Wittgenstein’s theory of judgment has other virtues

too. One is that he recognized, even before he had discussed it with Witt-genstein himself, how little Wittgenstein cared about describing the particularfeatures of the mental tokens involved in belief or doubt or judging. Anotheris that Ramsey recognized quite explicitly (again before he had discussed thematter with Wittgenstein) the close affinity Wittgenstein saw between his the-ory of judgment and his theory of meaning. In his critical notice on the Trac-tatus Ramsey remarks that Wittgenstein ‘explicitly reduces the question as tothe analysis of judgment, to which Russell has at various times given differentanswers, to the question “What is it for a proposition token to have a cer-tain sense?” ’10 If we ignore the feeling of belief or doubt or judging, thenfor someone to adopt a propositional attitude towards a proposition is for theconstituents of the proposition to be connected in the person’s mind so as toexpress the proposition concerned. Only thus can the verb of the propositionjudged occur as a verb rather than as a term.Because the judging subject A has disappeared, the kernel of Wittgenstein’s

theory of judgment is now indistinguishable from his theory of meaning. ‘Herewe have no co-ordination of a fact and an object, but a co-ordination of factsby means of a co-ordination of their objects.’11 The semantic task I face ifI am to judge something is to correlate the components of what I judge withsymbols that go proxy for them. Having done this, all that remains is for me toassemble those proxies so as to express what I am judging to be the case. Whatis operative in judgment is the relationship thus created between these proxies;this is just the same as the relationship that holds between the correspondingobjects if what is judged is the case.

10FoM, 274–5. 115.542.

Chapter "&"&"&

The picture theory

Everyone who writes about the Tractatus agrees that the picture theory is oneof its central themes. There is less agreement, though, about just what ismeant by the ‘picture theory’. Some use the phrase very broadly as a sort ofcatch-all for the logical doctrines of the book, others much more narrowly forthe specific proposal that propositions are pictures of reality. Exponents ofthe resolute reading of the Tractatus sometimes take ‘the picture theory’ to bejust what we recognize to be nonsense when we throw away the ladder. And asimilar vagueness of reference is widespread in the literature on Wittgenstein’slater writings, where it is commonly (and dubiously) asserted without furtherspecification that he ‘abandoned the picture theory’. The question whetherthe picture theory is already present in the Notes is correspondingly ambiguous.Here we shall piece together the various elements that make upWittgenstein’ssemantic theory in an attempt to disambiguate this question.

"&.!"&.!"&.! Coincidence of structure

It is of course trite to observe that the words we use might have meant some-thing different. It is, for instance, an accident of etymology that ‘black’ doesnot mean white, and a blessing I owe only to my parents’ good taste thatmy name is not ‘Rocky’ or ‘Dwayne’. But this trite observation immediatelyprompts a less trite question. If it is contingent which words mean what, whatensures that our words mean anything at all? What constraints are there onour choice of words? One first attempt at a sketch of Wittgenstein’s answerwould be to say that what constrains us is grammar. Grammar, according tohim, is not contingent as word selection is. But to recognize grammar as anessential feature of any language that succeeds in talking about the world isnot yet to offer an explanation for that success. We get closer to the essenceof the picture theory when we note Wittgenstein’s belief that the grammar oflanguage cannot help but be in a certain sense identical to the grammar of theworld.We noted earlier that Wittgenstein derived his distinction between names

and forms from Frege’s distinction between saturated and unsaturated ex-pressions. But, although the distinction between saturated and unsaturated

Coincidence of structure !!%

was prompted by linguistic considerations, and is at its most natural when ap-plied to language, Frege did not restrict its application to language. He heldthat the things in the world to which saturated and unsaturated expressionsrefer are correspondingly saturated or unsaturated. Names, which are satu-rated expressions, refer to objects, which are saturated; concept words andrelation words, which are unsaturated expressions, refer to concepts and rela-tions, which are unsaturated entities. In the same way, Wittgenstein intendedthe distinction between two kinds of components of symbolizing facts to tracka distinction between the components of facts in the world. When he said,‘Components are forms and constituents’,1 he intended the remark to applyto all facts, not just those that symbolize. In the case of a non-symbolic fact,the constituents would be objects and the form would be the relation betweenthem (or, in the case of one object, the property of it) whose holding consti-tuted the fact.The idea that there is a harmony between the structure of a fact and the

structure of the symbol that expresses it is quite explicit in the lectures Russellgave at Harvard in 1914, where he said that ‘the structure of the symbol mustbe identical with the structure of the symbolised’,2 and that ‘there is alwaysa sort of fundamental identity between symbol and symbolised’.3 In 1918,moreover, we find Russell in ‘The philosophy of logical atomism’ hoping topersuade his audience

that in a logically correct symbolism there will always be a certain fundamental identityof structure between a fact and the symbol for it; and that the complexity of the symbolcorresponds very closely with the complexity of the facts symbolized by it.4

In the Notes, on the other hand, Wittgenstein does not express the point inthis general manner. Instead he makes it laboriously case by case, at least inrelation to elementary propositions: if an atomic fact consists of a form andone object, the elementary proposition which expresses it must be of subject-predicate form; and similarly for other forms.

Every proposition which says something indefinable about a thing is a subject–pred-icate proposition; every proposition which says something indefinable about two thingsexpresses a dual relation between these things, and so on.5

The point, though, is at root the same: it is because the structure of the symbolmust be identical with the structure of what it symbolizes that a propositionwhich says something indefinable about a thing must be a subject-predicateproposition.The doctrine that ‘the structure of the symbol must be identical with the

structure of the symbolized’ is certainly a key component of the picture the-ory; moreover, it is present in the Notes on Logic, and directly traceable to thesymbolic turn that we noted in the letter Wittgenstein wrote to Russell shortly

1C50. 2Eliot’s notes, 7 Apr. 1914. 311 Apr. 1914. 4CP, VIII, 175. 5B77.

!!" The picture theory

after his visit to Frege at Christmas 1912. On the other hand, although it ispresent in the Notes, I would hesitate to say that it is fully secure there, becausethere is no sign that Wittgenstein yet had the resources to establish the har-mony between proposition and fact at any but the atomic level. Only when hehad a developed account that could explain away the case in which an appar-ently complex expression such as ‘the object on the table’ refers to somethingsimple would he be entitled to the claim in its full generality.

"&.""&.""&." The picturing analogyIt is, of course, natural to think of the essence of the picture theory as beingthe eye-catching claim6 that a proposition is a picture. But to decide whetherthat natural thought is right, we need to look at how far the eye-catching claimtakes us beyond Russell’s remark in 1914 that the structure of the symbol mustbe identical with the structure of the symbolized.The picturing analogy itself is not present in the Notes; the analogy that

guides Wittgenstein’s account of propositional expression there is more likethat of a compass needle being held next to a magnetic field to detect itsdirection (see §16.3). In the Tractatus, that analogy is almost wholly absent.Although propositions still have sense, the punning suggestion that this is tobe thought of as like the direction of an arrow has been all but dropped.7

What replaces it is a new analogy according to which a proposition is a pic-ture. Straightaway, though, the analogy has to be interpreted with care. Aproposition is still a fact, as it was in the Notes. And, it emerges,8 a picture isa fact too: it has to be, since propositions are a particular kind of picture (towit, the logical kind). So what Wittgenstein means by a picture cannot be apictorial complex, an arrangement of things on a canvas or wherever; it mustrather be a certain fact that we see exemplified in the arrangement when welook at it in a certain way.As we have seen, the idea that the structure of the proposition is the same

as the structure of what it represents is one that Wittgenstein acquired early.But so far the point is essentially negative. What the analogy of the pictureaims to add to it is something positive—an explanation for the harmony. It isby standing in a certain configuration that the components of a propositionrepresent that that is how things are.We can be fairly definite about when Wittgenstein came upon the idea of

using this analogy to express how a proposition represents a possible situation.It is not mentioned in the Notes on Logic or the Notes dictated to Moore, but itis at the centre of the discussions early in the wartime notebooks. ‘In theproposition a world is as it were put together experimentally. (As when inthe law-court in Paris a motor-car accident is represented by means of dolls,

64.021. 7Cf. 3.144. 82.141.

Truth !!&

etc.)’9 There is little reason to doubt his later recollection that he was led tothis thought by a description of a court case which he had recently read in amagazine. If we follow the popular practice of regarding this analogy as thecrux of the picture theory, therefore, we may say with some confidence that itwas born in August or September 1914.On the other hand, there is room to question how significant the analogy

really is for the Tractatus. Wittgenstein later remarked that the Tractatus con-tains kitsch,10 and there is something to be said for the view that the picturinganalogy is among the kitsch. For a spatial picture represents a way thingsmight be configured in space, and it does so by configuring representatives ofthese things in space in just this manner. But a proposition is a logical picture.What this means is that the manner of combination in a proposition is notspatial or musical, or of any other particular kind. Rather is it whatever allthese particular kinds of combination have in common. One way of puttingthis would be to say that the notion of a logical picture has a certain thinnessto it: there is something insubstantial about the manner of combination, incontrast to the case of a spatial picture, for instance, in which the manner ofcombination itself plays a representational role. And this thinness is essentialto Wittgenstein’s account. Only by insisting that there is nothing substantial tohow the components of a proposition are combined couldWittgenstein ensurethe generality of logic—keep faith, that is to say, with the motivating insightthat ‘logic cannot treat a special set of things’.11 If, for instance, what sym-bolized in a proposition were the spatial relationship between its components,what the proposition represents could only be a spatial relationship betweenthings—spatial things. And that would be to restrict the generality of logic.It is vital, therefore, that there should be nothing to the manner of combina-tion of the parts in a proposition—nothing, that is, except the bare minimumthat is present in any picture. But Wittgenstein does not explain why thereshould even be such a minimum. Why, that is, should we suppose that thereis anything various particular kinds of representation have in common?

"&.#"&.#"&.# Truth

I said earlier that in the Notes Wittgenstein did not aim to explain the truthor falsity of a proposition directly but went via a comparison between theproposition and the world. That is why he offered in the first instance anaccount of what it is for a proposition to be of like or contrary sense to thefacts rather than of what it is for a proposition to be true. So, to complete theexplanation, we need to add the stipulation that

(T) A proposition is true if and only if it is of like sense to the facts.

9NB, 29 Sep. 1914. 10Diary, 16 May 1930. 11C9.

!!' The picture theory

But at this point a worry emerges. I have just described (T) as a stipulation.Could we then have made the opposite stipulation?

(T!) A proposition is true if and only if it is of contrary sense to the facts.

If we adopted this amended truth scheme but left everything else in our lan-guage unchanged, the effect would be to reverse the sense of every proposi-tion.Although it is very similar, this is not in fact quite the same as the proposal

we considered in §10.2 of expressing ourselves by means of false propositionsrather than true ones. Both proposals result in its being appropriate to say‘not-q’ just in case in the standard interpretation it is appropriate to say ‘q’,but the reversals occur at different points: the earlier reversal attempted, ineffect, to swap the roles of truth and falsity; the one now under considerationleaves these unchanged but reverses the sense of every proposition.Wittgenstein correctly blocked the earlier reversal by insisting on the close

link between truth and assertion: a proposition is true just in case things are asit asserts them to be. This observation is of no help in blocking the stipulation(T!), however. If the link between truth and assertion is too tight for there tobe any room for choice in the matter, what has been said so far does not makeit clear that the same can be said of how the truth conditions for a propositiondepend on the meanings of its components. It seems, that is to say, as thoughwe have a genuine choice about how we fill in the dots in the truth-scheme

‘p’ is true if and only if . . .

There is plainly a difficulty here. At the very least, what we must not dois to think of the way in which the components of the proposition are put to-gether as a further substantial element in it. I mentioned in chapter 10 thatWittgenstein distinguished psychology, which attaches meaning to names andsenses to forms, from logic, which deals with how these fit together to formpropositions. In terms of this distinction the danger is that of seeing any-thing substantial in the logical part of the process. This, of course, is anotherapplication of Wittgenstein’s fundamental thought, that logic has no subjectmatter. But it is in any case urged on us by the need to avoid a version ofBradley’s regress. If we call the way in which the pieces of a proposition areassembled its structure, that should not be taken to reify the structure but onlyto introduce a variant way of talking about how the pieces are assembled.The difficulty, then, is that Wittgenstein’s account of sense has two parts.

First, a proposition is a standard to which facts behave. Wittgenstein spokeof facts as being either of like or of contrary sense to the proposition, but hemight just as well have used the language of correspondence: he might, that isto say, have talked of a proposition as corresponding or failing to correspondto the facts. Second, the proposition is true if the facts are of like sense to it,

The identity theory !!(

i.e. if they correspond to it. The proposition offers a standard to which wemay compare the world; and the proposition is true if the comparison comesout one way rather than the other.But this is wrong: where it has two parts, Wittgenstein’s account should

only have one. Although the variant truth-scheme (T!) is not quite the sameas the variant language of the inveterate liar, the mistake it exposes is thesame. What a proposition expresses is already true or false, and if I assertthe proposition, what I assert is just that—what it expresses. There cannotbe an intermediate stage in the explanation where a proposition represents apossible situation while yet remaining neutral as to whether it represents thesituation as obtaining or not: to represent a situation is the very same thing asto represent it as obtaining. And to assert the proposition is just to say that thesituation does obtain. Wittgenstein’s talk of a proposition as being true to thefacts is therefore faulty. What else could a proposition be true to?

"&.$"&.$"&.$ The identity theoryThe compass-needle analogy encouraged us to think of a proposition as ameasure that has to be held up against reality to see whether reality corre-sponds to it. That, we have now seen, was a weakness, because it suggestedthat there was a gap to be bridged between what the proposition expressesand how reality has to be if the proposition is to be true. But if this is an er-ror that the compass-needle analogy encouraged, the picturing analogy withwhich Wittgenstein replaced it does not on its own prevent the error. If wethink of the picture as a measure that we hold up against reality, we fall intothe very same mistake. We could just as well adopt the convention that pic-tures are to be read as saying ‘This is how things are’, or equally ‘This is howthings are not.’The difficulty is one that Frege pointed out eloquently.

It might be supposed . . . that truth consists in the correspondence of a picture withwhat it depicts. Correspondence is a relation. This is contradicted, however, by theuse of the word ‘true’, which is not a relation-word and contains no reference to any-thing else to which something must correspond. If I do not know that a picture ismeant to represent Cologne Cathedral then I do not know with what to compare thepicture to decide on its truth. A correspondence, moreover, can only be perfect if thecorresponding things coincide and are, therefore, not distinct things at all. . . But thisis not at all what is wanted when truth is defined as the correspondence of an idea withsomething real. For it is absolutely essential that the reality be distinct from the idea.But then there can be no complete correspondence, no complete truth. So nothing atall would be true: for what is only half true is untrue. Truth cannot tolerate a more orless.12

12‘Der Gedanke’, 59–60.

!#* The picture theory

Frege’s warning is directed so pointedly against the picturing analogy that itwould be easy to imagine he wrote it in response to something in one of hisconversations with Wittgenstein. But in fact the distinction Frege draws herebetween propositions and pictures is already present in the 1897 draft of ‘DerGedanke’. The most that Frege could have done in response to Wittgensteinis to expand the discussion a little.Eventually, Wittgenstein himself seems to have realized the point that Frege

was making, but it was not his adoption of the picturing analogy that led himto it. Even when that analogy was in place, Wittgenstein repeatedly worriedabout the apparent gap between the picture and what it represents.

If the proposition is given, and congruence, then the proposition is true if the situa-tion is congruent with it. Or: the proposition is given and non-congruence; then theproposition is true if the situation is not congruent with it.13

It took him some time to realize that the solution to the problem was to removethe gap that the notion of congruence (or non-congruence) was designed to fill.We might indeed ‘show how not to fence by means of fencing puppets’,14 butfencing puppets constitute only a pictorial complex, not a picture. The sensein which a proposition is a picture is a sense in which it already expresses howthings are if it is true: it is true, namely, just in case things are as it representsthem to be.It might well be argued, then, that what is essential to the picture theory of

the Tractatus is not its appeal to the picturing analogy on its own, but ratherthe manner in which the theory takes to heart Frege’s admonition that corre-spondence comes in degrees but truth does not. What is essential, that is tosay, is that it aims to be an identity theory, not a correspondence theory.

That the elements of the picture are combined with one another in a definite way,represents that the things are so combined with one another.In order to be a picture a fact must have something in common with what it pic-

tures.In the picture and the pictured there must be something identical in order that the

one can be a picture of the other at all.15

There is little sign in the surviving texts that Wittgenstein advanced this faruntil perhaps the very end of 1914. Only then does he seem to have becomequite clear that the representational form of the propositional sign does notmerely correspond, but is identical, to the situation it represents.16 ‘Non-truth,’he noted some months later, ‘is like non-identity.’17 And, he could presumablyhave added, truth is like identity. Or, as Ramsey crisply put it, ‘ “The fact thata has R to b exists” is no different from “a has R to b”.’18

13NB, 3 Nov. 1914. 14NB, 5 Nov. 1914. 152.15–2.161. 16NB, 18 Dec. 1914. 17NB, 27 Apr.1915. 18FoM, 143.

Possibility !#)

"&.%"&.%"&.% PossibilityThe picture theory claims, then, that language succeeds in representing theworld because the structure of the proposition is identical to the structure ofthe state of affairs it represents. But there is one last gap in the account thatneeds to be closed. The structure of a proposition, in the sense just referred to,is grammatical structure: each component of the proposition belongs to a gram-matical category, and they are combined together in accordance with gram-matical rules in whose expression these categories figure. It follows, therefore,that the state of affairs represented also has a grammatical structure: its com-ponents belong to grammatical categories just as the words that refer to themdo. But one might now be perplexed as to what this grammatical structureamounts to. What does it mean for the world to have a grammar?In the Tractatus Wittgenstein’s answer is that the grammar of the world

consists in the possibilities of combination of its parts. The grammatical typeof an object consists in ‘the possibility of its occurrence in atomic facts’,19 i.e.in its possibility of combination with other objects. The contrast betweenobject and fact is thus represented as being that between the necessary andcontingent features of the world: objects are what is constant; what can varyis which of these constant elements are configured to form atomic facts andwhich are not.20

But there is no trace of this answer in the Notes, where the notion of possibil-ity is not discussed at all. We saw in earlier chapters how convoluted were theexplanations Wittgenstein gave in an attempt to explain what different propo-sitions of the same form have in common: repeatedly he resorted to actual-world quantification to obtain the generality he needed. However naturallythe conception presented in the Notes of a world made up of facts, positive andnegative, invites a conception of possible worlds as varying only according towhich of the atomic facts obtain, there is no sign that Wittgenstein came uponsuch a conception until he was in Norway. If so, perhaps the timing was notcoincidental. For both the suspicion of possible worlds and the attempt to useactual-world quantification in their stead, which made Wittgenstein’s accountin the Notes so convoluted, are thoroughly Russellian. It was Russell, after all,who claimed that talk of possibility and necessity is simply a confused way ofexpressing the quantifiers: to say that f is possible, he maintained, is just tosay (

E

x) f x; and to say that f is necessary is just to say (x) f x. Perhaps, then, itwas only the move to Norway that gave Wittgenstein the distance from Rus-sell that he needed in order to see that the possibilities of configuration of theworld are encoded in the grammar of a proposition rather than expressed byit—shown by language, not said in it.

192.0141. 202.0271.

Chapter "'"'"'

Tractarian objects

As I stressed at the outset, the aim of this book has been to treat the Notes onLogic as a philosophical work in its own right, not merely as a source of someof the leading ideas of the Tractatus. Pursuant to that objective, I have notdevoted much space to discussing how the ideas in the Notes developed in thetransition to the Tractatus. Nonetheless, there is one change in Wittgenstein’sviews on which I think misunderstanding is sufficiently widespread to be worthclearing up here. It concerns the analysis of elementary propositions.Recall first that in the Notes Wittgenstein (or Russell on his behalf) drew

a distinction between components and constituents. ‘Components are formsand constituents.’1 ‘Every proposition which says something indefinable aboutone thing is a subject-predicate proposition.’2 Such a proposition contains,he says, ‘only one name and one form’,3 and therefore has one constituentbut two components. In 1913, then, Wittgenstein analysed ‘Socrates is mor-tal’ into two parts of fundamentally different kinds. The first is the name‘Socrates’: the second is what he called the form of the proposition. In the sameway ‘every proposition which says something indefinable about two things ex-presses a dual relation between these things’:4 Wittgenstein analysed the prop-osition ‘aRb’ into three components, two names ‘a’ and ‘b’ and a form. WhatI want to consider now is what happened later: did Wittgenstein continue toargue for this analysis in the Tractatus?

"'.!"'.!"'.! Relations as objectsWe have already seen that it is quite hard to think of good reasons for Witt-genstein’s claim that there cannot be different kinds of things. It is in a senseno great surprise, therefore, to find that he eventually gave up that part of theanalysis. Although he continued to hold that there are three components in‘aRb’, his view in the Tractatus was that all three, not just ‘a’ and ‘b’, are names:the third element is a name too (and, correspondingly, what it refers to is anobject).What I have just asserted is regarded in some quarters as controversial.

Various authors5 have claimed over the years that in aRb there are only two1C50. 2B77. 3C48. 4B77. 5E.g. Copi, ‘Objects, properties, and relations in the Tractatus’,Carruthers, Tractarian Semantics.

