Lutz, Sebastian (2013) the Semantics of Scientific Theories

8/13/2019 Lutz, Sebastian (2013) the Semantics of Scientific Theories

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The Semantics of Scientific Theories

Sebastian Lutz

Preprint: 20130315

Abstract

Marian Przeeckis semantics for the Received View is a good explica-tion of Carnaps position on the subject, anticipates many discussions andresults from both proponents and opponents of the Received View, and canbe the basis for a thriving research program.

Keywords:semantics of theories; logical empiricism; received view; theoret-ical terms; vagueness; analyticity; analytic-synthetic distinction

Contents

1 Introduction 2

2 The Received View in the philosophy of science 32.1 The observational-theoretical distinction . . . . . . . . . . . . . . . . 32.2 The analyticsynthetic distinction. . . . . . . . . . . . . . . . . . . . . 6

3 Przeeckis semantics for the Received View 83.1 Ostension, vagueness, and approximation. . . . . . . . . . . . . . . . 93.2 The analytic-synthetic distinction . . . . . . . . . . . . . . . . . . . . . 133.3 Przeeckis modifications of his semantics . . . . . . . . . . . . . . . . 18

4 After the Received View 204.1 Constructive empiricism. . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Suppe on partial interpretations . . . . . . . . . . . . . . . . . . . . . . 224.3 Andreas on the semantics of scientific theories . . . . . . . . . . . . . 24

5 Analyticity in vague languages 28

6 The semantics of scientific theories 30

Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universitt Mnchen. [email protected]. I thank Holger Andreas, Radin Dardashti, Krystian Jobczyk, and Alana Yufor helpful comments. Research for this article was supported by the Alexander von HumboldtFoundation.


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Sebastian Lutz The Semantics of Scientific TheoriesPreprint

Dedicated to Marian Przeecki

on occasion of his 90th

birthday.

1 Introduction

The Received View in thephilosophy of science was the logical empiricists frame-work for analyzing theories and related concepts. Developed mainly byCarnap(1939; 1966) andHempel (1958), and influenced by, for example,Reichenbach(1928),Neurath(1932),Feigl(1956), andNagel(1961), it arguably formed thecore of the logical empiricists philosophy. In 1969, it stood at a precipice. Car-nap had been concentrating on the philosophy of probability for years, withoutever having discussed in detail its connection to the Received View.1 From 1965

through 1969, Hempel had criticized and ultimately abandoned the ReceivedView in a series of talks which were just being published (Hempel 1969; 1970;1974). Hempels talk in 1969 played a central role in an influential conferencecritical of the Received View(Suppe 2000,S102S103). The proceedings of thatconference included what is now widely considered a canonical introduction tothe Received View (Suppe 1974a), which recommends its complete rejection. Thenext years would bring the Received Views spectacular fall into infamy (cf.Lutz2012b,79).

In the following, I will argue that this unequivocal dismissal of the ReceivedView after 1969 was far from justified. In fact, while sociologically it stood ata precipice, conceptually it had reached firm ground that could have ledand

might still leadto heights undiscovered by its chronological successors in thephilosophy of science. Specifically, I will argue that Marian Przeeckis outstand-ing2 monographThe Logic of Empirical Theoriesand his related articles gave theReceived View, for the first time, a natural and precise semantics that can capturemajor linguistic phenomena encountered in the analysis of scientific theories andprovides a unifying framework for discussions in the philosophy of science.3

Przeecki (1969,1) notes that a more adequate, though more cumbersometitle of his monograph would read: the logical syntax and semantics of the lan-guage of empirical theories. AndPrzeecki(1974b,402) adds:

The account of empirical interpretation of scientific theories ad-vanced therein is, as far as I can judge, not a new one; anyway it wasnot meant to be, as the main purpose of the monograph was to givea brief and elementary account of the current view of the subject.The view presented in the monograph is known under the name oftheStandard(orReceived)View of Scientific Theories.

1Carnaps work on probability may have been directly motivated by the need for probabilisticcorrespondence rules in the Received View(Lutz 2012b, 109110).

2Two other current philosophers of science have described the book as marvelous and won-derful, respectively, so I am in good company.

3I have argued elsewhere that the major criticisms of the Received View are spurious (Lutz2012b; Lutz 2012a,ch. 3, 4).

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The logical syntax that Przeecki lays out is a restriction of Carnaps higher or-

der syntax (cf.Carnap 1939, 1319) to first order logic, and thus is indeed notnew. The semantics, however, is Przeeckis own, though informed by the worksofKemeny(1956) andCarnap(1961), whichPrzeecki (1969, 107) cites as clas-sics. I will argue in the following that Przeeckis semantics completes the Re-ceived View in a natural way that fits with Carnaps informal descriptions andarguments (2), while going beyond Carnaps account both in its precision andin its content. Przeeckis semantics relies on classical model theoretic notionsdeveloped by Tarski and others, but expands them to deal with empirically inter-preted theories. It further permits straightforward generalizations without a lossof its basic insights (3). That Przeeckis semantics is very natural is indicatedby the sheer number of later results he anticipated, coming from both the Re-

ceived Views proponentsandits critics (4). And this treasure trove is far fromexhausted. As a simple example of a new result, I will point out an interestingrelation between vague languages and analyticity (5).

The overall picture that will emerge is this: Przeeckis semantics is a naturalformalism for the Received View, and provides a natural framework for under-standing the semantics of scientific theories in general. Furthermore, it pointsthe way to significant new research questions and results in the philosophy ofscience.

2 The Received View in the philosophy of science

Over the course of its development, the Received View has seen a variety of for-mulations and modifications by different authors. In the following, I will focuson Carnaps contributions, and specifically on those that play a major role inPrzeeckis semantics.