Widening the scope !##

objects, and that what differentiates it from aLb is only the manner in whichthese two objects are combined. Yet it is rather puzzling that there should bethis controversy, since the historical evidence that by the time of the Tractatushe took there to be three objects in aRb, not two, is overwhelming.Briefly, this evidence is as follows. First, in the early 1930s Desmond Lee

asked Wittgenstein to explain proposition 2.01 of the Tractatus, ‘An atomicfact is a combination of objects (entities, things).’ Wittgenstein’s response was,‘ “Objects” also include relations; a proposition is not two things connected bya relation. “Thing” and “relation” are on the same level. The objects hangas it were in a chain.’6 That is of course not in itself absolute proof of whathe thought when wrote the Tractatus: after all, it is not from Wittgenstein’sown pen, and it is no doubt prudent to leave room for the possibility that onsome aspects of the Tractatus his later memory about his intentions was notwholly accurate. On the other hand, Wittgenstein was certainly very readyin the 1930s to admit to changes of mind about some other things he said inthe book—earlier in the same conversation, for instance, he told Lee that 1.12was based on ‘an erroneous idea’—whereas in this case he seems to have beenquite definite about what he had meant.The second piece of evidence provides some reassurance on this point be-

cause it dates from significantly earlier. Surviving notes of Ramsey’s lecturesin 1925 show him unequivocally expounding, as if it is Wittgenstein’s, theview that the relation is an object. Ramsey had discussed the book in de-tail with Wittgenstein in September 1923, and it is surely very likely that thiswould have been one of the issues Ramsey raised then. His exegesis of the texttherefore has some claim to authority.The third piece of evidence is earlier still: Russell’s Introduction to the

Tractatus confidently asserts that ‘wise’ is a constituent of ‘Socrates is wise’.7

He too had discussed the book with Wittgenstein himself—they spent a weekin The Hague in December 1919 going through the whole book ‘point bypoint’8—and if Russell was by then in any doubt about the answer to thequestion, it is hard to believe he would not have asked it.The view that there are two objects in ‘Socrates is wise’, and three in ‘aRb’,

is also explicit in Wittgenstein’s own pre-Tractatus writing. In 1915, he wrotein his notebook, ‘Relations and properties, etc. are objects too.’9

"'.""'.""'." Widening the scopeWhat we can be certain of, therefore, is that something in Wittgenstein’s con-ception changed after 1913. In the Notes the relation is not a third object inthe fact that aRb; from 1915 onwards, it is. In the Notes Socrates is the onlyobject in the fact that Socrates is mortal; by 1915 he is not. Something had6Lectures, 1930–1932, 120. 7CP, IX, 104. 8BR to Constance Malleson, 16 Dec. 1919. 9NB, 16June 1915.

!#$ Tractarian objects

changed, then, but what? If we read the letter of January 1913 as addressingonly the question of whether mortality is an object, it may seem as if the Trac-tatus marks a return to the view Wittgenstein was rejecting in that letter. Butthis is wrong. The fundamental point of the 1913 letter was its rejection notof mortality but of the copula, and that rejection continued to hold force in theTractatus. The doctrine in the Tractatus continued to be, that is to say, that theproposition ‘Socrates is mortal’ contains only two components: one of themis the name ‘Socrates’; the change was that the other was now to be called aname rather than a form. What Wittgenstein had given up, therefore, was theclaim that there cannot be different kinds of things (and, correspondingly, thatthere cannot be different kinds of names). The second element in ‘Socratesis mortal’ continued to be of a different kind from ‘Socrates’, and it thereforecontinued to be ‘impossible to substitute the wrong way round’, because thetwo symbols are still ‘of a different kind themselves’.Indeed, there is every reason to think that the rejection of the copula was

now even more important to Wittgenstein than it had been in 1913. For hehad by now seen clearly the force of his conception of logic as contentless. Isuggested in §5.7 that in the Tractatus he used this conception as a way of cir-cumventing the problem which had led Kant to the transcendental deductionof the categories, and hence to transcendental idealism. But the rejection ofthe copula is simply a particular case of this. If there were such a thing asthe copula, a substantial element whose presence in a proposition is essentialto its unity, then it—the copula—could fairly be said to constitute a subjectmatter for logic. This would have two consequences. We would then requirea transcendental deduction to explain the application of logic to propositions.And, more importantly, logic would no longer be maximally general. Thetranscendental deduction would, simultaneous with demonstrating the valid-ity of logic in application to propositions, implicitly restrict logic to apply onlyto those propositions—only, that is to say, to those assemblies of names whoseunity is attributable to the presence in them of the copula. And that would beto undermine fatally logic’s status as being of ‘a totally different kind than anyother science’.The change after 1913 is thus in the first instance primarily a terminologi-

cal one: Wittgenstein’s 1915 statement that ‘relations and properties, etc. areobjects too’ was a declaration as to a widened use of the word ‘object’, toencompass properties and relations as well as what he had earlier called ob-jects. On the other hand, calling Wittgenstein’s change terminological runsthe converse danger of making it seem more trivial than it was. He did notwiden the scope of the term ‘object’ randomly. His purpose was to make histheory more general. In the sort of language countenanced in the Notes, ele-mentary propositions can have only two kinds of components, which he calls

Widening the scope !#%

names and forms. Such a language is not ruled out by the Tractatus: a lan-guage of this sort still counts as a Tractarian language in which both namesand forms are kinds of Tractarian names. What the Tractatus leaves open,however, is the possibility that there could be some further kinds of Tractariannames, or indeed that the unsaturatedness of the forms might be more widelydistributed, so that all the kinds of names in the language bear some measureof unsaturatedness.There are thus several stages inWittgenstein’s move away from the Fregean

categorization he used in the Notes. First he came to think that it is to some ex-tent arbitrary to analyse fa into a saturated name a and an unsaturated formf x, since we could equally analyse it into a name f and a form !a. But in thatcase, we could also analyse it in such a way that both elements are unsaturated.And if it is in principle possible for all the components of elementary prop-ositions to be unsaturated to various degrees, this finally removes the reasonwe offered earlier for supposing that there are only two kinds of component,since the bipartite classification was based on a distinction between the satu-rated and the unsaturated.Of course, this so far leaves open the possibility that there may be some other

reason for a bipartite classification: it might, for instance, bemerely a proxy foran independently motivated distinction between particulars and universals. Sowhat motivated Wittgenstein’s decision to give up Frege’s bipartite classifica-tion? One sort of motivation is of course the one to be found in ‘Universals’,10

where Ramsey adopted the essentially negative strategy of showing that noneof the arguments in the literature in favour of a bipartite classification is per-suasive. Ramsey made it quite clear in this paper that he regarded his generalapproach as Wittgenstein’s; and he was surely right11 to do so. It was indeed,as Ramsey said, Wittgenstein’s position in the Tractatus that ‘about the formsof atomic propositions we can know nothing whatever’.12 On the other hand,some of the details of Ramsey’s argument are perhaps not Wittgensteinian incharacter. Another sort of reason to abandon the bipartite classification mightbe provided by Wittgenstein’s picture theory, according to which a proposi-tion is a picture of an especially abstract kind. In pictures of other, less abstractkinds, such as photographs or gramophone records, we feel no temptation toimpose a bipartite classification of the components. In order that the picturinganalogy should help to explain the representational capacity of the proposi-tion, the form of a proposition had to be the most general pictorial form—aform that all pictures of whatever kind share. Wittgenstein therefore had areason to remove from his conception of the proposition those features of itsstructure that it does not share with photographs and gramophone records.

10FoM, 112–34. 11Contra Anscombe, IWT, 110. 12FoM, 134.

!#" Tractarian objects

"'.#"'.#"'.# Facts in the TractatusSo much for objects in the Tractatus. What about facts? I suggested earlier thatin the Notes Wittgenstein conceived of facts as sums of positive and negativefacts. It is not as clear as it might be, however, whether this was still whathe meant by ‘fact’ in the Tractatus. There is an alternative interpretation ac-cording to which he now restricted the word to apply only to sums of positivefacts.This alternative interpretation undoubtedly has textual support. For he says

in the Tractatus that what is the case, the fact, is the obtaining of Sachverhalte.13

But he also says that the obtaining of a Sachverhalt is a positive fact.14 So if wetake this strictly, a fact is a sum of positive facts. Moreover, a letter he wroteto Russell from the prisoner of war camp at the end of the war supports thatinterpretation.Sachverhalt is, what corresponds to an Elementarsatz if it is true. Tatsache is, what corre-sponds to the logical product of elementary prop[osition]s when this product is true.15

However, taking Wittgenstein at his word is distinctly problematic. For onething, it leads to the considerable linguistic awkwardness that negative factsare not now a kind of facts. For another, he says that the world is the totalof reality,16 and that reality is the existence and non-existence of Sachverhalte,i.e. positive and negative facts.17 So the world is the sum total of the positiveand negative facts. But it is also, famously, the totality of facts.18 This makessense only if a fact is taken to be any sum of positive or negative facts. If wetake this to be what he really meant, it at least has the virtue that it avoids therebarbative consequence that negative facts are not a kind of facts.Whatever its difficulties, though, we must grant that Wittgenstein did at

least flirt with the alternative interpretation according to which only sums ofpositive facts are facts. What might have led him to do this? It cannot simplyhave been an antagonism towards negative facts, since he explicitly mentionedthem in the book and allowed that they are part of reality. Perhaps it wasa desire to capture the idea that when what there is has been given, whatthere is not is implicitly determined by subtraction. Or perhaps it was aninept attempt to avoid the problem of the structure of negative facts discussedin §15.2. The account of the structure of the negative fact canvassed there,whether or not it was ever Wittgenstein’s, clearly could not survive when heno longer privileged one component of the positive fact as being its form andhence susceptible to pairing with a correspondingly privileged component ofthe negative fact. By focusing only on what there is, Wittgenstein may nowhave hoped somehow to dissolve this problem: what there is has a structure;what there is not has no structure, just because it is not.

132. 142.06. 1519 Aug. 1919. 162.063. 172.06. 181.1.

Confusion? !#&

Nonetheless, it has to be granted that the mere possibility of confusion overthis rather fundamental point of Tractarian interpretation is a little surprising.It is hard to avoid the feeling that Wittgenstein’s lack of clarity on the matterbetrays a lack of interest in a certain kind of constitutive metaphysics. If hehad thought the distinction mattered very much, he would surely have madesure that the Tractatus was clearer about it.

"'.$"'.$"'.$ Confusion?

It is a common view among commentators that Wittgenstein came in the1930s to think that he had been confused about the distinction between com-plex and fact at the time when he wrote the Tractatus. This view is based on areminiscence of Geach’s.

Wittgenstein told me once that after the Tractatus was published Frege asked himwhether a fact is bigger than each thing it is about: this criticism made no impressionat the time, but eventually led him to abandon the notion of facts as complexes.19

On the basis of this reminiscence Geach bluntly claims that in the Tractatus‘Wittgenstein confused a fact about A and B with a complex containing A andB’.20

This view evidently challenges what I have said here, both historically andphilosophically: historically, because if the remark of Frege’s that led Witt-genstein to the realization was made after the Tractatus was published, myaccount of how Frege might have influenced Wittgenstein’s treatment in theNotes is certainly wrong; and philosophically, because I have placed consider-able weight throughout my presentation of Wittgenstein’s ideas on a distinc-tion about which he was, according to Geach, still wholly confused. So it isimportant for us to examine Geach’s reminiscence in some detail.What is clear straightaway is that Geach’s report cannot be right just as it

stands. Frege never metWittgenstein after the Tractatuswas published (or evenafter it was written). The nearest in Frege’s surviving letters to the remarkGeach reports is the following, in a letter written after Frege had read thetypescript of the Tractatus (although not after it was published).

You write, ‘It is essential to a thing to be able to be a part of a Sachverhalt.’ Now can athing also be part of a fact? The part of the part is part of the whole. If a thing is partof a fact and every fact is part of the world, then the thing is also part of the world. . . Iwould like to have an example where Vesuvius is part of a Sachverhalt. Then also, itseems, parts of Vesuvius must be parts of this fact; the fact will also be made up ofsolidified lava. That just does not seem right to me.21

19‘Saying and showing in Frege and Wittgenstein’, 67–8. 20p. 67. 2128 June 1919.

!#' Tractarian objects

This does not quite amount to the question Geach reports concerning whethera fact is bigger than each thing it is about, but it is at least recognizably in thesame territory.What is significant for our purposes, however, is that neither Frege’s letter

nor Geach’s recollection amounts to the same point as the one that I quotedin §11.1 from Wittgenstein’s 1931 typescript on ‘Complex and fact’. Thatremark was about the mistake of regarding a red circle as a complex madeup of redness and circularity, and pointed to a confusion about complexes, notabout facts. No remark like this is to be found in any of Frege’s survivingletters toWittgenstein, and it is therefore much more likely that he made it in aconversation held not after the war but before it, during one of Wittgenstein’svisits to him.It is indeed very hard to see how Wittgenstein’s remarks about red circles

could amount to a criticism of the Tractatus. As Kenny gently observes, ‘Ifthey had been written by anyone else one would say they betrayed a misun-derstanding of that work.’22 Most of the typescript on ‘Complex and fact’surely consists of exegesis, not criticism, of the Tractatus. Wittgenstein makes thepoint, for example, that facts are not spatial: but it would be needlessly ob-tuse to read the Tractatus as suggesting that they are; and it would be curious ifWittgenstein had forgotten so completely his accusation in the Notes that it wasRussell who imagined every fact as a spatial complex.23 But if Wittgenstein wasnot in the Tractatus crudely imagining facts as spatial complexes, it does notfollow that he did not somehow allow intuitions drawn from the spatial caseto infect what he said about them. And there is one point in the note wherehe does seem to intend a correction of the Tractatus.

A chain, too, is composed of its links, not of these and their spatial relations.The fact that these links are so concatenated isn’t ‘composed’ of anything at all.The root of this muddle is the confusing use of the word ‘object’.The part is smaller than the whole: applied to fact and component part (constituent)that would yield an absurdity.

In the Tractatus (as in the Notes) Wittgenstein does talk of objects as constituentsof facts.24 So he does here intend what he says as a corrective to something hesaid in the Tractatus. And his talk of the part–whole relation is certainly redo-lent of Frege’s letter: the ‘absurdity’ Wittgenstein is referring to is thereforepresumably the one Frege mentions there, namely that a fact about Vesuviusshould have pieces of solidified lava as constituents. This makes it plausiblethat Frege’s letter was indeed what led him to the point he was making here.But what point was that? It cannot have been straightforwardly that he

was here rejecting ‘the notion of facts as complexes’, as Geach maintains.The distinction between a complex and the various facts that are exemplified

22Wittgenstein, 220. 23C26. 242.011.

Confusion? !#(

in it is something Wittgenstein explicitly comments on in the Tractatus. ‘Toperceive a complex,’ he says there, ‘means to perceive that its constituents arecombined in such and such a way.’ This, he suggests, may explain why wesee the Neckar cube in two ways as a cube. ‘For we really see two differentfacts.’25 Here, quite plainly, is an attempt to make the distinction between thecomplex and the various facts which we can see as exemplified in it.If there is room for dispute about quite what point Wittgenstein later ex-

tracted from Frege’s letter to him, there can be little doubt what Frege himselfintended: his target was not the distinction between complex and fact but theconception of simplicity. Indeed he had used an almost identical example inhis letter to Jourdain in 1914.

That part of the thought which corresponds to the name ‘Etna’ cannot be MountEtna itself; it cannot be the meaning of this name. For each individual piece of frozen,solidified lava which is part of Mount Etna would then also be part of the thoughtthat Etna is higher than Vesuvius. But it seems to me absurd that pieces of lava, evenpieces of which I have no knowledge, should be parts of my thought.26

Frege was here objecting to Russell’s conception of propositions as composedof pieces of the world, ‘even pieces of which I have no knowledge’, rather thanof the senses which according to Frege contribute to the structure of thoughts.Frege had made just the same point to Russell in 1904, although then his ex-ample was Mont Blanc and its snowfields rather than Vesuvius and its lavaflows. (When he reflected on simplicity, it seems, Frege’s thoughts alwaysturned to mountains.) At that time, a year before ‘On denoting’, Russell’sresponse (quoted in §7.1) was little more than to confirm that this was indeedhis view. By the time Frege made the point to Jourdain, however, there wasmore he could have said in response: once the proposition was correctly anal-ysed by means of the theory of descriptions, the simple entities of which it wascomposed would indeed turn out to be pieces of the world with which I am di-rectly acquainted. And in 1919 Wittgenstein could have responded to Frege’sletter in an analogous manner: the components of the fact are not Fregeansenses but Tractarian objects, whatever these turn out to be. If by 1931 Witt-genstein had come to think Frege’s criticism valid, that could well be becausehe no longer believed in the conception of simplicity on which this responserelies. As he justly observed, ‘The root of this muddle is the confusing use ofthe word “object”.’But, if this was Frege’s point, it need not have been Wittgenstein’s (not di-

rectly, at any rate). Something else that may by 1931 have begun to troublehim was his extension of the meaning of the word ‘object’ to cover both sat-urated and unsaturated constituents of an atomic fact—or, more precisely,the blurring of the distinction between saturated and unsaturated which this

255.5423. 26PMC, 79.

!$* Tractarian objects

change of terminology was designed to encourage. For, as we noted in the lastsection, that use is in danger of making the existence of negative facts seemespecially problematic. If we are not to reintroduce the copula, we must notregard the manner of combination as a substantial element in the fact thatsome objects stand in combination. But what should we then say about thenegative case? The more Wittgenstein stressed his rejection of the copula bymeans of the immediacy of the combination of the objects in an atomic fact,when it obtains,27 the less he could say in the case when it does not obtain.When the objects stand in combination, that they so stand is not a furthercomponent over and above them; and when they do not stand in combina-tion, that they do not so stand is equally not a further component. What, then,is the difference between the fact that they do not so stand and the fact thatsome other objects do not stand in combination? The problem of negativefacts rears its head yet again. On this second account, then, what Wittgen-stein came to regret was that the analogy with spatial pictures had led him toabandon Frege’s bipartite classification of the parts of a fact: Wittgenstein’suse of the word ‘object’ was confusing because it obscured this distinction andhence rendered puzzling the conception we must have of the structure of afact if we are to think of it as something which may or may not obtain.Each of the two diagnoses no doubt has something to be said for it. The first

makes Wittgenstein’s 1931 objection to his earlier conception rely on a view—the rejection of the Tractatus’s conception of simplicity—which we know he bythen held. The second is philosophically deeper and presents him as engagingmore fundamentally with the ideas that motivated Frege’s views, but it is per-haps harder to render consistent with the historical evidence. It would take ustoo far out of our way to arbitrate between them here. Nor do we need to. Allthat is necessary for our current purpose is to note that the mistake Wittgen-stein later believed he had made in the Tractatus does not on either accountlie on the surface. We are in no way compelled to represent it as merely acrude confusion over the distinction between complex and fact—a distinctionwhich, I have suggested, he first came to appreciate in 1913.

274.221.

Chapter "("("(

Philosophy

Outside the guilds of philosophical logicians and metaphysicians, the Tractatusis known almost exclusively for the allusive but elusive remarks that consti-tute its metaphilosophical frame—the Preface and the concluding couple ofpages. The insights to which these remarks try to give expression afford adistinctively problematic status to the book that they surround, and hencerender paradoxical the route by which these insights were acquired. Thereis no trace in the Notes of this attempt to render philosophy nonsensical, or ofthe relish for deliberate paradox that it betrays. Nonetheless, what is visiblealready is Wittgenstein’s concern for and distinctive conception of the role ofphilosophy, its limits, and its relationship to the natural sciences.

"(.!"(.!"(.! MetaphysicsPinsent once noted in his diary, in what one suspects was actually a report ofWittgenstein’s views, that ‘really logic is all Philosophy. All else that is looselyso termed is either metaphysics—which is hopeless, there being no data—orNatural Science, e.g. Psychology.’1 In the Notes, on the other hand, Witt-genstein was somewhat less pessimistic, observing that ‘philosophy consists oflogic and metaphysics: logic is its base’, but not going as far as to dismissmetaphysics as hopeless.It is true that there is rather little in the Notes that can be regarded as

straightforwardly metaphysical, but what that indicates is not so much thatWittgenstein thought metaphysics was hopeless as that he had a specific viewabout how metaphysical conclusions might be reached. What we see in oper-ation in the Notes—perhaps, indeed, because of the lack of relevant data—isa developing method of approaching metaphysics via the symbolism. This isan application of the symbolic turn mentioned in chapter 6. The features ofthe world are deduced from the features of the symbols used to represent it.The philosopher’s project of analysis, which we sketched in chapter 4, exhibitsvarious complex signs that occur in language as disguised definite descriptions(or other incomplete symbols). The sentences in which these signs occur areto be rewritten so that their occurrences are eliminated. The answer to the

125 Aug. 1913.

!$! Philosophy

metaphysician’s question, what there is, is revealed by this process: what thereis is whatever corresponds in the world to the simple signs that remain oncethe complex signs have been eliminated.Something like this conception, according to which metaphysical questions

are correctly pursued via, or can be seen as equivalent to, an analysis of lan-guage, can perhaps be discerned, at least dimly, in Boltzmann; but it is morelikely, I suspect, that Wittgenstein derived it from Frege. The idea certainlyplays a role, at least tacitly, in the Grundlagen, for instance. Dummett, in hisinterpretations of Frege, has gone much further, maintaining that this con-ception is not just tacitly but fully and centrally present in the Grundlagen. Onhis account, the role of the context principle in the Grundlagen is to rule out asillegitimate any attempt to ask what numbers are that goes beyond explainingthe contribution number-words make to the senses of sentences in which theyoccur. It is ‘part of Frege’s thesis’, he says,

that the nominalist is the victim of a superstition about what has to be done in order toconfer a reference on a name. If an expression satisfies the ‘formal’ criteria for being aname, then there is no further condition which needs to be satisfied for the expressionto be a name having a reference.2

The effect, on this view, is at a stroke to transform a metaphysical question—what are numbers?—into a linguistic one—how are number words used?Other readers, however, have hesitated to see in the Grundlagen such a

clear endorsement of the language-centred view. One might indeed won-der whether Dummett has been influenced by Wittgenstein’s more thorough-going application of this view into attributing to Frege a clearer grasp of itthan he really had. Even such an enthusiast as Dummett is willing to grant,indeed, that Frege ‘was not fully conscious of the thrust’ in the direction of ‘theinvestigation of thoughts through the analysis of language’.3 And Dummettacknowledges that Frege’s later writings hesitate over this question: Frege ad-mits, for instance (perhaps under the influence of the set-theoretic paradoxes),that ‘the main task of the logician consists in liberating himself from language’.4

Perhaps it is more accurate to regard this as another case of Wittgenstein’stendency (already visible, for instance, in his conception of logic as contentless)to see the implications of Frege’s views more clearly than Frege did himself.The insight which Frege drew from the context principle—that if a name hasa significant use, there can be no further condition it has to fulfil in order torefer to an object—is one that in the TractatusWittgenstein made his own. Thesymbolic turn, which in the Notes takes the simplicity of a name as a criterionfor the simplicity of the object it refers to, is there extended to encompass arejection of a metalinguistic perspective from which it would even make senseto ask about the latter independently of the former. The picture theory is not

2FPL, 497. 3OAP, 6. 4To Husserl, 30 Oct. to 1 Nov. 1906, my emphasis.

Psychology !$#

presented there as imposing a constraint which the world must satisfy if it is tobe representable in language. (For what sense could we make of the notion ofan unrepresentable world?)

"(.""(.""(." PsychologyI have already remarked on Wittgenstein’s insouciance about how logicalanalysis is actually to be carried out. He told Russell, for instance, that

I don’t know what the constituents of a thought are but I know that it must have suchconstituents which correspond to the words of language. Again the kind of relationof the constituents of a thought and of the pictured fact is irrelevant. It would bea matter of psychology to find out. . . Does a Gedanke consist of words? No! But ofpsychical constituents that have the same sort of relation to reality as words. Whatthose constituents are I don’t know.5

OnWittgenstein’s view, therefore, the analysis of language into its simple con-stituents corresponds to the analysis of thought into its simple constituents. Itis, it seems, for psychology to tell us what these constituents are.But what did Wittgenstein understand by psychology? In Cambridge there

were two rather different models available to him. Moore, whose lectures onpsychology tried with agonizing care to get clear about the correct use of termssuch as ‘psychology’, hoped to arrive at the nature of sense-data by enquiry,not by experiment. But Wittgenstein also spent a great deal of time at thenewly opened Psychological Laboratory experimenting on the perception ofrhythm. I am inclined to think that at this stage he had more sympathy withthe second model of psychological enquiry than with the first. If he regardedpsychology as part of natural science, as Pinsent’s diary entry suggests, thismight explain why he apparently thought philosophers were safe to ignore it.It is less clear, though, whetherWittgenstein really did ignore empirical psy-

chology as completely as he sometimes liked to claim. If it were, as a matter offact, possible for me to see the Neckar cube in both its configurations simulta-neously, would he still have felt able to maintain6 that my world is the totalityof facts, not of things? And Wittgenstein’s conception of what it is to see thesymbol in a sign surely depends for its plausibility on the phenomenology ofreading.What is notably absent in Wittgenstein’s letter to Russell just quoted, of

course, is any attempt to make a distinction between psychology and linguis-tics. The constituents of my mind, which are private to me, ‘have the samesort of relation to reality as words’, and yet words are communally available ina manner in which the constituents of my mind are not. The reason for Witt-genstein’s failure to distinguish between what is private and what is public is

519 Aug. 1919. 61.1.

!$$ Philosophy

one I discussed in chapter 7. One way of putting the point (a way that em-phasizes the private aspect) would be to say, as I did there, that Wittgensteinadhered to, or was at least indifferent to the consequences of, a certain kindof solipsism. It would equally be possible, though, to put the point so as toemphasize the public aspect: once a communally available realm of thoughtis acknowledged, it is dubious what point is served by attempting to contrastit with a private sphere of ideas available only to me. We do not find Wittgen-stein himself putting the point in this second way until the 1930s, of course,but the private language argument of the Investigations has its origins in such ashift. The transition between Wittgenstein’s early solipsism and this theme inhis later philosophy is most clearly visible in the Blue Book, where he exploredat length the idea, perhaps not fully recognizable in the Tractatus, that fromthis second perspective solipsism is not the outlandish claim that Russell al-ways regarded it as, but simply what common sense demands. Of course theworld is my world: who else’s could it possibly be?

"(.#"(.#"(.# EpistemologyEven if philosophers can ignore the details of what the simple signs in languageactually are, and hence what objects there are in the world, there remainsthe question of what the relationship is between the former and the latter(what Russell called ‘acquaintance’). Moreover, there is a further questionof how we acquire knowledge: Russell called the relationship by which weknow that an atomic fact obtains ‘perception’. WhenWittgenstein labelled thephilosophy of psychology epistemology,7 he presumably had in mind the taskof explaining these two relationships, acquaintance of objects and perceptionof facts. We have seen how little the Notes contribute to either of these tasks.But if this was the role of epistemology at the time of the Notes, it had un-

dergone a significant change by the time of the Tractatus. When the judgingsubject dropped out of the judgment relation, acquaintance became an un-sayable relation between a name and its bearer. In the Tractatus nothing thatis not merely programmatic is said about how this relationship is established.For this reason, perhaps, many commentators have offered a reading whichrenders epistemology significant in the Tractatus only by its absence. Accord-ing to Dummett, for example,

the Tractatus is a pure essay in the theory of meaning, from which every trace of episte-mological or psychological consideration has been purged as thoroughly as the houseis purged of leaven before the Passover.8

But this is too strong. It may indeed be the case that epistemology in the sensein which Russell used the term, is absent from the Tractatus, but what there

7B62. 8FPL, 679.

Value !$%

is instead is a recognition of the actual world as the world we perceive, andhence of perception as what distinguishes the actual world from other possibleworlds. Epistemology therefore makes its appearance when I note those fea-tures which the actual world has to possess by virtue of being my world. Themost striking of these is the one Wittgenstein used as an objection to Russell’stheory of identity. It is possible for two objects to have all the same properties.But a world in which this occurred could not be the actual world: if it were, thetwo objects could not have been recognized as being distinct, could not havebeen given different names, hence would not be for me distinct objects. TheTractatus, unlike the Notes, is built on a conception of our capacity to representthe world as based on the recognition of the possibility that it might have beendifferent. But Wittgenstein does not commit the modal fallacy of supposing,once these other ways the world could have been are recognized, that eachof them could have been the actual world. I can recognize the possibility ofthere being brains in a vat without being forced to admit that I might be onemyself. Epistemology, in the Tractatus, is what we glean when we take thatobservation at full value.It is clear, though, that this is very far from what Russell understood by the

term. In Wittgenstein’s hands epistemology is a way of privileging the actualworld over its non-actual alternatives—a way of explaining, in the absence ofanything in it amounting to a judging subject, what makes the world nonethe-less mine. What the Tractatus does not do is to contribute to the epistemolog-ical project which Russell had been attempting until Wittgenstein persuadedhim to abandon it in 1913. According to the picture theory, our task in rep-resenting the world through pictures presupposes that we have formed nameswhich can go proxy for the objects in these pictures. As already noted, theTractatus says very little about what these objects are. But it says even lessabout how the task of naming is to be effected. If, as has been suggested, Witt-genstein sometimes uses ‘kennen’ to translate Russell’s notion of acquaintance,he tells us nothing about what acquaintance with an object amounts to. Forhim, the relation of acquaintance is not (most of the time, at least) amenableto the kind of armchair experimental psychology so often favoured by Rus-sell; rather is it simply presupposed by meaningful discourse and mysteriouslyinaccessible to it.

"(.$"(.$"(.$ Value

What is common to Wittgenstein’s comment in the Notes and the entry fromPinsent’s diary entry quoted earlier is that they allow no place for a philosophyof value or of religion. And indeed whenever Russell strayed into this terri-tory (which, encouraged by Ottoline Morrell, he did with some frequency),

!$" Philosophy

Wittgenstein expressed his disapproval forcefully.Now there is of course a limited sense in which a rejection of the philosophy

of value remains in place in the Tractatus. Statements of value are accordingto its doctrines nonsensical, hence not for philosophy to adjudicate upon. Butthat is equally true of metaphysics (and indeed logic, insofar as it goes beyondthe mere enunciation of tautologies). Statements of value, although nonsen-sical and therefore in a sense insignificant, are nonetheless recognized in theTractatus as playing an important role in guiding our ethical lives. Moreover,although it is not for philosophy to adjudicate on them one by one, the Tracta-tus does discuss their nature. It thus implicitly concedes to a kind of meta-ethicsa place as part of philosophy—or, at any rate, as part of the prolegomenonto philosophy that the Tractatus represents itself as being. In what he wrote inthe Notes or said to Pinsent, Wittgenstein does not seem to have been willingto grant even this much.Something changed, therefore, in Wittgenstein’s attitude to the philosophy

of value. And it is clear, too, that Wittgenstein’s own religious views changedsomehow in the course of the war: when Russell met Wittgenstein in TheHague in 1919, his observation that Wittgenstein had turned into a mysticwas plainly a personal comment, and not merely an objection to his book’sattempt to connect mysticism with logic by means of the doctrine of sayingand showing. So it would be easy, no doubt, to succumb to the temptationof trying to correlate this change in his view of the philosophy of value with achange in Wittgenstein’s own religious views.Just what these views were when he was at Cambridge has not been trans-

mitted to us with any clarity, however. Although instructed as a child in theCatholic faith, he quite soon rejected it; but just what residue this rejectionleft is not clear.9 His horror when he discovered that one of his undergrad-uate acquaintances in Cambridge was a monk10 suggests a certain hostilityto organized religion; and other remarks in Russell’s letters (e.g. that he was‘terrible with Christians’)11 reinforce this impression. But it is difficult to dis-cern what the rational basis for Wittgenstein’s hostility to clerics really was(if indeed there was one). His charge against them seems, oddly enough, tohave been one of dishonesty.12 If this was not simply the childish assump-tion that anyone who disagreed with him was a hypocrite, perhaps it was thathe thought of religious experience as essentially private, so that any attempt tomanifest it publicly would inevitably fail. One of his objections to Russell’s1912 article on the essence of religion was ‘that such things are too intimatefor print’.13 (The idea that religion is inexpressible would of course represent apoint of continuity with the conception offered in the Tractatus.) On this view,the dishonesty of the cleric consisted in a failure to recognize this distinctively9McGuinness, Young Ludwig, 43. 10BR to OM, 17 Mar. 1912. 11To OM, 18 Mar. 1912.12Pinsent’s diary, 9 Nov. 1912. 13BR to OM, 11 Oct. 1912.

Value !$&

private character of his religion: the attempt to put his religious beliefs intowords could not possibly do them justice.Even if he was hostile to clerics, while he was at Cambridge Wittgenstein

was evidently far from denying the existence or the importance of a religiousaspect to man’s relationship with the world. Otherwise he would hardly haveadmired, as we know he did, the text, ‘What shall it profit a man if he gainthe whole world and lose his own soul?’14 He also recognized already the im-portance of private religious experience: in about 1910 he had had what heregarded as a sort of religious experience (a strong feeling of complete safety)while watching a play.15 And his diaries during the war are, from early on, suf-fused with religious language of a sort that suggests a long-standing familiaritywith religious concepts as modes for expressing his emotional life. Perhaps hisreligious understanding was deepened and enriched by prayer during periodsof personal danger, such as when he served on the Eastern Front during theBrusilov offensive of 1916; but there is nothing in his diary to suggest thathe underwent anything like a religious conversion or transforming experiencethat summer. By contrast, the philosophical change that occurred in 1916, visi-ble in his notebook entries, is clear and profound. This was the moment whenhis work ‘broadened out from the foundations of logic to the essence of theworld’.16 The insights he came upon that summer led him to make signifi-cant changes to the structure of the Tractatus, already largely written. Beforethat summer it was a work in what we would now call philosophical logic.(Wittgenstein, of course, professed to think that there is no such thing.) Itwas recognizably a descendant of the Notes, and might still be seen as Witt-genstein’s attempt to answer the question ‘What is logic?’—the very questionwhich Russell had abandoned in October 1912 when he realized that Witt-genstein was better able than he to answer it. Moreover, the book still hadas its concluding paragraph a technical claim in logic, a statement of the gen-eral form of proposition. Only after the summer of 1916 did Wittgensteintransform the book, by adding the 6s (especially the 6.5s), into a work withmuch wider ambitions, dealing also with ethics, mysticism, and the soul; nowtoo it concluded, as it had not done before, by throwing away the ladder andadmonishing us to silence.But this change in the ambitions of his philosophical work does not seem

to be straightforwardly correlated with any change in his own religious be-liefs. The fatalistic humanism that Wittgenstein wrote into the Tractatus after1916 is visible in the works of Tolstoy and Dostoevsky that he had read andbeen affected by before the war: he may not have come across The gospelsin brief, Tolstoy’s curiously fatalistic account of Christian teachings, until theAutumn of 1914, but he read Hadji Murat (which one might view as a sort of

14BR to OM, 29 May 1912. 15Malcolm,Memoir, 70. 16NB, 2 Aug. 1916.

!$' Philosophy

extended hymn to fatalism) in 1912 and thought it wonderful.17 The influ-ence of Tolstoy and Dostoevsky is detectable in his friends as well: almost theonly literary reference in Pinsent’s diary entries about Wittgenstein is to AnnaKarenina (‘very like Levin’,18 he perceptively observes); and it is probably not acomplete coincidence that Moore was reading Dostoevsky during the periodwhen his friendship with Wittgenstein was at its closest.

17LW to BR, [Summer 1912] (CL, no. 6). 182 Sep. 1913.

Chapter ")")")

Themes

It is hard to regard the Notes on Logic as a finished work, even in the Costello re-arrangement, and one can only guess at the struggles Johnson or Moore mighthave had if they had been asked to examine a BA dissertation based on them.The claim I made at the outset, however, was that treating the Notes as a workin its own right might pay both historical and philosophical dividends. Thephilosophical dividends are perhaps not of a kind to be swiftly summarizedin a concluding chapter: the philosophical illumination one hopes to cast ina study of this kind is not a single transforming insight but a series of smallclarifications of puzzles not hitherto sufficiently understood. The historical in-sights, on the other hand, are probably rather easier to identify. It is now timeto reflect on what we have learned here about Wittgenstein’s way of workingand of thinking, about the influences on and of his work, and about how theNotes influenced the Tractatus itself.

").!").!").! Working methods

Wittgenstein’s working methods seem to have remained remarkably constantthroughout his life. Central to them was his habit, his discipline, or even per-haps his obsession, of writing up his thoughts (almost always in German ratherthan English, however long he spent in Britain) in a philosophical journal day-by-day.The two methods by which Wittgenstein liked to mine these journals for

material are exemplified by the two parts of the Notes. The first method, bymeans of which I believe the Birmingham Notes were produced, was to markin his notebooks which remarks he thought worth preserving and then readthese out to a shorthand typist. Later in life he sometimes cut the resultingtypescripts up into slips which he could sort and rearrange, and it is surelyvery likely that some of the runs of consecutive remarks on particular topics inthe Prototractatus were first assembled in this way.The second method Wittgenstein had of advancing beyond his journal en-

tries was to dictate in English, not in general simply translations of remarksfrom the journals but often rephrasings or explanations of these remarks. The

!%* Themes

Cambridge Notes do not quite conform to the method he later found ideal,though. For one thing, he wrote part of them himself, and the remainder hedictated not to a student or friend, as he did later in his life, but to a short-hand secretary. For another, his interlocutor was Russell, who was capableon occasion of arguing back, whereas later his preference was for more qui-escent dictation partners—people who (so he presumed) understood what hewas saying well enough to look quizzical from time to time, but not the sortby whom he might ever feel philosophically threatened.The fact that it was Russell he was explaining his ideas to rather than some-

one more willing to listen dutifully (Moore, perhaps) is probably part of the ex-planation for the rather chaotic character of the Cambridge Notes, especiallyMS2. It is worth stressing, however, that Wittgenstein’s working method wasnormally far from chaotic. The Birmingham Notes, in contrast to the Cam-bridge ones, are a methodical report of the logical results he had obtained sofar. Closely keyed as it was to his interest in achieving philosophical under-standing without advancing arguments, Wittgenstein’s later working methodplaced a premium on the ability to rearrange his thoughts so as to create jux-tapositions that achieve certain effects. The Birmingham Notes were already ina form that would make it possible for him to draw on them for this purpose(as he eventually did when he came to compile the Prototractatus).It is worth noting, too, that Wittgenstein had literary models for the form

of expression that resulted from this method of working. The collection ofaphorisms was a familiar literary form in Wittgenstein’s Germanic tradition;perhaps the fact that he conceived a philosophical work with a highly non-narrative structure is partly a result of his exposure to, and hence his appre-ciation of the cumulative effect of, collections of aphorisms. Lichtenberg, forinstance, was celebrated for such a collection (a copy of which Wittgensteingave to Russell in the summer of 1913), and he will have been aware of nu-merous other examples (Schlegel, Nietzsche, Schopenhauer, Kraus).

")."")."")." Characteristics

Wittgenstein had, it must be said, a particular, if not peculiar, sort of mind.Again and again we see in his work ideas that have a forcibly striking combi-nation of depth and simplicity. This combination, indeed, is so characteristicof his ideas, both early and late, that respecting it may be taken to be aninterpretative constraint on the commentator. Wittgenstein’s ideas are oftennot easy to grasp, but once grasped they usually have a sort of directedness—obviousness, almost—that makes them, if not compelling, then at any ratecompulsive. Accounts which represent the arguments for his claims as compli-cated are therefore rarely convincing.

Characteristics !%)

A running theme in Wittgenstein’s work is his refusal to argue for his ideasin anything like the conventional manner of philosophers. This, as we can seefrom his period in Cambridge, started early. Russell tried to persuade him ofthe importance of argument to philosophy, but in vain. ‘He said argumentsspoil its beauty, and he would feel as if he was dirtying a flower with muddyhands.’1 Nor was Russell the only one who tried: Frege recommended him torework the Tractatus as a series of articles arguing for specific conclusions.2

In the interpretative literature there has been a tendency to confuse thiswith a different strand in his later philosophy, the strand which emphasizesthat Wittgenstein was not concerned with advancing theses or making claims.Sometimes his aim is described as being that of encouraging us to give upfalse (or, more usually, nonsensical) claims that we have entertained in thepast, without putting other (equally false or nonsensical) claims in their place.On other occasions the purpose is presented more positively as being that ofleading us into a certain way of viewing a problem, where what we must un-derstand is that the ‘way of viewing’ in question is by its nature not somethingthat can be put into words.Whichever of these two ways of conceiving his project is stressed, however,

the nature of the goal he was pursuing is taken by interpreters to explain whyhe did not proceed by the philosopher’s normal method of advancing argu-ments. Wittgenstein’s delphic mode of expression is represented as being hisresponse to the peculiar difficulty faced by anyone who tries to construct anargument directed towards what cannot really be described as a conclusion atall.Just for this reason the later Wittgenstein has occupied a strikingly contro-

versial place in the wider taxonomy of philosophy. If it is taken to be char-acteristic of analytic philosophy that it is concerned with making claims andassessing arguments, Wittgenstein is presented as a paradigm case of a non-analytic philosopher. And yet it is equally often held that what is characteristicof analytic philosophy is the pursuit of clarification, and from this perspectivethe later Wittgenstein may, equally paradigmatically, be presented as belong-ing firmly in that tradition, even if the methods by which he pursued the goalof clarification were atypical.The tension between these two views of the later Wittgenstein has had its

influence on assessments of his early work too. The early Wittgenstein wasfor a long time treated, of course, as an analytic philosopher––as co-founderalong with his two great inspirations, Frege and Russell, of the tradition. Inthe years immediately after his death the connections between the Tractatusand the posthumously published Philosophical Investigations were not detectable

1To OM, 27 May 1912. 2To LW, 30 Sep. 1919.

!%! Themes

by most readers. No great attention was given to his own request that his laterwork be viewed next to the early, with the result that the ‘two Wittgensteins’view dominated the interpretative literature for more than thirty years.What made this view increasingly untenable was the gradual emergence

into the public domain of Wittgenstein’s middle period works, the Blue andBrown Books, and especially the Philosophical Remarks and Philosophical Grammar,together with notes of his Cambridge lectures during the 1930s. Once thecontinuities of thought between early and late began to be understood, it be-came increasingly uncomfortable to treat early Wittgenstein as a founder ofthe analytic tradition and late Wittgenstein as its first postmodernist subverter.It became popular, therefore, to look for signs in Wittgenstein’s early espousalof a version of Russell’s analytic method of an intention to deconstruct it fromwithin.To the extent that a study of the Notes can be expected to contribute to

these larger debates, what it offers is a reminder of much more modest andless philosophically resonant interpretative points. The most obvious of theseis that Wittgenstein’s tendency towards the oracular was not a response to thepeculiar difficulty involved in arguing for conclusions that cannot, by theirnature, be put into words. Philosophically unsatisfying as it may be, the truthis that he did it because he was constitutionally incapable of doing anythingelse.The inability to argue for their conclusions, the inability to explain what

they mean, are not such rare phenomena in budding philosophers. When Icome across cases in which this tendency is not corrected early, my inclinationis to blame the supervisor. In Russell’s case, though, I suspect that this isunfair. If he had resolutely withheld fatherly approval until Wittgenstein hadsupplied an argument for every claim hemade, the result would no doubt havebeen only that Wittgenstein left philosophy and went back to engineering,with undoubted loss to the former and (judging by his 1910 patent application)no great gain to the latter.To the wider debate about the role of the unsayable in the Tractatus, on the

other hand, a study of the Notes can contribute comparatively little. The Notesdo at least show, however, that Wittgenstein did not always believe that theclaims made in the text of the Tractatus were nonsense. For many of them arealready contained, sometimes in the very same words, in the Notes, and theyare not there advanced ironically, ‘transitionally’, or for purely literary effect:in the Notes, if not in the Tractatus, Wittgenstein said what he did because hebelieved it was true. Pinsent reported in 1912 that Wittgenstein had solveda problem in the most fundamental logic that had puzzled Russell and Fregefor years; and solving such problems was just what throughout his time atCambridge Wittgenstein saw himself as doing.

What if !%#

").#").#").# What ifIn chapter 11, I mentioned Adolf Reinach, a student of Husserl and nearcontemporary of Wittgenstein who fought in the German army and was killedin 1917. Nowadays Reinach merits barely a footnote in the history of earlyanalytic philosophy. It is perhaps one of the idler what-ifs of history to wonderhow much attention Wittgenstein would now be accorded if he too had beenkilled during the war.More relevant to our current purpose, I suppose, is the corresponding ques-

tion in the event that Wittgenstein had committed suicide in Norway, as Rus-sell evidently feared.3 Whether this was really as serious a risk as Russellsupposed is open to doubt. Talk of suicide seems to have been used muchmore frequently then than now as a sort of rhetorical trope. (Russell, for ex-ample, assures us in his autobiography,4 with patent implausibility, that hepersuaded his wife not to cite Ottoline in their divorce by the simple device ofassuring her he would commit suicide if she did.) On the other hand, threeof Wittgenstein’s brothers did actually commit suicide, which would on itsown be sufficient to explain the frequency with which he himself consideredit. Nonetheless, it needs to be remembered that considering suicide—evenfirst-personally—is not quite the same as being close to doing it.What we can say for certain is that if Wittgenstein had died in Norway that

winter, Russell would have done what he could to publish Wittgenstein’s writ-ings and bring them to wider attention. What effect that would have had ismoot, however. Even as late as 1916 the Prototractatus was a radically differentwork from the Tractatus as it was eventually published. The transforming in-sight that his work had ‘extended from the foundations of logic to the natureof the world’5 had yet to turn the book into anything other than the narrowaccount of the nature of the proposition, and hence of the nature of logic, forwhich his work in Cambridge with Russell had been a preparation.But the Prototractatus, even in its 1916 form, at least contained several im-

portant theories which have had an important influence on twentieth-centuryphilosophy. It presented, for example, a conception of the role of a propo-sition as being not merely to say how things stand but to contrast that withother ways they might have stood but do not. The book works out many con-sequences of this conception, the most prominent of which is that objects areto be thought of as what remain constant as the possibilities propositions allowfor vary. But none of this is mentioned in the Notes. For that reason the Noteswould surely not have attracted the attention of the Vienna Circle in anythinglike the way that the Tractatus did, since there is nothing in the Notes even tosuggest (let alone confirm) that Wittgenstein might have had any sympathywith their positivist agenda.

3To OM, 2 Oct. 1913. 4Autobiography, 213. 5NB, 2 Aug. 1916.

!%$ Themes

On the other hand, the Notes do reach several conclusions which are surelyof central importance to that part of philosophical logic which Frege and Rus-sell had been developing. They point out, for example, the incoherence ofRussell’s attempt to treat propositions as incomplete symbols when they oc-cur in the context of belief attributions. And with equal finality they exposeFrege’s bewildering error of treating sentences as names of truth-values.At other points the Notes are more tentative. Wittgenstein had seen the need

for his account of the sense of a proposition to make manifest the manner inwhich !(a) says about a just what !(b) says about b, but he had not reallygot clear about how to meet that need. And the difficulty there is in statingthe theory of types without violating its very principles is so far only gesturedat and not thought through. More generally, one has a sense that althoughWittgenstein had already come to conclusions—that ‘no name is the name of aform’, for example, or that ‘facts cannot be named’—which point to instancesin which language seems inevitably to stumble; and, although he regardedsuch cases as rather more than awkwardnesses to be treated with a pinch ofsalt, he did not yet appreciate the full significance of this observation.Moreover, some gaps in the Notes are obvious, and were so to Wittgenstein

himself. Having dealt adequately, as he thought, with the process by whichmolecular propositions are formed from atomic ones, he needed to extendthe treatment to include an account of quantified propositions. That account,when he produced it, might be expected quickly to expose the inadequacy ofwhat he had so far said about Russell’s paradox and hence to point up theneed for a more nuanced account of what it is for one proposition to dependon, or be contained in, another. Even if quantification could be dealt with,that would still leave identity to be explained (or, as it turned out, explainedaway). And, having demolished Russell’s theory of judgment, Wittgensteindid not yet have more than a plainly inadequate sketch6 of a theory of his ownto put in its place.

").$").$").$ Fundamental thoughtsWhat is most striking about the Notes, however, is not the particular claimsthere made which survive into the Tractatus, but rather the general principleswhich inform that book and which we find already guiding Wittgenstein’swork in the Notes.The most evident of these principles, perhaps, is what Wittgenstein himself

regarded as his Grundgedanke, namely that the logical constants do not repre-sent. Already in the Notes this principle led him to reject Russell’s use of realvariables and to conceive of the logical structure of a proposition as implicitin the structure of its components rather than as a further element set against

6B55.

Influences on Wittgenstein !%%

them. Soon the same principle would lead him to his account of tautologiesas essentially contentless and hence trivial, and to a rejection of Frege’s con-ception of numbers as objects.Equally important, however, was what I have called Wittgenstein’s sym-

bolic turn, his understanding of the criterion for an object to be simple asbeing no more than that it be the reference of a simple symbol. This un-derstanding gave prominence in his work to the problem of detecting whichfeatures of language symbolize—which features, that is to say, have logical asopposed to psychological significance. It was this that gave rise to the centralrole played in the Tractatus by the distinction between sign and symbol, thetransition between which is the contribution made by the self to the process ofrepresentation.The search for what is symbolically significant in the proposition—for the

symbolizing fact—may be characterized as a search for a logically perfect lan-guage. At what stage Wittgenstein began to see this holy grail as unattainableis a central question for Tractarian exegesis. It is plain, though, that it is aquest which he had not yet abandoned when he wrote the Notes. For himthis quest depended on a distinctive conception of analysis which he explic-itly enunciated in the Notes and which in turn depended on the eliminativetechnique introduced by Russell in ‘On denoting’.The analysis itself Wittgenstein did not attempt, and this too was a conse-

quence of one of his guiding thoughts. Which words have which meaningsis a matter for psychology, and the philosophy of psychology is epistemology,which was Russell’s business. What is a matter for philosophy is thereforeform, not content: logic is the study only of the structure of propositions andof how that structure enables them to mean what they do; or, to put it anotherway, of how, rather than what, they symbolize.Another central thought in the Notes is that what symbolizes in a propo-

sition is not the sentential complex by means of which we express it, how-ever that may be individuated, but a fact which that complex may be seenas exemplifying. It is this conception of the proposition as a fact which en-abled Wittgenstein, so he thought, to combine what is attractive in both thefunction–argument and the part–whole models of propositional structure.

").%").%").% Influences on Wittgenstein

In 1931 Wittgenstein sought in his diary to explore the question—alien, per-haps, to many modern readers—of what his Jewishness amounted to. ‘It istypical for a Jewish mind,’ he suggested, ‘to understand someone else’s workbetter than he understands it himself.’

!%" Themes

I think there is some truth in my idea that I really only think reproductively. I don’tbelieve I have ever invented a line of thinking: I have always taken one over fromsomeone else. I have simply straightaway seized on it with enthusiasm for my workof clarification. That is how Boltzmann, Hertz, Schopenhauer, Frege, Russell, Kraus,Loos, Weininger, Spengler, Sraffa influenced me.7

We are not compelled, of course, to take such musings at face value, but inthis case there does seem to be some truth behind the less than convincingfacade of Wittgenstein’s thinking. None of the fundamental thoughts I havejust mentioned is wholly original with Wittgenstein, but each of them is a wayof bringing to prominence what is significant in an earlier idea.Overwhelmingly, though, the origin of these ideas lies with either Frege or

Russell, and not with the other authors cited. Hertz is often quoted as a sourcefor the picture theory and his claim that ‘there musts be a certain conformitybetween nature and our thought’8 may well have influenced the view implicitin the Notes that there is an essential harmony between the structure of a factand the structure of any proposition that represents it. But if there is an in-fluence at just this point, it seems to me to be rather superficial: for Hertz theconformity between nature and thought is something which we learn fromexperience and which could in principle fail to hold, whereas for Wittgensteinit is a condition on anything that may properly be called thought at all. Thedeeper influence of Hertz on the Notes surely lies not so much in the picturetheory itself as in a methodological principle which one might think of as asort of correlate of it, namely that philosophical problems are to be solved byfinding an analysis in which the number of primitives corresponds precisely tothe number of degrees of freedom of the system being described.Boltzmann may have had an influence on the conception the Notes offer

of the place of philosophy above or below the natural sciences. Above all,though, what Wittgenstein drew from him (as from Hertz) was, I suspect, notso much any specific doctrine as a method—in Boltzmann’s case, the methodby which philosophical problems are not to be solved but rather dissolved.One example is Boltzmann’s demonstration that what is said by a certainkind of solipsist need not differ from the ordinary way of expressing things.9

Another is his attempt to show that various apparent philosophical questionsresult from not knowing what we were really asking: even the theist and theatheist may, Boltzmann suggests, ‘think the same thought without suspectingit’.10

It has often been suggested that Wittgenstein was careful to list the influ-ences on him in chronological order. If so, then Wittgenstein had not beeninfluenced by Weininger by the time he was at Cambridge, since he did notmeet Loos until 1914. Nor, I think, is it necessary to search the Notes for traces7CV, 18–19. 8Prinzipien der Mechanik, 1. 9Theoretical Physics and Philosophical Problems, 61–2.10Ibid. 75.

Influences on Wittgenstein !%&

of Kraus: Wittgenstein had already read Die Fackel by then, no doubt, butKraus’s influence on him surely lies some time later. And although Wittgen-stein had read Schopenhauer in his youth (at sixteen, according to Anscombe),and returned to some of his ideas in the summer of 1916, the subject matterof the Notes is rather distant from Schopenhauer’s concerns, and it is thereforeno great surprise to detect no Schopenhauerian influence on them.Although the later development of Wittgenstein’s ideas is not my main con-

cern here, I nonetheless want to caution against a faulty contrast betweenhis period in Cambridge with Russell and what followed thereafter. I havewarned already against overstating the degree to which the 1916 transforma-tion of Wittgenstein’s work—the extension of its scope beyond the confines ofphilosophical logic—is narrowly a response to the Brusilov offensive, or cor-relates crudely with a transformation in his religious views. It seems likelythat his move to Norway was a more or less deliberate attempt, unexpect-edly lengthened by the war, to move away from Russell’s sphere of influence.The work Wittgenstein did during the war was not a response to a whollynew stream of ideas: the influences it integrates into his work—Dostoevsky,Tolstoy, Schopenhauer—are ones to which he had already been exposed longbefore. What the war did, however, was to give him a space in which he couldintegrate these influences into his philosophy away from a climate which hehad begun to find restrictive.That both Frege and Russell influenced Wittgenstein’s thinking in the

Tractatus is too obvious to need much elaboration. Not only is that what Witt-genstein himself said in the Preface, but it is manifest in the text: both authorsare targets for substantial criticism there, but both are also responsible forsome of the book’s central ideas. One school of interpretation has been keento emphasize Frege’s influence on Wittgenstein rather than Russell’s. Dum-mett, for instance, regards the point as obvious. ‘Everyone knows,’ he says,

that Wittgenstein was soaked in Frege’s writings and Frege’s thought. Doubtless manyphilosophers unnamed byWittgenstein can be shown to have given him ideas. Others,to whom he does refer, provided him with material that he found interesting to reflector comment on: but Frege is very nearly the only one whom he quotes with approval.11

It may be significant, at least if Wittgenstein’s own later view of the matter hasany relevance, that those who knew him later in his life have generally sharedDummett’s opinion. Geach, for example, said that ‘the influence of Frege onWittgenstein was pervasive and life-long’.12 (Part of the reason Geach knewthis was that Wittgenstein had been instrumental in persuading him to trans-late Frege into English.) And only a few days before his death Wittgensteinwrote, ‘Frege’s style of writing is sometimes great.’13

11FOP, 237. 12‘Saying and showing in Frege and Wittgenstein’, 55. 136 Apr. 1951 (CV, 99).

!%' Themes

Another school, aiming perhaps to rehabilitate Russell’s place as the truefather of analytic philosophy, has treated pronouncements such as Dummett’sas a sort of challenge, and endeavoured instead to detect Russell’s influencein Wittgenstein’s works at every possible turn. Recognizing, perhaps, that itwould be wildly implausible to deny Frege’s influence completely, this schoolof interpretations has instead tried to place that influence as late as possible,suggesting, for instance, thatWittgenstein did not seriously study Frege’s worksuntil he arrived at Olmütz in Autumn 1916. This school has also, by pointingout places where Wittgenstein allegedly misunderstood Frege’s ideas,14 en-couraged the idea that whatever influence Frege had on him was not merelylate but superficial.Perhaps the most dubious technique that has been used to support this

view, however, is to find in Russell’s writings from 1914–19 prefigurations ofthe Tractatus, ignoring the fact that in the writings in question Russell oftenexplicitly acknowledges Wittgenstein’s influence.15 Russell’s influence on theTractatus was undoubtedly significant, but it is implausible that it could beestablished by citing chapter 2 of Our Knowledge of the External World, the lectureson logical atomism, or section 1 of ‘On propositions’. In all of these places theinfluence was plainly in the opposite direction: Russell was here very largelyreporting the ‘vitally important discoveries’ of his erstwhile student, not hisown ideas.Debates about whether one influence was greater than another are often

sterile, especially when, as here, it is so evident that both were formative andimportant. ‘It is obvious,’ said Pinsent, ‘that Wittgenstein is one of Russell’sdisciples and owes enormously to him.’16 Nonetheless, I hope that one thingthis book has done is to make clear how misbegotten is the attempt to mini-mize the effect of Frege on Wittgenstein’s thought, not just in the Tractatus butfrom the summer of 1912 onwards. Indeed, Frege’s influence on the Notes is sopervasive and so manifest that it is almost superfluous to supply an argumentfor it. The main reason that I have nonetheless thought it worth providingsuch an argument here is because I wanted to bring to light one suggestivefeature that this argument possesses. When we look for passages in Frege’spre-1913 writings that illustrate the influence he had on Wittgenstein’s think-ing as it is expressed in the Notes, it is notable how often it is the PosthumousWritings that we turn to rather than his published works. What this does isto hint at the enormous effect that his few visits to Frege must have had onWittgenstein. He seems to have absorbed Frege’s conception of philosophi-cal method at least as much through their conversations during these visits asthrough a careful study of his ‘great works’.

14Goldfarb, ‘Wittgenstein’s understanding of Frege: The pre-Tractarian evidence’; Kenny, ‘TheGhost of the Tractatus’ (although Kenny’s aim is not to promote Russell’s influence overFrege’s). 15E.g. Grayling, Russell, 96–102. 1625 Aug. 1913.

Influence on Russell !%(

The allegation that Wittgenstein mistook Frege’s ideas is in any case a dan-gerous one to wield as a way of showing that his engagement with those ideaswas superficial. Wittgenstein, like many of the best philosophers, was not acareful reader of texts. By that I mean that his technique in reading a text wasto think through its ideas for himself, as if from the inside. The practice of thistechnique may occasionally lead to exegetical errors, no doubt; but it is theopposite of superficial.

").&").&").& Influence on RussellFor a time, Wittgenstein’s influence on Russell was substantial. ‘Wittgenstein’sdoctrines influenced me profoundly,’ he later recalled. ‘I have come to thinkthat on many points I went too far in agreeing with him.’17 The war splitthis influence into ‘two waves’.18By the time he wrote Analysis of Mind and theIntroduction to the 2nd edition of Principia, Russell had read the Tractatus anddiscussed it in detail with its author. Yet the influence of the Tractatus itselfon these works is curiously muted. What we can certainly detect is the influ-ence of ideas Wittgenstein had in Norway in 1913–14: this influence is visible,for instance, in Russell’s discussion of propositional attitude statements, in hisattempt to do without the axiom of reducibility, and in the disappearance ofthe self from his account of mind, which in turn was a prerequisite for hisretreat into neutral monism. And the account in Analysis of Mind of proposi-tions ‘pointing towards’ or ‘pointing away from’ facts is based19 on things inthe Notes which Wittgenstein had all but abandoned by the time he finishedthe Tractatus. Much of what Wittgenstein added to the book during the war,on the other hand, always struck Russell as radically mistaken. He was nevertempted by Wittgenstein’s doctrine of unsayability, for instance, preferringinstead to avoid the paradoxes by the device, nowadays usually associatedwith Tarski rather than Russell, of an ever-expanding hierarchy of metalan-guages;20 and the book’s mysticism always struck him as wrong-headed. Ifthe specifically Tractarian influence on Russell is slight, this need not mean,however, that Russell did not study the Tractatus carefully, or that he did notunderstand it at all. It is at least as likely, it seems to me, that this is to beinterpreted in the opposite sense.There is room, therefore, to question how profoundly the Tractatus itself

influenced Russell. The effect on him of the prewar Wittgenstein, althoughcertainly profound, has perhaps been misrepresented. While Wittgenstein wasin Cambridge, Russell published (by his hectic standard) rather little in main-stream analytic philosophy: his two main projects, on matter and epistemol-ogy, were both stymied by his pupil’s opposition, and he in any case diverted

17MPD, 112. 18Ibid. 19The Analysis of Mind, 272 n. 20CP, IX, 111.

!"* Themes

much of his energy into other projects which he hoped would please his lover,Ottoline. Once Wittgenstein was in Norway, Russell was able to complete hisproject on matter free from that stultifying presence, but in the resulting paper‘On the relation of sense-data to physics’ Wittgenstein would surely only haveagreed with the rejection of the inference from sense-data to matter, not withthe attempt to replace it with a logical construction. Where Russell’s writingdoes show his debt to Wittgenstein is in the three texts cited earlier: chap-ter 2 of Our Knowledge of the External World, the 1918 logical atomism lectures,and the first section of ‘On propositions’. But here what Russell says containsso much that is straightforwardly a report of Wittgenstein’s views that it doesnot seem quite right to describe it without qualification as ‘influence’. In the1918 lectures, especially, Russell was reporting Wittgenstein’s logical resultsin order to bring them to public attention while Wittgenstein was unable todo it himself. We should represent these lectures as a phase in Russell’s ownintellectual history, therefore, only in a heavily qualified sense.The influence of the prewar Wittgenstein on Russell was in fact largely neg-

ative. When Russell described Wittgenstein’s criticisms as ‘an event of first-rate importance in his life’, he was right, although (as I argued in chapter 13)what was important was not the one paralysing criticism which led him toabandon the book he was writing, but the cumulative effect of their collabora-tion. Russell’s move into the psychology of language use in 1918 is explicablepartly as an attempt to find a research area to which he could still hope, afterWittgenstein’s destructive criticisms, to make a distinctive contribution.

").'").'").' Influence on Frege

The other person on whom we might hope to detect a direct Wittgenstein-ian influence is of course Frege, but in truth it is hard to find. Frege’s 1919article ‘Der Gedanke’, for instance, has surely struck many readers as con-taining some of the most Wittgensteinian passages in his writing, and it wouldtherefore be forgivable to speculate that it is influenced by the Tractatus. Butit is not: Frege submitted it for publication shortly before he saw the Tractatusfor the first time. Wittgenstein’s wartime letters to Frege have survived onlyin tantalizingly brief summaries, but they can hardly have been enough toexplain the material in question. And even the prewar Wittgenstein cannothave influenced the article very much, since all its main contentions are al-ready present in an earlier draft that was almost certainly written in 1897. Soif parts of ‘Der Gedanke’ (roughly, those not devoted to the rather inept attackon a naive version of idealism) strike the reader as Wittgensteinian, that canonly be further evidence of Frege’s influence on Wittgenstein rather than theconverse.

Conclusion !")

Much the same is true of the other parts of the Logische Untersuchungen: theyvery largely expound views that Frege held long before he met Wittgenstein;the only significant stretch of new material in them, a lengthy and franklyrather tedious discussion of negation, is a response not to Wittgenstein but toBruno Bauch (a neo-Kantian colleague of Frege at Jena and founder of thejournal in which the Untersuchungen were published).21

Of course, that is not to say that Wittgenstein did not influence Frege atall. It is notable, for instance, that in his late writings Frege hesitated overthe existence of logical objects. And he no longer laid any stress on the ideathat the reference of a sentence is a truth-value. These could conceivablybe signs that their prewar conversations had had some effect on him. Butwhen Frege read the Tractatus, most of his criticisms of it were ploddinglypedantic, depressingly reminiscent of the catalogue of confusions that, he toldJourdain, prevented him from progressing further than the first few pages ofPrincipia. What is clear at any rate is that if Wittgenstein did influence Frege,then Frege, unlike Russell, did not have the grace to admit it. The reference hewas prepared to write to Bauch on Wittgenstein’s behalf was by any standardscrushingly lukewarm.

I could tell him that I have got to know you as a thoroughly serious thinker. On thebook itself I cannot express a judgment, not because I don’t agree with its content, butbecause it is too unclear to me.22

Frege plainly felt affection for Wittgenstein and respected his abilities enoughto want to discuss philosophy with him, but there is no sign in the survivingcorrespondence that he had as much regard for him philosophically as Russelldid (or, indeed, that he even realized in how much esteem Wittgenstein heldhim).

").(").(").( ConclusionWittgenstein did not commit suicide in Norway nor die on the Eastern Front,and Russell was not required to publish the Notes posthumously as Wittgen-stein’s only philosophical legacy. They have therefore been viewed, eversince they were eventually published (at first only in Costello’s rearranged ver-sion) in 1957, through the lens of the book for which they were preliminarysketches. More recently, and less plausibly, that book in turn has increasinglybeen viewed as a sort of prolegomenon to his later writings rather than as aphilosophical terminus of its own. My aim here, by contrast, has been to ex-plain the Notes and assess them on their own merits—aware, of course, of whatthey later became, but not guided solely by that awareness.21See Schlotter, ‘Frege’s anonymous opponent in Die Verneinung’. 22Frege to LW, 30 Sep.1919.

!"! Themes

Russell thought Wittgenstein’s ideas in the Notes were ‘as good as anythingthat has ever been done in logic’.23 He devoted considerable time and ef-fort to coaxing those ideas onto paper in the form of the Cambridge Notes, totranslating the Birmingham Notes into English, and then to bringing the viewsboth sets of notes contain to wider attention at Harvard and in London. Hisattitude to the Notes was no doubt complicated by the deep personal (and fa-therly) affection he felt for their author. But he was not a sentimental old fool.What he had recognized, and had the grace to admit, was that Wittgensteinwas already closer to a correct account of the nature of propositions and theirrelationship to the world than he would ever be.And yet, however persuaded Russell was by the details of Wittgenstein’s

conception, most of the underlying principles which guided it were so farfrom Russell’s way of thinking that he never quite understood what they were.The nearest he came, perhaps, was to be troubled by an ominous sense thatWittgenstein had seen further and more clearly than he had. These guidingprinciples Wittgenstein owed to Frege, not to Russell. Some we see presentedin the Begriffsschrift and in the account of definitions in the Grundgesetze. Oth-ers emerge clearly only in the late flowering of Frege’s work in ‘Der Gedanke’.Others still lie just below the surface of his published work and are explicit onlyin his unpublished writings. What is notable about these principles, though, isnot merely that they continued to guide Wittgenstein in the Tractatus as muchas in the Notes, but that they constitute so much of what it seems right to regardas that book’s philosophical legacy.The Tractatus, it is worth recalling, is riddled with implausibilities: Witt-

genstein later rejected several of the book’s central contentions, most notablyits atomism, and hence its conception of analysis; the picture theory, its mostrenowned logical proposal, is highly problematic, the more so if the analogywith pictures is made to do any argumentative work; the book’s atomism is indanger of resurrecting Zeno’s paradox (which Russell hoped to have buried)and thereby rendering motion impossible; the account of the will leaves theconnection between its transcendental and empirical manifestations certainlyunexplained and probably inexplicable; the book’s insistence on the trivial-ity of logic is in blatant tension with its manifest complexity; and the notionthat the only part of mathematics whose application in real-world reasoningrequires philosophical explanation is elementary arithmetic is patently absurd.What is of lasting worth in the Tractatus is therefore not these theories, nor

even, perhaps, the book’s eventual (somewhat contrived) self-immolation, butrather the insights into the central themes of philosophical logic which it offers,insights which it derives preponderantly from Frege and which, I hope to haveshown, Wittgenstein had already in very large part grasped by the time he leftfor Norway in October 1913.

23To OM, 3 Oct. 1913.

Appendix A

History of the text

Much of this book has been concerned with explaining the content of the Noteson Logic. In this appendix, on the other hand, my aim is to reconstruct thecircumstances of composition of Wittgenstein’s first surviving philosophicalwork. Because none of the versions of the Notes that have survived is in Witt-genstein’s hand (barring corrections and emendations to one typescript), weshall have to engage in a considerable amount of detailed detective work. Thepurpose will be to settle which of these versions best represents his intentions,and indeed what those intentions were. What will emerge as a consequence,I hope, is that understanding how the Notes on Logic were composed is a usefulaid to understanding their content.

A.1 Narrative

In September 1913 Wittgenstein took a three-week holiday in Norway withhis university friend, David Pinsent. During the holiday he made plans to re-turn to Norway almost immediately and spend the following year there. Hewrote to Russell without yet telling him about these plans but asking to see himas soon as possible. ‘Give me time enough to give you a survey of the wholefield of what I have done up to now and if possible let me make notes for youin your presence.’1 Russell suggested a time (4 October at 1pm), but in theevent Wittgenstein evidently could not wait that long and went to see him assoon as he got back to Cambridge on Thursday 2 October. Part of this meet-ing they spent reading together the manuscript of Whitehead’s latest work onspace; and part on discussing Wittgenstein’s plan to spend the following yearin Norway.2 Then, Russell said, Wittgenstein ‘stayed late . . . and read me bitsof the work he has done. I think it is as good as anything that has ever beendone in logic.’3 He was ‘explaining a number of very difficult logical ideaswhich I could only just understand by stretching my mind to the utmost’.4 Itis worth noting, though, that, despite Russell’s pleadings for Wittgenstein towrite some of these ideas down, there is no mention yet that he did: perhaps

120 Sep. 1913. 2BR to OM, 2 Oct. 1913. 3To OM, 3 Oct. 1913. 4To OM, 6 Oct. 1913.

!"$ History of the text

this was the occasion when ‘his artistic conscience got in the way, and becausehe couldn’t do it perfectly he couldn’t do it at all’.5

Over the next four daysWittgenstein metMoore on three occasions (Thurs-day, Friday, and Sunday).6 Russell, on the other hand, seems to have beenaway in London from Friday afternoon until Sunday, and we have no evi-dence that Wittgenstein saw him again before he himself went off to Birm-ingham on Monday 6 October to stay with Pinsent’s family for a couple ofnights. By then he had promised Russell that he would ‘leave a written state-ment of what he [had] already done before he start[ed] for Norway’.7 Thefollowing day (Tuesday 7th) he went to the Berlitz school of languages next toNew Street railway station in the centre of Birmingham, where he succeededin finding a German shorthand writer to whom he could dictate ‘extracts fromhis note-book’. Pinsent’s diary records that he left the house at 5.30pm andreturned around 8.30, so we may guess that the dictation session itself lastedfrom 6pm until about 8.OnWednesday morningWittgenstein returned to Cambridge: he will have

arrived shortly before lunch. Russell had arranged to walk Wittgenstein outto Jourdain’s for tea, but apart from that, and a brief period when he hadto deal with another visitor, they were together (presumably from lunchtime)until ‘near midnight’, he told Ottoline. (Moore’s diary records that he sawWittgenstein again from 9.30 to 10.30 that evening: perhaps that is whenRussell saw his other visitor.) There were, Russell said,

newer things [than what Wittgenstein had dictated in Birmingham], and things notsufficiently explained. He said he would make a statement of them, and sat down todo it. After much groaning he said he couldn’t. I abused him roundly and we hada fine row. Then he said he would talk, and write down any of his remarks that Ithought worth it, so we did that, and it answered fairly well. But we both got utterlyexhausted, and it was slow.8

Before this marathon session Russell had thought it would be ‘a last dose’ ofWittgenstein, but as things turned out they arranged to meet one more timethe following day. In the interim Russell managed to obtain the services ofan (English) secretary who could take dictation in an attempt to speed up theprocess of getting Wittgenstein’s explanations written down.Jourdain’s secretary (the one who is prettier than Waterlow’s bride) is coming to takedown our conversation in short-hand. Mercifully Jourdain sent her this morning toborrow a book of mine so I grabbed her. It is early closing day so no one can be gotexcept as a favour. TomorrowWittgenstein goes to London, and Saturday to Norway.Today in the middle I have to have Lucy Donnelly’s young lady to tea—she will givea breathing-space.9

The secretary (the Miss Harwood Wittgenstein refers to in a later letter)10

5BR to OM, 9 Oct. 1913. 6Moore’s diary. 7BR to OM, 6 Oct. 1913. 8To OM, 9 Oct. 1913.9To OM, 9 Oct. 1913. 10To BR, [Nov. 1913] (CL, no. 29).

The manuscripts !"%

will no doubt have delivered the result of this last dictation session to Russell afew days later. He also soon received the typescript dictated in Birmingham.Perhaps Wittgenstein had asked the German stenographer to send it direct toRussell, but this would mean that he trusted the stenographer to have copiedthe logical notation correctly. It is more probable, I suspect, that the typescriptwas posted to Wittgenstein, who quickly copied in the logical formulae andposted it on to Russell just before leaving for Norway. Wittgenstein wrote toRussell from the boat, ‘Hope you have got typewritten business all right.’11

However, he forgot to post the letter—someone else did so on his behalf—and he therefore wrote again a few days later. ‘Did you get the copy of mymanuscript?’ he asked.12 The first of these enquiries might, I suppose, beabout either of the two typescripts, but the second can surely only refer to thetypescript dictated in Birmingham.Shortly thereafter, then, Russell must have had in his possession two type-

scripts, one produced in Birmingham by the German stenographer and theother in Cambridge by Jourdain’s secretary, together with whatever man-uscript Wittgenstein had managed to produce by the end of Wednesday’smarathon session with Russell. On 25 October Russell sent the Cambridgetypescript to Wittgenstein, who had by then arrived in Norway, for him tocorrect, together with a list of questions that had occurred to him since theirmeetings; we also know that he had the Birmingham typescript by then, sinceone of his questions quotes from it.

A.2 The manuscripts

The typescript dictated in Cambridge on the Thursday has survived: it hasbeen labelled by Russell as a ‘Summary’, and contains corrections in his handas well as in Wittgenstein’s own. But the Birmingham typescript and Wittgen-stein’s manuscript from the Wednesday are now lost. What we have insteadis a manuscript in Russell’s hand, written in English, which evidently datesfrom February 1914. This was when he was, as he told Ottoline, ‘translatingand copying and classifying the notes of Wittgenstein’s work, as I shall wantthem for lecturing on logic at Harvard—that takes a lot of time, but is nowfinished’.13

Russell’s manuscript is in four parts, which he has labelled ‘MS1’, ‘MS2’,‘MS3’, and ‘MS4’. One question that we need to address, then, is whichparts of the manuscript correspond to which of the sessions described earlier.The view I shall argue for is that MSS 1,3, and 4 are Russell’s translationinto English of the typescript dictated in Birmingham, while MS2 is Russell’s

1117 Oct. 1913. 1229 Oct. 1913. 1328 Feb. 1914.

!"" History of the text

transcription of what Wittgenstein had written out (mostly, at least, in English)in his presence on Wednesday 8 October.One reason for believing this is that there are some notable differences be-

tween MS2 and the other manuscripts. Certainly the versions of MSS 1, 3,and 4 in Russell’s hand all contain clear signs that they are translations: wordsare crossed out and amended in ways that would be odd if all he was do-ing was copying out a text in English. No such emendations are present inMS2. Yet Russell’s own description of his activities in February 1914 refersto both translating and copying: the summary did not need to be copied as itwas already typed, so MS2 is left as the only piece we have that Russell mighthave copied. Stylistically, too, MS2 is closer to the Summary than it is to therest of the Notes: much of it consists of reformulations of things in MSS 1, 3,and 4 (the things, presumably, that Wittgenstein thought were ‘not sufficientlyexplained’); and it reads as halting and repetitive in comparison to them.A further feature which distinguishes MS2 and the Summary, on the one

hand, from MSS 1, 3, and 4, on the other concerns how Wittgenstein laterused the material they contain. There is a section of the Prototractatus14 whichdraws very closely, with only minor changes of wording, on remarks fromMSS 1, 3, and 4 (broadly in that order). But the Prototractatus never draws onMS2 or the Summary word-for-word in this manner: there are occasionallysentences expressing the same ideas, but the formulations are always a littledifferent. This fits well with the notion that MS2 and the Summary representWittgenstein’s attempts to explain his ideas to Russell in English, to be writtendown in one case by Wittgenstein himself and in the other by Miss Harwood,whereas MSS 1, 3, and 4 are Russell’s translations of remarks taken fromWittgenstein’s notebooks. Whether or not Wittgenstein was in possession ofa copy of MS2 and the Summary when he compiled the Prototractatus, he didnot make direct use of them in it because they had a wholly different origin.Did Wittgenstein and Russell usually speak to each other in German? Rus-

sell’s recollection forty years later (commenting on a German radio broadcastabout their relationship) certainly suggests that they did.

At a certain p[oin]t in our acquaintance we began to say ‘du’ to each other and in ourcorrespondence we always did so. In the broadcast it is right to keep ‘sie’ down to andincluding the point at wh[ich] I told him not to become an aviator, but after that itw[oul]d be more true to history to say ‘du’ instead of ‘sie’.15

But the suggestion, implicit in Russell’s recollection, that he and Wittgensteinhabitually spoke to each other in German is quite hard to accept. When Witt-genstein first arrived in Cambridge, we know that they conversed in Englishat Wittgenstein’s insistence, even though, according to Russell, Wittgensteinspoke ‘very little English’.16 (Since Wittgenstein had by then been living in

14pp. 28(3)–34(3). 15To O’Rourke, 22 Sep. 1955. 16To OM, 18 Oct. 1911.

The manuscripts !"&

England for three years, this was presumably just a sign of his nervousnesson meeting Russell for the first time; at any rate we have no evidence fromany other source that Wittgenstein’s spoken English was weak enough to bea problem.) It seems very likely that they continued to speak to each otherin English for the remainder of Wittgenstein’s time in Cambridge. For hisletters to Russell continued to be written in English until some time after hehad arrived in Norway; and when he finally wanted to switch to German,17

he realized that he did not know whether to address Russell familiarly or not.This would be strange if they were by then used to calling each other ‘Du’ inspeech. The presumption that they always spoke to each other in English is inany case independently likely: Russell had spent a year in Germany as a youngman, but that was now almost twenty years in the past, so it is hard to imag-ine that his spoken German was by 1913 nearly as fluent as Wittgenstein’sEnglish. (Of course, this does not exclude the possibility that Wittgensteinsometimes read out to Russell sentences in German from his notebooks.)Nonetheless, it is a little odd that on the account I am offering Russell chose

to copy out MS2 by hand before sending it to be typed. Why did he dothis? One possibility—conjectured, in effect, by McGuinness18—is that hedid not: that the document we possess was written by Russell in October 1913to Wittgenstein’s dictation. But at which meeting? Not, presumably, on theWednesday, when Russell was explicit that it was Wittgenstein who did thewriting, nor on Thursday, when Miss Harwood was there to take dictation.And which, on McGuinness’s account, was the document Russell ‘copied’ thefollowing February?If we therefore reject McGuinness’s conjecture and agree that the docu-

ment we possess is Russell’s transcription of an original, now lost, written byWittgenstein himself at the Wednesday meeting, we do admittedly have toexplain why Russell bothered to copy it out. Certainly Wittgenstein’s hand-writing was in normal circumstances quite legible. One possibility is that themanuscript he produced on this occasion, even if it was legible, was so disorga-nized that Russell did not feel Miss Harwood would be able to make enoughof it to be able to type it out. The manuscript as we have it is, as noted ear-lier, halting and repetitive; but the account I am now offering suggests, evenso, that it may already have undergone one stage of editing by Russell. An-other possibility is that even if, for the reasons just explained, Wittgenstein’smanuscript was very largely in English, it may in a few places have lapsedinto German (presumably where Wittgenstein was simply quoting from hisnotebooks) and hence required partial translation.If this deals adequately with MS2, let us turn now to the other three man-

uscripts. We can be confident at least that MS3 was dictated in Birmingham,

17[Nov. 1913] (CL, no. 30). 18Young Ludwig, 186.

!"' History of the text

for we have the anecdote, recollected by Russell much later,19 that whenWitt-genstein came to the sentence ‘ “A” is the same letter as “A” ’ (which occursonly in MS3), the stenographer remarked, ‘Well, that is true anyway.’ Weknow also that Russell had received the typescript of MS3 by the time he sentWittgenstein his list of questions, since (as noted earlier) one of these ques-tions quotes a sentence from the German of MS3. And although we cannotbe equally certain that the typescript Russell received from Birmingham in-cluded MS1 and MS4 as well, my main reason for thinking that it did is that itis hard to think of any other hypothesis which explains the available informa-tion. At any rate the total length of MSS 1, 3, and 4 (about 4,000 words) couldcomfortably be dictated to a competent stenographer in two hours, even al-lowing for the occasional passages of logical notation which the stenographerwould have had to copy out longhand.If this is right, the productivity of Wittgenstein’s session in Birmingham was

in marked contrast to the following day, when he managed to write down only600 words in something over six hours. Wittgenstein’s ‘artistic conscience’,which ‘prevents him from writing anything until he has got it perfect’,20 wasevidently something that struck only selectively, and was especially debilitat-ing in Russell’s presence. It is worth noting, though, that if my hypothesesare correct, the circumstances were very different on the two occasions: inBirmingham Wittgenstein was simply reading out to a stenographer markedpassages from his notebooks; in Cambridge he was explaining his ideas labo-riously to Russell, arguing about many of them, and writing down, often in extempore formulations, those Russell thought worth preserving.

A.3 Russell’s labellingWhy, then, did Russell label the sections of the Notes as he did? The labellingof the Summary suggests that Wittgenstein had brought with him to that dic-tation session several of his notebooks. And a comparison with the other partsof the Notes does indeed provide us with passages in the Summary which areclose enough in wording to show that he was translating (if perhaps freely)from the notebooks from which MSS 3 and 4 are drawn.Turning now to MSS1–4, we should note that Russell probably saw the

text we are currently discussing only as an intermediate stage in an editing taskthat would eventually turn Wittgenstein’s apparently random arrangement ofmaterial into what has become known as the Costello version of the Notes (ofwhich more shortly). With this in mind, the labelling of MSS 1–4 may indeedhave been no more than that—a labelling, whose only purpose was to corre-late parts of Russell’s version with the corresponding parts of Wittgenstein’s

19McGuinness, Approaches, 257. 20BR to Lucy Donnelly, 19 Oct. 1913.

Russell’s labelling !"(

text; it may therefore simply have reflected the order in which Wittgenstein’stexts lay in the file when he drew it out in February 1914. On the other hand,it is worth noting that the order in whichMSS 1, 3, and 4 are numbered corre-sponds to the order in which Wittgenstein drew on the propositions containedin them when he composed the Prototractatus. So Russell does seem at least tohave kept the three parts of the Birmingham typescript in their original orderwhen he labelled his translations of them.The labelling of MS2 suggests that perhaps on the Wednesday, unlike the

following day, Wittgenstein only had one of his notebooks with him. Indeed,although MS2 contains reformulations of various remarks in MSS1, 3, and 4,none is close enough in wording to show that Wittgenstein must have had therelevant notebook with him.But that does not explain why Russell numbered it as he did, interposing

MS2 into the sequence of the Birmingham typescripts. In this connection itis perhaps worth noting that MS1 is only about 600 words long and need nothave taken a competent typist more than quarter of an hour to type. Onepossible explanation, therefore, is that the Birmingham stenographer typedMS1 straightaway on the Tuesday evening and gave it to Wittgenstein thenand there (or, I suppose, did it after Wittgenstein had gone and left it some-where for him to pick up on his way to New Street station the next morning).Wittgenstein would then have given this typescript to Russell at the beginningof their meeting the following afternoon. On this hypothesis the Birminghamstenographer finished the rest of the typing and posted it to Russell later, sothat the ordering of MSS 1–4 simply reflects the order in which Russell re-ceived them.It is difficult, however, to supply a similar hypothesis to explain the divi-

sion of MS3 and MS4 into two parts. At a pinch, I suppose, it might simplybe that the Birmingham stenographer typed up his shorthand notes in twofurther batches and forwarded them separately to Russell. A more likely ex-planation, I think, is that Wittgenstein had instructed the typist to keep themdistinct because they were drawn from different notebooks. We know thatWittgenstein had taken with him on his holiday in Norway a ‘portmanteauwith all his manuscripts inside’.21 We know, too, that Wittgenstein often re-ferred to his notebooks as ‘manuscripts’. We may conclude, therefore, thatthere were at this point several Cambridge notebooks that Wittgenstein stillregarded as worth consulting (and indeed whose temporary loss threw himinto a panic, as Pinsent relates in his diary). It would surely have been natu-ral for Wittgenstein to extract remarks from more than one of these, and toinstruct the stenographer to type them separately.However, there is a difficulty with this hypothesis. Wittgenstein certainly

had the text of the Birmingham typescript available to him when he was com-21Pinsent’s diary, 30 Aug. 1913.

!&* History of the text

piling the Prototractatus: as noted earlier, there is a section of that work in whichalmost every remark is drawn from it. And yet we know, for reasons I neednot go into here,22 that the only prewar notebook Wittgenstein had in Aus-tria during the war was the one he kept during his year in Norway. For thisreason McGuinness has suggested that the Notesmust be derived directly fromthis notebook.I am inclined to reject this suggestion, for a number of reasons. As just

mentioned, if all of MSS 1, 3, and 4 were drawn from the same notebook, itis puzzling why Russell labelled them as distinct texts. Moreover, it is quiteimplausible that all these notes are the product of a short creative burst dur-ing Wittgenstein’s three-week holiday in Norway with Pinsent. They readmuch more like extracts from notebooks written over a considerable period—not, certainly, the whole of Wittgenstein’s period in Cambridge, but the con-tent suggests that they extend back perhaps as far as the previous Febru-ary. If so, McGuinness’s hypothesis requires us to suppose that one notebooklasted Wittgenstein from early in 1913 until the summer of 1914. This is notimpossible—we know, after all, that it was a ‘large notebook’23—but it doesseem a little unlikely.Something else worth noting is the pattern detectable during the war ac-

cording to which Wittgenstein started a new notebook when his life entered anew phase. Of course, this pattern is scarcely decisive on its own—one reasonhe had to start new notebooks during the war was so that the old ones couldbe deposited in Vienna, from where they were to be dispatched to Russell inthe event of his death in action—but it is at least suggestive. Wittgenstein’sbehaviour during his short period in England between the Norwegian holidayand the beginning of his longer Norwegian exile speaks very much of someonetidying up the loose ends of one stage of his life in order to start a new stagewith as little baggage as possible, and it is quite plausible that he would havestarted a new notebook at this point.If this consideration carries any weight, though, it is open to question ex-

actly when Wittgenstein saw the new phase of his life as starting. There isa hint in Pinsent’s diary that perhaps Wittgenstein had already formed hisplan to spend a year in Norway earlier in the summer and treated the holi-day in September as a means of investigating the practical implementation ofthe plan. An hypothesis I find quite attractive, therefore, is that Wittgensteinstarted his Norwegian notebook at the beginning of the holiday and continuedit through the succeeding year. If that is what happened, the ‘newer things’not in the Birmingham dictation which Wittgenstein explained to Russell inCambridge will presumably be the product of the Norwegian holiday, writtenup in the new Norwegian notebook.

22See McGuinness, Approaches, 261. 239 Aug. 1914.

The Costello version !&)

This, though, is a point of relative detail. More important are the gen-eral conclusions we have reached, namely that MSS 1, 3, and 4 are drawnfrom Wittgenstein’s Cambridge notebooks, which he left behind when he de-parted for Norway and which after the war he instructed Russell to destroy;and that the dictation in Birmingham was an attempt to extract from thesenotebooks—for Russell’s benefit, no doubt, but also (perhaps primarily) forhis own—everything he thought worth preserving in them.If this is broadly correct, and if MSS 1, 3, and 4 correspond to three dis-

tinct Cambridge notebooks, the route by which extracts from them reachedthe Prototractatus cannot be directly from the notebooks, to which he had no ac-cess for the duration of the war, but must rather be via the typescripts them-selves. One possibility is that Russell passed them back to Wittgenstein afterhe had translated them: a natural opportunity for him to have done this wasvia Moore, who departed for his visit to Wittgenstein in Norway shortly af-ter Russell had finished his translation. However, even if Russell did pass hiscopy of the Birmingham typescript back to Wittgenstein in this manner, it isunlikely to have been the only copy: it would be natural, especially if this type-script was the summation of his work in Cambridge, for Wittgenstein to haveensured that the Birmingham typist made two copies of the typescript, one forRussell, the other for Wittgenstein himself. By one or other of these routes (orperhaps both), therefore, I believe Wittgenstein obtained in Norway a copy ofthe Birmingham typescript, from which he later extracted remarks to includein the Prototractatus.

A.4 The Costello versionWe noted earlier that in the last week of February 1914 Russell was ‘trans-lating and copying and classifying’ Wittgenstein’s notes. The manuscript inRussell’s hand discussed in the previous section represents, as we have seen,the translating and copying. A surviving typescript (which McGuinness callsthe ‘second-stage typescript’ or T) provides us with evidence of how he thenwent about the classifying stage. This typescript was prepared for Russell byMiss Harwood, the secretary who had taken dictation in October: in the caseof the Summary she was therefore redoing what she had already typed inOctober, but now incorporating Russell’s and Wittgenstein’s corrections toit; in the case of the remainder of the Notes she was working from Russell’smanuscripts. Russell now wrote a list of numbered headings on a separatesheet and on the new typescript put corresponding numbers against the para-graphs, classifying them according to their subject matter; two paragraphs24

he marked for deletion.

24B49, B64.

!&! History of the text

Russell told Ottoline that he was doing this editorial work onWittgenstein’snotes because ‘I shall want them for lecturing on logic at Harvard’. And cer-tainly he took either the marked up second-stage typescript just described oranother prepared from it with him to Harvard. There Russell was allocatedas his teaching assistant for his logic course a graduate student called HarryCostello; and as Costello later recalled, he ‘had with him some notes andexcerpts, giving the opinions of a brilliant student of his, named Ludwig Witt-genstein. . . I copied this manuscript’.25

This copy, which was first published in 1957 (by Costello himself) has cometo be known as the Costello version of the Notes. Producing it from the second-stage typescript Russell had marked up was not a purely mechanical process,though. The Costello version not only divides the Notes into seven sectionsaccording to Russell’s classification; it also rearranges the paragraphs withineach section; in some (but by no means all) of the cases where the Birminghamand Cambridge versions duplicate one another it chooses which to include(more often the Birmingham version); and it makes various minor changes inwording.What is clear is that whoever first performed this task had not only the

surviving copy of the second-stage typescript with Russell’s annotations but acarbon copy as well, since there are no signs on the surviving typescript of thesort of tick marks that would have been necessary in order to confirm that eachparagraph had been copied across. The method used was probably to cut upthe carbon copy into paragraphs, sort these into seven piles correspondingto Russell’s classificatory markings, reorder each pile so as to make the bestsense, adapt the prose by hand in a handful of places, and then either copy theresult out by hand or give the ordered piles of paper slips to a typist to retype.Who performed this editing process? The only two plausible candidates are

evidently Russell himself and Costello. According to Costello’s own reminis-cences Russell did hand round to those who attended his logic lectures piecesof Wittgenstein’s work, and there may well have been various opportunitiesfor them to copy parts of them. If it was Costello who performed the editing,however, he can only have done it with Russell’s knowledge and approval.For, as just noted, it involved at the very least marking, and probably cuttingup, Russell’s carbon copy of the notes. The process must have cost someonea great deal of work and study,26 and there is therefore a certain plausibil-ity in the idea that Russell—a man who, as McGuinness drily observes, ‘hadnever tied up a parcel in his life’27—might have asked his teaching assistant,Costello, to perform it: the purpose, presumably, was to make the resultingdocument available for further consultation by other students attending Rus-sell’s logic course.28 In that case, though, it is somewhat curious that Russell

25Costello, ‘Notes on Logic’, 230. 26See Approaches, 252. 27Ibid. 258. 28I have been unable totrace what happened to Costello’s copy after he published it in 1957, and do not know whetherit was typed or handwritten.

The Costello version !&#

himself should have broken off the editing process when he did, with all theparagraphs classified but not yet rearranged: he told Ottoline on 28 February1914, six days before he was due to sail for America, that the editing, althoughit had taken a lot of work, was ‘now finished’; yet on this account it was reallyonly half done.The alternative hypothesis is that Russell himself, having classified the para-

graphs of the second-stage typescript according to subject, completed the taskby performing the rearrangement of the carbon copy just described. It willthen presumably have been Miss Harwood who produced, just in time forRussell’s departure, the final typescript of this rearrangement. On this ac-count Russell may well have taken only this typescript to Harvard, where hemade it available for Costello to copy. This hypothesis has the advantage thatit conforms with Costello’s later description of his version of the Notes as onehe had ‘copied’ (not ‘compiled’) at Harvard. On this account, too, the remainsof the process which have survived among Russell’s papers are explicable asbeing just those he left behind in Cambridge when he departed for Americain March 1914. When he returned to England in June, he may simply haveleft behind at Harvard the copy of the Costello version which he had takenwith him.The textual differences between the surviving typescript and the Costello

version as it has been transmitted to us may also be thought to lend someplausibility to this second hypothesis. Some of these differences (‘causes’ for‘cases’, ‘cannot’ for ‘can not’, a whole phrase omitted) are evidently just copy-ing errors, for which either Costello or an intervening typist was no doubtresponsible; the logical symbols have been especially badly mauled (‘q’ for‘!’, existential quantifiers omitted). Other changes are the sort of unimpor-tant minor variations (‘where’ for ‘when’) that many copyists make almostunconsciously. But some (‘may be symbolized’ for ‘symbolized’, putting ‘log-ical objects’ in scare quotes, or changing ‘know the meaning of names’ to thesomewhat different ‘know that our names have meaning’) suggest a certainconfidence, a degree of ownership over the text, that would be a little sur-prising in a graduate student. And one or two are explanatory rephrasingswhich surely betray a more expert hand than Costello’s. Would a graduatestudent producing a text for circulation have felt certain enough to gloss ‘abfunctions’29 as ‘functions with sense’, ‘logical types’30 as ‘forms’, ‘constituent’as ‘particular’, and ‘component’ as ‘particular or relation, etc.’?31

None of these considerations is decisive on its own, of course, but cumu-latively they make it more likely than not, I believe, that the Costello versionresulted from his copying of a typescript, now lost, which Miss Harwood com-piled on the basis of Russell’s own instructions and which Russell then tookwith him to Harvard.

29B73. 30B16. 31B37.

!&$ History of the text

A.5 Wittgenstein’s dissertation

At just the time when Russell was engaged in ‘translating, copying, and classi-fying’ the Notes, Moore was preparing to visit Wittgenstein in Norway. Russellmet Moore to discuss Wittgenstein’s ideas and lent him a copy of the Notesto read.32 If Russell had by then completed the Costello version, it would bevery natural for him to have given a carbon copy of it to Moore to take toWittgenstein. One obvious reason for him to do this would have been thatRussell intended to show the Notes to his students at Harvard, and one mightthink that courtesy required him to ensure that Wittgenstein saw the versionto be circulated.The question of who compiled the Costello version is therefore not of purely

antiquarian interest. For if it was a carbon copy of the Costello version thatMoore took to Norway, it is possible that this is whatWittgenstein then wantedto submit as a BA dissertation at Cambridge. If so, then perhaps the rear-rangement of Wittgenstein’s material that the Costello version represents hadWittgenstein’s blessing. This might explain, too, Wittgenstein’s touchinesswhen Moore told him that he would have to supply a signed declaration ofthe extent to which the dissertation was his own work. The degree of editingwhich the Costello version had undergone very plainly represents a greaterinput than one would normally expect from a supervisor, and so it would nothave been straightforward for Wittgenstein to make the declaration in just theform stipulated in the Cambridge regulations.But none of this is certain; the available evidence does not exclude the

alternative that it was the notes Wittgenstein dictated to Moore in Norwaythat he wanted to submit as a dissertation. And whoever rearranged the Notesto form what we now know as the Costello version, what is certain is thatit was not Wittgenstein. For this reason I have chosen to reprint the earlierversion of the Notes in this book and not to attempt to draw conclusions aboutWittgenstein’s views from the arrangement of the Costello version.

A.6 Conclusion

The central point I have been concerned to establish in this Appendix is thatthe Notes should be thought of as consisting of two distinct texts of rather dif-ferent characters, one produced in Birmingham and the other in Cambridge.The first of these, the Birmingham Notes on Logic, consists of MSS 1, 3, and

4. They are extracts from Wittgenstein’s Cambridge notebooks which he dic-tated in German to a stenographer in Birmingham; the version we have ofthese Birmingham Notes is a translation made by Russell four months later.

32Moore’s diary, 28 Feb. 1914.

Conclusion !&%

The three parts most likely correspond to three distinct notebooks. We can-not now determine for certain whether the Birmingham Notes preserve thechronological order of the entries in these notebooks, but I see no strong rea-son to suppose that they do not.The second text, the Cambridge Notes on Logic, consists of MS2 and the

Summary. MS2 was written (largely, if not entirely, in English) by Wittgen-stein himself in Russell’s rooms in Cambridge; the Summary he dictated thefollowing day entirely in English. This Cambridge text seems to contain afew extracts from Wittgenstein’s notebooks, and in the case of the Summary,at least, these will presumably be Wittgenstein’s own impromptu translations.Very largely, though, the document is a commentary on the BirminghamNotes. Only a very few of its remarks are not either translations of or ex-planatory glosses on remarks in the Birmingham version: these few additionalremarks may be what Russell called ‘newer things’—ideas Wittgenstein hadhad only very recently, during his Norwegian holiday.Much of the route we have taken here in order to reach these conclusions is

perhaps at best of biographical rather than philosophical interest. (If nothingelse, it provides eloquent testimony of Russell’s extraordinary patience in deal-ing withWittgenstein and of the esteem in which he heldWittgenstein’s work.)Nonetheless, the outline just given is worth bearing in mind when studying theNotes, since it suggests the approach that we should take when reading them:read the Birmingham Notes first and treat the Cambridge Notes, as Wittgen-stein intended them, mainly as a source of alternative formulations that mayin some cases be more illuminating or explanatory than the originals.

Appendix B

The Notes on Logic

The Birmingham Notes

First MS.B1. Indefinables are of two sorts: names, and forms. Propositions cannotIIconsist of names alone; they cannot be classes of names. A name can not onlyoccur in two different propositions, but can occur in the same way in both.B2. Propositions [which are symbols having reference to facts] are themselvesIIfacts: that this inkpot is on this table may express that I sit in this chair.B3. It can never express the common characteristic of two objects that we des-Vignate them by the same name but by two different ways of designation, for,since names are arbitrary, we might also choose different names, and wherethen would be the common element in the designations? Nevertheless one isalways tempted, in a difficulty, to take refuge in different ways of designation.B4. Frege said “propositions are names”; Russell said “propositions corre-Ispond to complexes”. Both are false; and especially false is the statement“propositions are names of complexes”.B5. It is easy to suppose that only such symbols are complex as contain namesVof objects, and that accordingly “(

E

x,!) !x” or “(

E

x,R,y) xRy” must be simple.It is then natural to call the first of these the name of a form, the second thename of a relation. But in that case what is the meaning of (e.g.) “!(x,y) xRy”?Can we put “not” before a name?B6. The reason why “!Socrates” means nothing is that “!x” does not ex-IIIpress a property of x.B7. There are positive and negative facts: if the proposition “this rose is notIred” is true, then what it signifies is negative. But the occurrence of the word“not” does not indicate this unless we know that the signification of the prop-osition “this rose is red” (when it is true) is positive. It is only from both, thenegation and the negated proposition, that we can conclude to a character-istic of the significance of the whole proposition. (We are not here speakingof negations of general propositions, i.e. of such as contain apparent variables.Negative facts only justify the negations of atomic propositions.)B8. Positive and negative facts there are, but not true and false facts.IB9. If we overlook the fact that propositions have a sense which is independentIof their truth or falsehood, it easily seems as if true and false were two equally

The Birmingham Notes !&&

justified relations between the sign and what is signified. (We might then saye.g. that “q” signifies in the true way what “not-q” signifies in the false way.) Butare not true and false in fact equally justified? Could we not express ourselvesby means of false propositions just as well as hitherto with true ones, so longas we know that they are meant falsely? No! For a proposition is then truewhen it is as we assert in this proposition; and accordingly if by “q” we mean“not-q”, and it is as we mean to assert, then in the new interpretation “q” isactually true and not false. But it is important that we can mean the same by“q” as by “not-q”, for it shows that neither to the symbol “not” nor to themanner of its combination with “q” does a characteristic of the denotation of“q” correspond.

3rd MS.B10. An analogy for the theory of truth: Consider a black patch on white Ipaper; then we can describe the form of the patch by mentioning, for eachpoint of the surface, whether it is white or black. To the fact that a point isblack corresponds a positive fact, to the fact that a point is white (not black)corresponds a negative fact. If I designate a point of the surface (one of Frege’s“truth-values”), this is as if I set up an assumption to be decided upon. But inorder to be able to say of a point that it is black or that it is white, I must firstknow when a point is to be called black and when it is to be called white. Inorder to be able to say that “p” is true (or false), I must first have determinedunder what circumstances I call a proposition true, and thereby I determinethe sense of a proposition. The point in which the analogy fails is this: I canindicate a point of the paper without knowing what white and black are, butto a proposition without sense nothing corresponds, for it does not designatea thing (truth-value), whose properties might be called “false” or “true”; theverb of a proposition is not “is true” or “is false”, as Frege believes, but whatis true must already contain the verb.B11. The comparison of language and reality is like that of retinal image Iand visual image: to the blind spot nothing in the visual image seems to cor-respond, and thereby the boundaries of the blind spot determine the visualimage—as true negations of atomic propositions determine reality.B12. Logical inferences can, it is true, be made in accordance with Frege’s or IIIRussell’s laws of deduction, but this cannot justify the inference; and thereforethey are not primitive propositions of logic. If p follows from q, it can also beinferred from q, and the “manner of deduction” is indifferent.B13. Those symbols which are called propositions in which “variables occur” IVare in reality not propositions at all, but only schemes of propositions, whichonly become propositions when we replace the variables by constants. Thereis no proposition which is expressed by “x=x”, for “x” has no signification; butthere is a proposition “(x) x=x” and propositions such as “Socrates = Socrates”etc.

!&' The Notes on Logic

B14. In books on logic, no variables ought to occur, but only the generalIVpropositions which justify the use of variables. It follows that the so-calleddefinitions of logic are not definitions, but only schemes of definitions, and in-stead of these we ought to put general propositions; and similarly the so-calledprimitive ideas (Urzeichen) of logic are not primitive ideas, but the schemesof them. The mistaken idea that there are things called facts or complexesand relations easily leads to the opinion that there must be a relation of ques-tioning to the facts, and then the question arises whether a relation can holdbetween an arbitrary number of things, since a fact can follow from arbitrarycases. It is a fact that the proposition which e.g. expresses that q follows fromp and p ! q is this: p p ! q !p,q q.B15. At a pinch, one is tempted to interpret “not-p” as “everything else, onlyInot p”. That from a single fact p an infinity of others, not-not-p etc., follow,is hardly credible. Man possesses an innate capacity for constructing symbolswith which some sense can be expressed, without having the slightest idea whateach word signifies. The best example of this is mathematics, for man has untillately used the symbols for numbers without knowing what they signify or thatthey signify nothing.B16. Russell’s “complexes” were to have the useful property of being com-IIpounded, and were to combine with this the agreeable property that theycould be treated like “simples”. But this alone made them unserviceable aslogical types, since there would have been significance in asserting, of a sim-ple, that it was complex. But a property cannot be a logical type.B17. Every statement about apparent complexes can be resolved into theIIlogical sum of a statement about the constituents and a statement about theproposition which describes the complex completely. How, in each case, theresolution is to be made, is an important question, but its answer is not un-conditionally necessary for the construction of logic.B18. That “or” and “not” etc. are not relations in the same sense as “right”IIIand “left” etc., is obvious to the plain man. The possibility of cross-definitionsin the old logical indefinables shows, of itself, that these are not the right inde-finables, and, even more conclusively, that they do not denote relations.B19. If we change a constituent a of a proposition !(a) into a variable, thenVIthere is a class

p{( Ex) !(x) = p}.

This class in general still depends upon what, by an arbitrary convention, wemean by “!(x)”. But if we change into variables all those symbols whose sig-nificance was arbitrarily determined, there is still such a class. But this is nownot dependent upon any convention, but only upon the nature of the symbol“!(x)”. It corresponds to a logical type.B20. Types can never be distinguished from each other by saying (as is oftenVI

The Birmingham Notes !&(

done) that one has these but the other has those properties, for this presupposesthat there is a meaning in asserting all these properties of both types. But fromthis it follows that, at best, these properties may be types, but certainly not theobjects of which they are asserted.B21. At a pinch, we are always inclined to explanations of logical functions Iof propositions which aim at introducing into the function either only theconstituents of these propositions, or only their form, etc. etc; and we overlookthat ordinary language would not contain the whole propositions if it did notneed them: However, e.g., “not-p” may be explained, there must always be ameaning given to the question “what is denied?”B22. The very possibility of Frege’s explanations of “not-p” and “if p then IIIq”, from which it follows that “not-not-p” denotes the same as p, makes itprobable that there is some method of designation in which “not-not-p” cor-responds to the same symbol as “p”. But if this method of designation sufficesfor logic, it must be the right one.B23. Names are points, propositions arrows—they have sense. The sense of Ia proposition is determined by the two poles true and false. The form of aproposition is like a straight line, which divides all points of a plane into rightand left. The line does this automatically, the form of proposition only byconvention.B24. Just as little as we are concerned, in logic, with the relation of a name IIto its meaning, just so little are we concerned with the relation of a propo-sition to reality, but we want to know the meaning of names and the senseof propositions—as we introduce an indefinable concept “A” by saying: “ ‘A’denotes something indefinable”, so we introduce e.g. the form of propositionsaRb by saying: “For all meanings of “x” and “y”, “xRy” expresses somethingindefinable about x and y”.B25. In place of every proposition “p”, let us write “a

b p”. Let every correlation IIIof propositions to each other or of names to propositions be effected by acorrelation of their poles “a” and “b”. Let this correlation be transitive. Thenaccordingly “a#a

b#b p” is the same symbol as “ab p”. Let n propositions be given.

I then call a “class of poles” of these propositions every class of n members,of which each is a pole of one of the n propositions, so that one membercorresponds to each proposition. I then correlate with each class of poles oneof two poles (a and b). The sense of the symbolizing fact thus constructed Icannot define, but I know it.B26. If p=not-not-p etc., this shows that the traditional method of symbol- IIIism is wrong, since it allows a plurality of symbols with the same sense; andthence it follows that, in analyzing such propositions, we must not be guidedby Russell’s method of symbolizing.B27. It is to be remembered that names are not things, but classes: “A” is Vthe same letter as “A”. This has the most important consequences for every

!'* The Notes on Logic

symbolic language.B28. Neither the sense nor the meaning of a proposition is a thing. TheseIwords are incomplete symbols.B29. It is impossible to dispense with propositions in which the same argu-Vment occurs in different positions. It is obviously useless to replace !(a,a) by!(a,b) a = b.B30. Since the ab-functions of p are again bi-polar propositions, we can formIIIab-functions of them, and so on. In this way a series of propositions will arise,in which in general the symbolizing facts will be the same in several members.If now we find an ab-function of such a kind that by repeated application ofit every ab-function can be generated, then we can introduce the totality ofab-functions as the totality of those that are generated by application of thisfunction. Such a function is !p v !q.B31. It is easy to suppose a contradiction in the fact that on the one handIIIevery possible complex proposition is a simple ab-function of simple proposi-tions, and that on the other hand the repeated application of one ab-functionsuffices to generate all these propositions. If e.g. an affirmation can be gen-erated by double negation, is negation in any sense contained in affirmation?Does “p” deny “not-p” or assert “p”, or both? And how do matters stand withthe definition of “!” by “v” and “!”, or of “v” by “.” and “!”? And how e.g.shall we introduce p | q (i.e. !p v !q), if not by saying that this expression sayssomething indefinable about all arguments p and q? But the ab-functions mustbe introduced as follows: The function p | q is merely a mechanical instrumentfor constructing all possible symbols of ab-functions. The symbols arising by re-peated application of the symbol “|” do not contain the symbol “p | q”. Weneed a rule according to which we can form all symbols of ab-functions, in or-der to be able to speak of the class of them; and we now speak of them e.g. asthose symbols of functions which can be generated by repeated application ofthe operation “|”. And we say now: For all p’s and q’s, “p | q” says somethingindefinable about the sense of those simple propositions which are containedin p and q.B32. The assertion-sign is logically quite without significance. It only shows,Iin Frege and Whitehead and Russell, that these authors hold the propositionsso indicated to be true. “#” therefore belongs as little to the proposition as(say) the number of the proposition. A proposition cannot possibly assert ofitself that it is true.B33. Every right theory of judgment must make it impossible for me to judgeIthat this table penholders the book. Russell’s theory does not satisfy this re-quirement.B34. It is clear that we understand propositions without knowing whetherIthey are true or false. But we can only know the meaning of a propositionwhen we know if it is true or false. What we understand is the sense of theproposition.

The Birmingham Notes !')

B35. The assumption of the existence of logical objects makes it appear re- IIImarkable that in the sciences propositions of the form “p v q”, “p ! q”, etc.are only then not provisional when “v” and “!” stand within the scope of agenerality-sign [apparent variable].

4th MS.B36. If we formed all possible atomic propositions, the world would be com- IIpletely described if we declared the truth or falsehood of each. [I doubt this.]B37. The chief characteristic of my theory is that, in it, p has the same meaning Ias not-p.B38. A false theory of relations makes it easily seem as if the relation of fact IIand constituent were the same as that of fact and fact which follows from it.But the similarity of the two may be expressed thus: !a ! !,a a = a.B39. If a word creates a world so that in it the principles of logic are true, it IIthereby creates a world in which the whole of mathematics holds; and sim-ilarly it could not create a world in which a proposition was true, withoutcreating its constituents.B40. Signs of the form “p v !p” are senseless, but not the proposition “(p) p v III!p”. If I know that this rose is either red or not red, I know nothing. Thesame holds of all ab-functions.B41. To understand a proposition means to know what is the case if it is true. IHence we can understand it without knowing if it is true. We understand itwhen we understand its constituents and forms. If we know the meaning of“a” and “b”, and if we know what “xRy” means for all x’s and y’s, then wealso understand “aRb”.B42. I understand the proposition “aRb” when I know that either the fact thataRb or the fact that not aRb corresponds to it; but this is not to be confusedwith the false opinion that I understand “aRb” when I know that “aRb ornot-aRb” is the case.B43. But the form of a proposition symbolizes in the following way: Let us IIconsider symbols of the form “xRy”; to these correspond primarily pairs ofobjects, of which one has the name “x”, the other the name “y”. The x’sand y’s stand in various relations to each other, among others the relationR holds between some, but not between others. I now determine the senseof “xRy” by laying down: when the facts behave in regard to “xRy” so thatthe meaning of “x” stands in the relation R to the meaning of “y”, then I saythat they [the facts] are “of like sense” [“gleichsinnig”] with the proposition“xRy”; otherwise, “of opposite sense” [“entgegengesetzt”]; I correlate the factsto the symbol “xRy” by thus dividing them into those of like sense and those ofopposite sense. To this correlation corresponds the correlation of name andmeaning. Both are psychological. Thus I understand the form “xRy” when Iknow that it discriminates the behaviour of x and y according as these standin the relation R or not. In this way I extract from all possible relations the

!'! The Notes on Logic

relation R, as, by a name, I extract its meaning from among all possible things.B44. Strictly speaking, it is incorrect to say: we understand the proposition pIwhen we know that ‘ “p” is true’ " p; for this would naturally always be thecase if accidentally the propositions to right and left of the symbol “"” wereboth true or both false. We require not only an equivalence, but a formalequivalence, which is bound up with the introduction of the form of p.B45. The sense of an ab-function of p is a function of the sense of p.IIIB46. The ab-functions use the discrimination of facts, which their argumentsIIIbring forth, in order to generate new discriminations.B47. Only facts can express sense, a class of names cannot. This is easilyVshown.B48. There is no thing which is the form of a proposition, and no name whichIIis the name of a form. Accordingly we can also not say that a relation which incertain cases holds between things holds sometimes between forms and things.This goes against Russell’s theory of judgment.B49. It is very easy to forget that, though the propositions of a form can beeither true or false, each one of these propositions can only be either true orfalse, not both.B50. Among the facts which make “p or q” true, there are some which makeIII“p and q” true; but the class which makes “p or q” true is different from theclass which makes “p and q” true; and only this is what matters. For weintroduce this class, as it were, when we introduce ab-functions.B51. A very natural objection to the way in which I have introduced e.g.IVpropositions of the form xRy is that by it propositions such as (

E

x,y) xRy andsimilar ones are not explained, which yet obviously have in common with aRbwhat cRd has in common with aRb. But when we introduced propositions ofthe form xRy we mentioned no one particular proposition of this form; andwe only need to introduce (

E

x,y) !(x,y) for all !’s in any way which makes thesense of these propositions dependent on the sense of all propositions of theform !(a,b), and thereby the justness of our procedure is proved.B52. The indefinables of logic must be independent of each other. If an in-definable is introduced, it must be introduced in all combinations in which itcan occur. We cannot therefore introduce it first for one combination, thenfor another; e.g., if the form xRy has been introduced, it must henceforth beunderstood in propositions of the form aRb just in the same way as in propo-sitions such as (

E

x,y) xRy and others. We must not introduce it first for oneclass of cases, then for the other; for it would remain doubtful if its meaningwas the same in both cases, and there would be no ground for using the samemanner of combining symbols in both cases. In short, for the introductionof indefinable symbols and combinations of symbols the same holds, mutatismutandis, that Frege has said for the introduction of symbols by definitions.B53. It is a priori likely that the introduction of atomic propositions is fun-III

The Birmingham Notes !'#

damental for the understanding of all other kinds of propositions. In fact theunderstanding of general propositions obviously depends on that of atomicpropositions.B54. Cross-definability in the realm of general propositions leads to quite sim- IVilar questions to those in the realm of ab-functions.B55. When we say “A believes p”, this sounds, it is true, as if here we could Isubstitute a proper name for “p”; but we can see that here a sense, not a mean-ing, is concerned, if we say “A believes that ‘p’ is true”; and in order to makethe direction of p even more explicit, we might say “A believes that ‘p’ is trueand ‘not-p’ is false”. Here the bi-polarity of p is expressed, and it seems thatwe shall only be able to express the proposition “A believes p” correctly bythe ab-notation; say by making “A” have a relation to the poles “a” and “b”of a–p–b.The epistemological questions concerning the nature of judgment and be-

lief cannot be solved without a correct apprehension of the form of the prop-osition.

Aa p b

B56. The ab-notation shows the dependence of or and not, and thereby thatthey are not to be employed as simultaneous indefinables.B57. Not: “The complex sign ‘aRb’ ” says that a stands in the relation R to b; Vbut that ‘a’ stands in a certain relation to ‘b’ says that aRb.B58. In philosophy there are no deductions: it is purely descriptive. PB59. Philosophy gives no pictures of reality. PB60. Philosophy can neither confirm nor confute scientific investigation. PB61. Philosophy consists of logic and metaphysics: logic is its basis. PB62. Epistemology is the philosophy of psychology. PB63. Distrust of grammar is the first requisite for philosophizing. PB64. Propositions can never be indefinables, for they are always complex.That also words like “ambulo” are complex appears in the fact that their rootwith a different termination gives a different sense.B65. Only the doctrine of general indefinables permits us to understand the IInature of functions. Neglect of this doctrine leads to an impenetrable thicket.B66. Philosophy is the doctrine of the logical form of scientific propositions P(not only of primitive propositions).B67. The word “philosophy” ought always to designate something over or Punder, but not beside, the natural sciences.B68. Judgment, command and question all stand on the same level; but all Ihave in common the propositional form, which does interest us.B69. The structure of the proposition must be recognized, the rest comes ofitself. But ordinary language conceals the structure of the proposition: in it,relations look like predicates, predicates like names, etc.

!'$ The Notes on Logic

B70. Facts cannot be named.IB71. It is easy to suppose that “individual”, “particular”, “complex” etc. areVIprimitive ideas of logic. Russell e.g. says “individual” and “matrix” are “prim-itive ideas”. This error presumably is to be explained by the fact that, byemployment of variables instead of the generality-sign, it comes to seem asif logic dealt with things which have been deprived of all properties exceptthing-hood, and with propositions deprived of all properties except complex-ity. We forget that the indefinables of symbols [Urbilder von Zeichen] onlyoccur under the generality-sign, never outside it.B72. Just as people used to struggle to bring all propositions into the subject-IVpredicate form, so now it is natural to conceive every proposition as expressinga relation, which is just as incorrect. What is justified in this desire is fullysatisfied by Russell’s theory of manufactured relations.B73. One of the most natural attempts at solution consists in regarding “not-Ip” as “the opposite of p”, where then “opposite” would be the indefinablerelation. But it is easy to see that every such attempt to replace the ab-functionsby descriptions must fail.B74. The false assumption that propositions are names leads us to believeIthat there must be logical objects: for the meanings of logical propositions willhave to be such things.B75. A correct explanation of logical propositions must give them a uniquePposition as against all other propositions.B76. No proposition can say anything about itself, because the symbol of theVIproposition cannot be contained in itself; this must be the basis of the theoryof logical types.B77. Every proposition which says something indefinable about a thing is aVIsubject-predicate proposition; every proposition which says something inde-finable about two things expresses a dual relation between these things, and soon. Thus every proposition which contains only one name and one indefin-able form is a subject-predicate proposition, and so on. An indefinable simplesymbol can only be a name, and therefore we can know, by the symbol of anatomic proposition, whether it is a subject-predicate proposition.

The Cambridge Notes

2nd MS.C1. We must be able to understand propositions which we have never heardIIbefore. But every proposition is a new symbol. Hence we must have generalindefinable symbols; these are unavoidable if propositions are not all indefin-able.C2. Whatever corresponds in reality to compound propositions must not beIIImore than what corresponds to their several atomic propositions. (B36)

The Cambridge Notes !'%

C3. Not only must logic not deal with [particular] things, but just as little withrelations and predicates.C4. There are no propositions containing real variables. IVC5. What corresponds in reality to a proposition depends upon whether it Iis true or false. But we must be able to understand a proposition withoutknowing if it is true or false. (B41)C6. What we know when we understand a proposition is this: We know what Iis the case if the proposition is true, and what is the case if it is false. But wedo not know [necessarily] whether it is true or false. (B41)C7. Propositions are not names. (B4)C8. We can never distinguish one logical type from another by attributing aproperty to members of the one which we deny to members of the other. (B20)C9. Symbols are not what they seem to be. In “aRb”, “R” looks like a sub- IIstantive, but is not one. What symbolizes in “aRb” is that R occurs betweena and b. (B57) Hence “R” is not the indefinable in “aRb”. Similarly in “!x”,“!” looks like a substantive but is not one; in “!p”, “!” looks like “!” butis not like it. This is the first thing that indicates that there may not be logicalconstants. A reason against them is the generality of logic: logic cannot treata special set of things.C10. Molecular propositions contain nothing beyond what is contained in IIItheir atoms; they add no material information above that contained in theiratoms. (B36)C11. All that is essential about molecular functions is their T-F schema [i.e. IIIthe statement of the cases when they are true and the cases when they arefalse].C12. Alternative definability shows that the indefinables have not been Vreached. (B18)C13. Every proposition is essentially true–false: to understand it, we must Iknow both what must be the case if it is true, and what must be the case if it isfalse. Thus a proposition has two poles, corresponding to the case of its truthand the case of its falsehood. We call this the sense of a proposition.C14. In regard to notation, it is important to note that not every feature of Va symbol symbolizes. In two molecular functions which have the same T-Fschema, what symbolizes must be the same. In “not-not-p”, “not-p” does notoccur; for “not-not-p” is the same as “p”, and therefore, if “not-p” occurredin “not-not-p”, it would occur in “p”. (B22)C15. Logical indefinables cannot be predicates or relations, because proposi- IIItions, owing to sense, cannot have predicates or relations. Nor are “not” and“or”, like judgment, analogous to predicates or relations, because they do notintroduce anything new.C16. Propositions are always complex even if they contain no names. (B5) IVC17. A proposition must be understood when all its indefinables are under- IIstood. The indefinables in “aRb” are introduced as follows:

!'" The Notes on Logic

“a” is indefinable;“b” is indefinable;Whatever “x” and “y” may mean, “xRy” says something indefinable abouttheir meanings. (B24)

C18. A complex symbol must never be introduced as a single indefinable.V(Thus e.g. no proposition is indefinable.) For if one of its parts occurs also inanother connection, it must there be re-introduced. And would it then meanthe same? (B52)C19. The ways by which we introduce our indefinables must permit us toVconstruct all propositions that have sense [? meaning] from these indefinablesalone. It is easy to introduce “all” and “some” in a way that will make theconstruction of (say) “(x,y) xRy” possible from “all” and “xRy” as introducedbefore. (B51)

SummaryC20. One reason for thinking the old notation wrong is that it is very unlikelyIIIthat from every proposition p an infinite number of other propositions not-not-p, not-not-not-not-p, etc., should follow. (B15)C21. If only those signs which contain proper names were complex thenIVpropositions containing nothing but apparent variables would be simple. Thenwhat about their denials? (B5)C22. The verb of a proposition cannot be “is true” or “is false”, but whateveris true or false must already contain the verb. (B10)C23. Deductions only proceed according to the laws of deduction, but theselaws cannot justify the deduction. (B12)C24. One reason for supposing that not all propositions which have moreIthan one argument are relational propositions is that if they were, the relationsof judgement and inference would have to hold between an arbitrary numberof things.C25. Every proposition which seems to be about a complex can be anal-IIysed into a proposition about its constituents and about the proposition whichdescribes the complex perfectly; i.e., that proposition which is equivalent tosaying the complex exists. (B17)C26. The idea that propositions are names of complexes suggests that what-Iever is not a proper name is a sign for a relation. Because spatial complexes1

consist of Things and Relations only and the idea of a complex is taken fromspace.C27. In a proposition convert all its indefinables into variables; there thenremains a class of propositions which is not all propositions but a type. (B19)C28. There are thus two ways in which signs are similar. The names SocratesVI

1Russell—for instance imagines every fact as a spatial complex.

The Cambridge Notes !'&

and Plato are similar: they are both names. But whatever they have in com-mon must not be introduced before Socrates and Plato are introduced. Thesame applies to a subject-predicate form etc. Therefore, thing, proposition,subject-predicate form, etc., are not indefinables, i.e., types are not indefin-ables.C29. When we say A judges that etc., then we have to mention a whole prop- Iosition which A judges. It will not do either to mention only its constituents,or its constituents and form, but not in the proper order. (B55) This showsthat a proposition itself must occur in the statement that it is judged; however,for instance, “not-p” may be explained, the question, “What is negated” musthave a meaning. (B21)C30. To understand a proposition p it is not enough to know that p implies I‘ “p” is true’, but we must also know that !p implies “p is false”. This showsthe bi-polarity of the proposition. (B55)C31. To every molecular function a WF2 scheme corresponds. Therefore IIIwe may use the WF scheme itself instead of the function. Now what the WFscheme does is, it correlates the letters W and F with each proposition. Thesetwo letters are the poles of atomic propositions. Then the scheme correlatesanother W and F to these poles. In this notation all that matters is the corre-lation of the outside poles to the pole of the atomic propositions. Thereforenot-not-p is the same symbol as p. And therefore we shall never get two sym-bols for the same molecular function.C32. The meaning of a proposition is the fact which actually corresponds Ito it.C33. As the ab functions of atomic propositions are bi-polar propositions IVagain we can perform ab operations on them. We shall, by doing so, corre-late two new outside poles via the old outside poles to the poles of the atomicpropositions. (B30)C34. The symbolising fact in a–p–b is that, say3 a is on the left of p and b on IIIthe right of p; then the correlation of new poles is to be transitive, so that forinstance if a new pole a in whatever way i.e. via whatever poles is correlatedto the inside a, the symbol is not changed thereby. It is therefore possible toconstruct all possible ab functions by performing one ab operation repeatedly,and we can therefore talk of all ab functions as of all those functions which canbe obtained by performing this ab operation repeatedly.4 (B25)C35. Naming is like pointing. A function is like a line dividing points of a III

2W-F= Wahr-Falsch 3This is quite arbitrary, but if we once have fixed on which sides the poleshave to stand we must of course stick to our convention. If for instance “a p b” says p then b p asays nothing. (It does not say "p) But a–a p b–b is the same symbol as a p b (here the ab functionvanishes automatically) for here the new poles are related to the same side of p as the old ones.The question is always: how are the new poles correlated to p compared with the way the oldpoles are correlated to"p. 4[Note by B.R. abmeans the same as WF, which means true-false.]

!'' The Notes on Logic

plane into right and left ones; then “p or not-p” has no meaning because itdoes not divide the plane. (B23)C36. But though a particular proposition “p or not-p” has no meaning, aIIIgeneral proposition “for all p’s, p or not-p” has a meaning because this doesnot contain the nonsensical function “p or not-p” but the function “p or not-q”just as “for all x’s xRx” contains the function “xRy”. (B40)C37. A proposition is a standard to which facts behave, with names it is oth-Ierwise; it is thus bi-polarity and sense comes in; just as one arrow behaves toanother arrow by being in the same sense or the opposite, so a fact behaves toa proposition.C38. The form of a proposition has meaning in the following way. Considera symbol “xRy”. To symbols of this form correspond couples of things whosenames are respectively “x” and “y”. The things x/y stand to one another inall sorts of relations, amongst others some stand in the relation R, and somenot; just as I single out a particular thing by a particular name I single out allbehaviours of the points x and y with respect to the relation R. I say that if anx stands in the relation R to a y the sign “xRy” is to be called true to the factand otherwise false. This is a definition of sense. (B43)C39. In my theory p has the same meaning as not-p but opposite sense. TheImeaning is the fact. The proper theory of judgment must make it impossibleto judge nonsense. (B33)C40. It is not strictly true to say that we understand a proposition p if we knowthat p is equivalent to “p is true” for this would be the case if accidentally bothwere true or false. What is wanted is the formal equivalence with respect toIthe forms of the proposition, i.e., all the general indefinables involved. (B44)The sense of an ab function of a proposition is a function of its sense. (B45)There are only unasserted propositions. Assertion is merely psychological.(B32) In not-p, p is exactly the same as if it stands alone; this point is absolutelyfundamental. Among the facts which make “p or q” true there are also factswhich make “p and q” true; if propositions have only meaning, we ought, insuch a case, to say that these two propositions are identical, but in fact, theirsense is different for we have introduced sense by talking of all p’s and allq’s. (B50) Consequently the molecular propositions will only be used in caseswhere their ab function stands under a generality sign or enters into anotherfunction such as “I believe that, etc.”, because then the sense enters.C41. In “a judges p” p cannot be replaced by a proper name. This appearsIif we substitute “a judges that p is true and not p is false”. The proposition “ajudges p” consists of the proper name a, the proposition p with its 2 poles, anda being related to both of these poles in a certain way. This is obviously not arelation in the ordinary sense. (B55)C42. The ab notation makes it clear that not and or are dependent on one an-

The Cambridge Notes !'(

other and we can therefore not use them as simultaneous indefinables. Sameobjections in the case of apparent variables to the usual old indefinables, as IVin the case of molecular functions: The application of the ab notation toapparent-variable propositions becomes clear if we consider that, for instance,the proposition “for all x, !x” is to be true when !x is true for all x’s and falsewhen !x is false for some x’s. We see that some and all occur simultaneously inthe proper apparent variable notation.C43. The Notation is: IV

for (x)!x : a%(x)%a !x b%( Ex)%b

andfor (

E

x)!x : a%( Ex)%a !x b%(x)%b

Old definitions now become tautologous.C44. In aRb it is not the complex that symbolises but the fact that the symbol Va stands in a certain relation to the symbol b. Thus facts are symbolised byfacts, or more correctly: that a certain thing is the case in the symbol says thata certain thing is the case in the world. (B57)C45. Judgment, question and command are all on the same level. Whatinterests logic in them is only the unasserted proposition. (B68) Facts cannotbe named. (B70)C46. A proposition cannot occur in itself. This is the fundamental truth of VIthe theory of types. (B76)C47. Every proposition that says something indefinable about one thing is a VIsubject-predicate proposition, and so on. (B77)C48. Therefore we can recognize a subject-predicate proposition if we know VIit contains only one name and one form, etc. This gives the construction oftypes. Hence the type of a proposition can be recognized by its symbol alone.(B77)C49. What is essential in a correct apparent-variable notation is this:– VI

(1) it must mention a type of propositions;

(2) it must show which components of a proposition of this type are constants.

C50. [Components are forms and constituents.]C51. Take (!) !!x. Then if we describe the kind of symbols, for which !! VIstands and which, by the above, is enough to determine the type then auto-matically “(!) !!x” cannot be fitted by this description, because it contains “!!x”and the description is to describe all that symbolizes in symbols of the !!-kind.If the description is thus complete vicious circles can just as little occur as forinstance in (!) (x)! (where (x)! is a subject-predicate proposition).

!(* The Notes on Logic

Textual notesThis text is based on three sources:R A 23-page manuscript of MSS1–4 in Russell’s hand (RA1.710.057823);S A 7-page typescript of the Summary with corrections by both Russell andWittgenstein (RA1.710.057822);

T A typescript prepared from both R and S as corrected, containing furthercorrections by Russell (RA1.710.057824).

The text generally follows T as corrected by Russell. In the very few caseswhere T differs from the corrected versions of R and S from which it wascompiled, I have chosen whichever made the best sense. In a few places (B10,B17, C12) clear errors which remained in T have been corrected. Readersseeking full diplomatic and normalized editions of the texts are recommendedto consult those prepared by the Wittgenstein Archives at the University ofBergen, which are available online at http://wab.aksis.uib.no/.The text is presented here in two parts, which are labelled the Birmingham

and Cambridge versions of the Notes: the justification for this presentationis contained in Appendix A. Paragraph numbers, as well as the parentheti-cal references in the Cambridge Notes to paragraphs of the Birmingham Notescovering similar material, have been added. Square brackets are present inthe original texts and evidently represent Russell’s own insertions. Italiciza-tion of letters used as logical signs (p, q, r, etc.), which in the source texts iscarried out only fitfully, has here been silently made consistent. In the notesbelow, alterations made by Wittgenstein to the typsecript S are labelled LW;those made by Russell to S or to his manuscript R are labelled BR1; thosemade by Russell to the typescript T are labelled BR2. The sentences within aparagraph are referred to, where necessary, by letter.

First MS.

B3 designate] after deleted denote BR1B7 what it] itsB7 the proposition] inserted

3rd MS.

B10 An analogy] after deleted A comparisBR1B10g in] after deleted on BR1B10 fails] after deleted depends BR1B10 without knowing what white andblack are] what is white and black R,TB14 (Urzeichen)] inserted BR1B14 questioning] emphasized with query inmargin BR2

B16 like] above deleted as BR1B17 sum] product R,TB21 logical] inserted BR1B21 aim at introducing into the function]insertedB21 only] before deleted contain BR1B21 form] forms RB23 propositions] after deleted sentencesBR1B24 propositions—] propositions TB30 introduce] after deleted define BR1B31 every] after deleted all BR1B32 only] inserted BR1B35 in the sciences] inserted BR1B35 p v q] p or q

http://wab.aksis.uib.no/

Textual notes !()

B35 are] inserted BR1

4th MS.

B42 understand] R understood TB43 now] after deleted knowB44 we] We RB49 very] insertedB50 only] insertedB51 justness] replaces justification BR2B52 combinations] over deleted classesB54 quite] after deleted the BR2B68 command and] inserted BR1B68 all] after deleted and BR1B69 structure] above deleted constructionB69 proposition] above deleted sentenceB71 sign] signs BR1B77 symbol] above deleted sign

2nd MS.

C12 definability] indefinability R,T

Summary] added BR1

C20 One] after deleted The BR1C21 those] written over these BR1C23 Deductions] replaces The deductionsBR1C24 One] above deleted The BR1C24 if] inserted BR1C24 were,] comma insertedC24 would] above deleted that BR1C25 its] after deleted those BR1C25 the] above deleted a (twice)C25 the] after deleted aC26 suggests] replaces deleted between sug-gestions LWC26 spatial complexes] footnote added LWC26fn. Russell—for instance imagines]You—for instance imagine BR2C26 Because . . . space.] added LWC27 is] above deleted has BR1C28 a] omitted SC29 A judges] a judge is (twice) BR1C29 constituents] constituent (twice)C29 meaning.] before deleted Always aquestion that is negated must have ameaning. LW adds in margin Rott!C29 the question . . . a meaning] added

LWC30 p implies ‘ “p” is true’] replaces “p im-plies p” is true BR1C30 "p implies “p is false”] after deleted palso implies ‘ “not-p” is false’C30 bi-polarity] polarityC31 W] w (four times) BR1C31 F] f (four times) BR1C31fn. W-F= Wahr-Falsch] added BR1C31 the scheme correlates] correspondsC31 W] fC31 F] wC31 matters] after deleted it BR1C31 the] itsC31 And] insertedC31 function] functions BR1C33 bi-polar] by polarC33 shall, by] will be LWC34 say] footnote inserted LWC34fn. arbitrary, but] arbitrary but SC34fn. (here] omitted SC34fn. "p] p SC34 so] suchC34 for instance] added LWC34 i.e. via whatever poles] added LWC34fn. Note by B.R.] inserted BR1C34fn. ab] after NB. SC35 of a plane] added LWC35 the] a LWC36 “p or not-p”] replaces “p” or “not-p”LWC36 “for all p’s, p or not-p”] replaces forall p’s, “p” or “not-p”C36 the] over a (twice)C36 “p or not-q”] replaces p or “not-q”C37 facts] after deleted all BR2C37 with names it is otherwise] thatnames it otherwiseC37 thus bi-polarity] then by polarityC37 arrow] error (twice)C38 with respect to] inserted after deleted theone betweenC38 R] inserted before deleted of the otherLWC38 R] inserted after deleted of LWC38 “xRy”] x or y LWC40 In not-p, p is] If not-p is

!(! The Notes on Logic

C40 true] inserted LWC40 have only meaning] over deleted doonly mean BR1C40 in such a case, to say] to know sucha case, say LWC40 their] there SC40 then] inserted LWC41 a relation] the relation LWC42 makes] and apparent variables makeLWC42 Same objections . . . indefinables]Some objections to old indefinables LW

C42 the usual] inserted BR2C42 apparent-variable] apparently vari-able LWC42 becomes] becomeC44 more correctly] after deleted the BR1C47 indefinable] above deleted importantLWC49 apparent-variable] apparentC51 for which !! stands and] inserted LWC51 fitted] filledC51 in] omitted TC51 because . . . proposition).] inserted LW

The Costello versionBelow is an analysis of how the Costello version, as published in 1957, re-lates to the typescript T from which it was derived. As was explained in Ap-pendix A, Russell allocated paragraphs to sections by markings (reproducedabove) in the margins of T. Paragraphs B49 and B64, which Russell theremarked for deletion (although they do not duplicate remarks elsewhere in thetext), are omitted from the Costello version. Paragraphs C3, C7, and C23,which are not there clearly assigned to sections, are likewise omitted. Para-graphs B21c, B41ab, B42, B69, C8, C22, C38, C40ae, C42a, and C45 (all ofthem more or less duplicates of other paragraphs elsewhere in the Notes) areomitted from the Costello version even though they have been assigned tosections in T. Elsewhere the Costello version follows Russell’s classificationsin T. In the analysis that follows, minor differences in punctuation have beenignored.

Preliminary

B58 it] itB67B59B60 Philosophy] andB61 logic] the formerB62B63B66 only of primitive propositions] prim-

itive propositions onlyB75 the] omitted

I. Bi-polarity of propositions: sense and meaning,truth and falsehood.

B4B70 named] namedB74 logical objects] “logical objects”C5C6 know[necessarily] ] necessarily know

true or false] after inserted actuallyC13 to understand . . . false] omitted

falsehood] falsityC32B37 not-p] added (constituent=particu-

lar, component=particular or rela-tion, etc.)

B28B34 meaning] meaning

The Costello version !(#

sense] senseC30B41cdB42B44C40bB7 to] about

negations] the negationsB8B9 then] deletedB10 that it is white] it is white

what] whichB11 retinal] a retinal

as] just asB15 At a pinch] deleted

lately] recentlyB32C40dB68 which does interest us] and that

alone interests usC45b

in them] omittedis . . . proposition]are. . . propositions

C29 statement that] statement to the ef-fect thatHowever, for instance,] For in-stance, however

C41 a judges p] A judges (that) p (twice)appears] is apparenta being] A’s being

B33C24C26 suggests] has suggested

Because . . . taken from space] Rus-sell, for instance, imagines every factas a spatial complex, and since spa-tial complexes consist of things andrelations only, therefore he holds alldo.

B21abAt a pinch we are]We are very oftenoverlook that] overlook the fact that

B23B73 One of the most natural attempts at

solution] It is wrong to conceive ev-ery proposition as expressing a rela-tion. A natural attempt at such a so-lution

the ab-functions] functions withsense (ab functions)

B55 here we could] we could hereab notation] before added (later ex-plained)say by making] by, say, making

C37acbwith] but with

C39 my theory] this theoryThe proper theory] a proper theory

C40cC40f–j

for] and

II. Analysis of atomic propositions: general inde-finables, predicates, etc.

B36 If . . . each [I doubt this] ] It may bedoubted whether, if . . . each (Rus-sell).

B39 If a word creates a world so that init] If there were a world created inwhichit thereby creates a world in which]in that worldand similarly it could not create aworld] No world can be createdwithout creating its constituents] un-less the constituents of the proposi-tion are created also

B1 can not] cannotB2C1B65C17B24 Just as little as we are] We are not

a name] any specific namejust so little are we concerned] andjust as littleproposition] given propositionbut we want to know the mean-ing of names and the sense ofpropositions—as we introduce] Wedo want to know that our nameshave meanings and propositionssense, and we thus introduceso we introduce e.g. the form] or theform

!($ The Notes on Logic

B43 But] deletedsymbolizes] may be symbolizedto these] to whichamong others] and among other re-lationsthey (the facts)] these facts

B48C9 is not] it is not

symbolizes] is symbolizedB16 logical types] before inserted (forms)

would] would thenB38 the similarity of the two may be ex-

pressed] there is a similarity of thetwo, expressible

B17 apparent] deletedC25 Every proposition] To repeat: every

proposition

III. Analysis of molecular propositions: ab-functions.

C2C10C11 when] where (twice)B53B22B26C35C36B12B6B40B35B18C15B25B45B46C31 WF] TF (three times)

scheme] (or ab) schemeit] that it

C33 ab] ab (TF)C34 Then] Then, given a p b,

repeatedly] before inserted (cf., Shef-fer’s work)

B50B30B31 “"”] “ ”

IV. Analysis of general propositionsB72C21 were] areC16C4B13 only . . . when] do not . . . unlessB14 the schemes] schemes

cases] causesB54C42b–d

Same objections] There is the sameobjectionold indefinables] indefinables

C43B51 justness] justification

proved] established

V. Principles of symbolism: what symbolizes ina symbol. Facts for facts.B5 suppose that] suppose

(

E

x,!) !x] (x,!) !x(

E

x,R,y) xRy] (

E

x,y)xRyof (e.g.)"(x,y) xRy] e.g. of"( Ex,y) xRy

C12 shows that] showsnot been] not yet been

B52 would be] could beB3 but by] but otherwise byB27C14 note] observe

schema] schemeC18 its parts] the parts of the complex

symbolC19B57 Not:] One must not sayB47C44

VI. TypesB71 deprived of all properties except

thinghood, and with propositions]omitted

B77 and so on] etc.simple symbol] symbol

C46B19C28 thus] omittedC46 and so on] etc.

The Costello version !(%

says something] says something in-definable

C47C49 components] components (forms

and constituents)

C51 complete] completedfor instance in] can for instance

B20 Types can never be distinguishedfrom each other] We can never dis-tinguish one logical type from an-other

Citations

Notes on Logic

B1 111B3 82, 201B4 135B5 110B7 106B9 73, 97B10 89, 97, 172B11 143B13 55, 197B16 200, 273B17 44, 45B18 159, 175B19 178, 188B21 67, 175, 190B22 166, 213B23 154B24 99B26 213B27 64B29 208

B30 159B31 159, 171,

189B32 95, 97B33 122B36 146, 148B37 132, 273B40 140B41 151, 152B42 151B43 99, 100,

153, 154B44 152, 153B47 111B48 115, 117,

125B50 138B51 153, 199B55 220, 254B57 113

B61 57B62 99, 244B63 68B64 92B67 41, 57B69 68B71 56, 199B73 273B76 187B77 116, 189,

225, 232C2 148C5 73, 151C6 151C9 57, 227C10 148C11 170C13 174C14 170, 209,

210

C15 175, 220C25 44C26 238C28 84C31 163, 170,

173C33 170, 173C36 139C38 153, 154C39 138C40 97, 99, 153,

175, 190C41 219C43 180C44 113C48 232C49 178C50 225, 232C51 190, 195

Tractatus

1.1 150, 236,243

1.12 2332 2362.01 2332.011 2382.0141 2312.021 722.0271 2312.06 141, 2362.063 2362.141 2262.15 2302.16 230

2.225 303.12 2103.14 2113.142 1113.143 2113.1432 1133.144 2263.203 643.3 13, 1783.322 82, 2013.332 187, 1894.002 684.021 2264.024 151

4.0312 614.0621 1414.063 89, 1724.1273 1934.221 2404.26 1464.431 1664.441 1744.442 1605.02 1135.134 135.23 1655.461 1765.502 191

5.512 2145.514 2155.522 1785.5302 2055.532 2085.542 221, 2235.5422 1235.5423 2395.5521 575.5563 2125.64 766.1203 164, 1746.375 1486.3751 46

Index

a priori, 27, 30, 40, 137, 149, 150, 193, 205,206, 282

synthetic, 149, 150, 205aboutness shifter, 14, 21Alexander, S., 133analogyblack patch, 89, 277compass needle, 155, 156, 226

Anscombe, E., 90, 98, 235, 257arithmetic, see mathematicsassertion, 55, 56, 86, 87, 89–91, 94–101, 166,

177, 196, 198, 199, 228sign for, 280

atomism, logical, 36, 37, 76, 133, 134,148–150, 205, 207, 225, 258, 260, 262

Baker, G., 112Bauch, B., 261Beethoven, L., 20Berkeley, G., 11, 32Blackwell, K., 33Boltzmann, L., 242, 256Bradley, F., 11–13, 114, 116, 228Brentano, F., 105Brusilov, A., 247, 257

Carruthers, P., 232Chadwick, J., 7Coffey, P., 65common sense, 25, 34, 37, 40, 74, 102, 244concept, 11–14, 21, 81, 113, 117, 225denoting, 20, 21, 42

conjunction, 44, 45, 51, 145, 161, 164, 166constructional conjecturestrong, 26, 28, 29weak, 28, 39

context principle, 64, 65, 178, 242contradiction, 15, 16, 67, 84, 99, 143, 185,

216, 280Copi, I., 232Costello, H., 133, 249, 261, 268, 271–274,

292

Davidson, D., 99

De Hevesy, G., 7Dedekind, R., 101deduction, see inferencedefinite description, 241Demos, W., 144denotation, 12, 14, 20–23, 71, 159, 179, 213,

277–279, 290Descartes, R., 32descriptiondefinite, 23, 69, 71, 135disguised, 23, 25, 71, 135, 136, 241

Donnelly, L., 106, 264, 268Dostoevsky, F., 247, 248, 257Dummett, M., 67, 91, 98, 99, 112, 156, 242,

244, 257, 258

Eccles, W., 9, 18eliminativism, 102Eliot, T. S., 123, 133, 225Epimenides, 16epistemology, 10, 14, 23, 33, 34, 37, 43, 61,

99, 120, 121, 126, 130, 133, 149, 206,219, 244, 245, 255, 259, 283

Erdmann, J., 65ethics, 246, 247meta-ethics, 246

experience, 11, 24, 26, 30, 32, 35–37, 42, 57,69, 73, 75, 103, 140, 149, 193, 215,246, 247

external world, 12, 23, 24, 26, 32, 37

factas a complex, 237, 238atomic, 133, 137, 140, 143, 145–150, 157,205, 206, 225, 231, 233, 239, 240

complex, 144–146conjunctive, 142disjunctive, 142–145, 214general, 147–150logical, 156, 157negative, 127, 137, 141, 143–147, 154,176, 206, 217, 236, 240, 276, 277

not complexes, 108, 114, 211positive, 127, 137, 154, 206, 236, 277signifying, 210–212, 214, 215

#** Index

symbolizing, 93, 115, 170, 171, 175, 182,189, 190, 209–212, 214–216, 220–222,225, 255, 279, 280

totality of, 236fatalism, 247, 248Fine, K., 142forceassertoric, 87, 89, 94, 97, 100indicator of, 94, 98

formalism, 43, 47, 91Frege, G.Begriffsschrift, 15, 55, 61, 67, 86, 87, 91, 95,112, 178, 213, 262

Grundgesetze, 15, 53, 59, 86, 88, 95, 156,199, 262

Grundlagen, 64, 178Posthumous Writings, 59, 60, 67, 91, 93, 97,98, 100, 101, 258

‘Der Gedanke’, 60, 72, 229fundamental thought, 49, 54, 61–63, 79, 100,

143, 150, 228, 256

Gödel, K., 16, 88, 186incompleteness theorems, 16

Geach, P., 95, 106, 112, 220, 237, 238, 257Geiger, H., 7geometry, see mathematicsGoldfarb, W., 98, 101, 258grammar, 23, 68, 78–81, 83–85, 87, 88, 94,

95, 99, 103, 117, 121, 137, 180, 199,209, 211, 222, 224, 231, 283

Grattan-Guinness, I., 16Gray’s Elegy argument, 21, 71Grayling, A., 258Grelling, K., 185, 194Griffin, N., 12, 118, 129, 130

Hacker, P., 112Hamilton, K., 7Hardy, G., 18Harwood, Miss, 264, 266, 267, 271, 273Haworth, W., 7Hegel, G., 11Hertz, H., 33, 84, 85, 159, 256Hesperus, 13, 71Hicks, D., 30, 31Hide, O., 11Hitler, A., 6Holmes, S., 23humanism, 247Hume, D., 31Husserl, E., 66, 67, 157, 253

idealism, 10–12, 32, 61, 62, 66, 72, 74, 76,77, 136, 234, 260

identity, 245Russell’s theory of, 245

incomplete symbols, 22, 42, 51–53, 57, 69,111, 119, 184, 241, 254, 280

indefinables, 84, 85, 92, 153, 159, 199, 278,282–289, 292–294

inference, 198deductive, 50, 61, 62, 234, 277, 283, 286,291

inductive, 27logical, 26, 40, 43, 95, 277

infinity, 14, 92, 101, 142, 150, 171, 200, 202,214, 278, 286

internal relations, 12, 13intersubjective, 99intuition, 61, 190, 238

Johnson, W., 249Jourdain, P., 16, 22, 50, 70, 71, 75, 101, 186,

239, 261, 264, 265judgment stroke, 87, 88, 94, 96, 98, 99

Kant, I., 11, 61, 64, 205, 234, 261Kaplan, D., 63Kenny, A., 238, 258Kienzler, W., 91Klagge, J., 85knowledge, 3, 7, 12, 23, 25, 31, 32, 37, 40, 59,

89, 129, 149, 195, 239, 242, 244, 272Kraus, K., 250, 256, 257Kremer, M., 98Kripke, S., 25

Lamb, H., 7, 8, 18, 47Landini, G., 118languageand the world, 68, 69, 72, 176formal, 182, 199, 201logically perfect, 67, 213, 215, 216, 255ordinary, 23, 26, 43, 64, 66–68, 83, 87, 93,119, 156, 175, 185, 190, 211–213, 279,283

structure of, 65, 66Lee, D., 233Lewis, D., 146Lichtenberg, G., 250linguistic turn, 64–66linguistics, 68, 99, 243Linsky, B., 15Littlewood, J., 8Locke, J., 31logicas a science, 58as contentless, 58, 60, 61, 234, 242foundations of, 247, 253generality of, 57–59, 227, 285laws of, 58, 59nature of, 53, 56–58, 121, 253polyadic, 61, 183

Index #*)

quantified, 182, 214universality of, 58

logicalanalysis, 23, 43, 67, 243connective, 158, 175constant, 51–54, 57, 61, 79, 143, 145, 254,285

construction, 26, 27, 34, 39–41, 260form, 22, 55, 218, 265, 283positivism, 65type, 184, 187–189, 197, 199, 200, 273,278, 284, 285, 294, 295

logicism, 13–15, 157Long, P., 100, 113Loos, A., 256Lotze, H., 65

Marbe, K., 105, 106Marsden, E., 7mathematics, 7–10, 14–19, 23, 40–42, 49, 50,

55, 61, 73, 114, 119, 134, 157, 160,195, 206, 262, 278, 281

arithmetic, 13, 14, 88, 101, 262geometry, 7, 58, 59, 67

McGuinness, B., 6–8, 53, 59, 202, 246, 267,268, 270–272

McTaggart, J., 18, 106meaning, theory of, 140, 141, 223, 244Meinong, A., 22, 90mereological sum, 104, 145, 146meta-ethics, see ethicsmetalanguage, 168, 201, 203, 259metaphysical subject, 77, 134metaphysics, 29, 57, 65, 142–150, 206, 237,

241, 246, 283Miah, S., 40Milkov, N., 24Monk, R., 6Moore, G., 1, 11–14, 18, 19, 24, 31, 33, 42,

43, 46, 51, 63, 68, 71, 72, 83, 90, 96,106, 130–133, 136, 162, 173, 194, 204,206, 226, 249, 250, 264, 271, 274, 303

Morrell, O., 19, 20, 30, 47, 107, 116, 118,120, 121, 126, 128, 129, 135, 245, 253,260, 264, 265, 272, 273

Mozart, W., 107multiple relation theory of judgment, 19, 118,

120, 129, 218, 222mystical, 216

name, logically proper, 23, 25, 73, 135natural science, 41, 46, 57, 61, 241, 243, 256,

283Neckar cube, 239, 243Nietzsche, F., 250nonsense, 16, 80, 82, 123, 124, 126–128, 197,

221, 222, 224, 241, 246, 251, 252, 288

Nordmann, A., 85

Occam’s razor, 34, 40, 41, 71, 102, 143Ogden, C., 47, 195ontology, 39, 42, 102, 105

paradox, 57, 95, 184–196, 201, 203, 218, 242Burali-Forti’s, 16Grelling’s, 185, 186, 194liar, 218Russell’s, 15–17, 80, 81, 184, 186, 254set-theoretic, 242Zeno’s, 262

Parak, F., 65Peano, G., 15perception, 26, 27, 31, 37, 103, 243–245Perkin, W., 7Petavel, J., 8phenomenology, 214, 243philosophical logic, 17, 135, 241, 247, 254,

257, 262philosophy of mind, 19philosophy of psychology, 99, 244, 255, 283philosophy of value, 245, 246Phosphorus, see Hesperusphysics, 7, 26–30, 33, 34, 36, 38, 39, 45, 46,

48, 57–59, 260pictorial form, 235picture theory, 1, 141, 216, 224–227, 230,

231, 235, 242, 245, 256, 262Pinsent, D., 17, 50, 106, 118, 180, 186, 241,

243, 245, 246, 248, 252, 258, 263, 264,269, 270

Platonism, 21possible world, 138, 152, 207, 231, 245postmodernism, 252predicate calculus, 22, 196present King of France, 20, 22, 23, 39, 144,

145private language argument, 244propositionas a complex, 51, 109, 110, 118as a name, 92, 135, 136, 276, 284, 286asserted, 95–98, 288, 289atomic, 92, 103, 110, 116, 137, 143–146,148, 162, 163, 171, 181, 188, 219, 276,277, 281–284, 287, 293

complex, 171, 280compound, 22, 146, 148, 174, 284elementary, 13, 110, 116, 141, 148, 170,171, 181, 205, 207, 225, 232, 234, 235

false, 96, 97, 119, 120, 132, 155, 228, 277general, 147, 179, 192, 278, 283, 288, 294generalized, 191molecular, 141, 148, 171, 175, 181, 189,191, 192, 219, 254, 288, 294

primitive, 55, 128, 198, 277, 283, 292

#*! Index

true, 21, 56, 96, 119, 132, 138, 140unasserted, 95–97, 288, 289unity of, 79, 80, 87, 109–111, 114, 115

propositional function, 15, 25, 44, 55, 56, 83,102, 154, 177–179, 184–186, 188,190–192, 194, 195, 202, 205

propositional meaning, 137, 141propositional sign, 171, 210–213, 216, 230psychologism, 59, 64–66, 100, 178psychology, 12, 19, 31, 37, 47, 59, 72, 95, 96,

98–101, 118, 135, 181, 228, 243–245,255, 260, 281, 283, 288

quality, 33, 34, 47, 59, 81, 82, 96, 108, 119quantifier, 22, 23, 44, 52, 55, 81, 139, 163,

179, 181, 195, 198, 208, 231, 273universal, 52, 55, 81, 198

Quine, W., 4

Ramsey, F., 18, 188, 213, 214, 221–223, 230,233, 235

realism, 11, 28, 42, 66, 71–77reducibility, axiom of, 40, 197, 198, 204, 259referenceand sense, 70, 74, 75of a name, 73of a simple symbol, 66, 255theory of, 74

Reinach, A., 105, 106, 253religion, 245–247, 257Rhees, R., 10Robinson, R., 7Russell, B.Introduction to Mathematical Philosophy, 134Our Knowledge of the External World, 133, 258,260

Principia Mathematica, 26, 40, 47, 49, 50, 55,56, 82, 97, 127–129, 134, 170, 174,177, 179, 181, 190, 195–197, 201–204,213, 218, 259, 261

Principles of Mathematics, 10, 11, 14, 15, 18,20, 21, 50, 51, 57, 63, 90, 96, 97, 118,119, 127, 153

Problems of Philosophy, 26, 30, 32Theory of Knowledge, 110, 116, 120, 122, 219

Rutherford, E., 7, 8

saturated, 85, 110, 113–117, 224, 225, 235,239

saying and showing, 194, 246scepticism, 31, 33, 41, 42, 102Schlegel, F., 250Schlotter, S., 261Schopenhauer, A., 10, 250, 256, 257World as Will and Idea, 10

semantic externalism, 73semantic theory, 12, 70–72, 87, 215, 224

one-step, 12, 72, 215two-step, 12, 71, 215

semantic value, 152–154, 215sensation, 24, 32, 34, 69sense-data, 24–41, 45, 63, 69, 73–75, 205,

243, 260sensibilia, 33–36, 40, 43, 63, 76Sheffer, H., 159, 160, 165, 191, 294simples, 25, 26, 37, 44, 46, 69, 103, 278solipsism, 74–77, 244Spengler, O., 256Spielt, P., 7Sraffa, P., 256Stevens, G., 118Stumpf, C., 105, 106subjective, 99, 120substance, argument for, 72, 136Sullivan, P., 105, 112symbolic turn, 63–70, 78, 80, 81, 93, 116,

187, 209, 225, 241, 242, 255symbolism, 49, 80, 82, 83, 86, 109, 110, 113,

116, 178, 180, 182, 187, 213, 214, 217,225, 241, 279, 294

tautology, 134, 156, 164, 174, 198, 213, 216,217, 255

theory of descriptions, 22, 23, 42, 73, 207,239

theory of types, 15, 19, 80, 82–84, 124, 129,180, 184–187, 189–197, 199–201, 203,211, 254, 289

Tolstoy, L., 247, 248, 257transcendental, 77deduction, 61, 62, 234idealism, 61, 62, 234

truthand assertion, 97, 228concept of, 61, 169, 173, 218correspondence theory of, 141identity theory of, 71, 96, 119laws of, 59logical, 30, 49, 56, 139, 140, 156, 157,208, 216

truth-diagrams, 160, 162–164, 173, 174, 180,181, 204, 208

truth-maker, 132, 144truth-operation, 165–168, 170, 171, 173truth-tables, 160, 164, 173truth-value, 53, 54, 86–94, 139, 158, 160,

165, 166, 173, 254, 261, 277turnstile, 86, 87, 94, 95, 97, 98typical ambiguity, 81, 196, 197, 200–202, 211

universals, 25, 73, 235unsaturated, 85, 110, 113–117, 224, 225,

235, 239unsayability, 194, 200, 216, 259

Index #*#

value-ranges, 53verification, 28, 29, 41vicious circle principle, 185–187, 189, 192Vienna Circle, 46, 47, 253von Wright, G., 10

Wahl, R., 118Ward, J., 15Weininger, O., 256Weiss, B., 118Whitehead, A., 18, 27, 33, 40, 47, 49, 54, 55,

82, 129, 177, 195–197, 200–203, 263,280

Wittgenstein, H., 8

Wittgenstein, L.Blue Book, 244, 252Brown Book, 252Investigations, 244Notebooks, 45, 53, 61, 110, 111, 193Notes dictated to Moore, 1, 51, 174, 192, 207,226

Philosophical Grammar, 252Philosophical Investigations, 251Philosophical Remarks, 252Prototractatus, 164, 192, 194, 210, 249, 250,253, 266, 269–271

Wrinch, D., 219

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Wittgenstein s Notes on Logic Michael Potter (1)

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