2.1 The observational-theoretical distinction

The central component of the Received View on scientific theories is its dis-tinction between observational and non-observational (theoretical) sentences.This distinction can be found in Carnaps work from his earliest contributionto the Received View(Carnap 1923,99100) to his last(Carnap 1966,ch. 23).

In some of his works, the observational and theoretical languages are taken tobe completely distinct, where the theoretical language is used as a metalanguageof the observational language, with a translational scheme from observational(object-) sentences to theoretical (meta-) sentences(Carnap 1932, 216217). How-ever,Neurath(1932,207) suggested distinguishing between observational andnon-observational sentences in the same language (cf.Carnap 1932, 215216);4

the translational scheme can then be realized by sentences of that same language,

4Carnaps conjecture that Neurath was the first to suggest this treatment of the sentences rela-tion is somewhat puzzling, sinceCarnap (1928)himself had already used this method.

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theoretical terms:

electric field, ...

temperature, . . .length, . . .

interpretation oftheoretical terms:

determined

through

observational terms: yellow, hard, . . .

interpretation of

observational terms: determined

directly

observable propertiesof things

Figure 1: Giving an empirical interpretation to theoretical terms (loosely based on a diagram byCarnap 1939,205).

known byCarnap (1966,ch. 24) as correspondence rules. One typically makes adistinction between the setCof correspondence rules and the setTof sentencesof the theory proper, although for the logical analysis within a single language, it

is often convenient to combine them into a set T Cof a single new theory.Given the initial distinction between observational and non-observationalsentences, it is typical to distinguish between the set of observational termsand the set of non-observational (theoretical) terms.5 Together, they make upthe whole of theory s vocabulary = .6 The observational sentencescontain only observational vocabulary and may be further restricted in their log-ical strength, for example to first order logic, finitely quantified first order logic,or molecular sentences(Carnap 1956,41).Carnap(1963,959) calls the languagewhose sentences contain only -terms but that is otherwise unrestricted thelogi-cally extended observation language.

The -terms are directly interpreted, either by observation or by simple, un-

controversial measurements(Carnap 1966, 226227).7

The-terms are not di-rectly interpreted (cf.Carnap 1956,47) but rather interpretedonlythrough thedirect interpretation of the -terms and the relations of the -terms with the -terms given by (see figure 1). When interpreting the -terms one might need to

5As is traditional in the philosophy of science, I will use term to refer to any non-logicalconstant. This fits well with general usage and related terms like terminology, but unfortunatelynot with the usage in symbolic logic.

6Throughout, I refer only to non-logical constants as the vocabulary of a theory. Theoriesand sentences can, of course, always contain logical constants and variable names.

7Chang(2005)gives a contemporary defense of such a direct interpretation of-terms.

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introduce new objects that are not observable.Carnap(1958,237238, 242243)

assumes that such newly introduced objects can be taken as mathematical objects:Assuming that there are at most countably many observable objects, he suggestsmapping them injectively to the natural numbers, and treating theoretical termsas applying only to the natural numbers and objects that can be constructed fromthe natural numbers with the help of Cartesian products or powersets. Carnapnever spelled out this schema in much detail, although it seems that the formal-ization of a statement like Some red object has a temperature of 0C would beas follows:

x.Rx t mx = 0, 8 (1)

wherem is the mapping from observational objects to natural numbers,t is the function assigning the temperature in degrees Celsius, andR refers tored objects. The analogue of the temperature concept of natural language (whichassigns a value in degrees Celsius directly to an observable object) is thus not t(which assigns such a value to a natural number), but rather t m. Since m isa mapping from observable objects to unobservable ones and its extension canthus not be determined by observation or simple measurement, it is a theoreticalterm as well. New, unobservable objects are introduced simply as other numbersor more complicated mathematical constructs. Theoretical terms hence apply tothem in the same way that they apply to the mathematical representations ofobservable objects underm.9

The bipartition between -terms and -terms is not fixed. In fact, accordingtoCarnap (1932,224) one can choose the observation terms depending on thecontext:

Let G be a law [ . . . ]. To check G, derive concrete sentences thatrelate to specific space-time points [ . . . ]. From these concretesentences, derive further concrete sentences using other laws andlogico-mathematical rules of derivation, until one reaches sentencesthat one wishes to accept in the specific case. And it is a matter ofchoice which sentences one intends to use as these endpoints of thereduction[ . . . ]. Whenever one wants tofor instance, if there aredoubts or one wants to consolidate the scientific hypotheses moresecurelyone can reduce those sentences previously accepted as end-

points again to other ones and choose those to be endpoints.[T]hereare no absolute primary sentencesfor the construction of science.10

8Throughout, I assume that quantifiers have minimal scope unless followed by a dot, in whichcase they have maximal scope.

9Carnap introduces this rather circuitous way of interpreting theoretical terms to address theworry that it may be impossible to interpret or discuss unobservable objects or non-observationalterms. However, it is at least not obvious why the mapping function m is needed, as Carnapsformalism also works for a mappingt that assigns temperatures directly to observational objects(so thatt = t mfor observable objects and t = tfor unobservable objects).

10Es seiG ein Gesetz[ . . . ]. Zum Zweck der Nachprfung sind aus G zunchst konkrete, auf

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This notion of a hierarchy of the scientific language is one of the central assump-

tion of Carnaps account of scientific theories(Lutz 2012a, 124). In figure1,thisassumption could be drawn by partitioning the theoretical terms into a se-ries1, 2, . . . , nfrom lower (more observational) to higher (more theoretical)terms, where the interpretation of the higher terms k is determined solely bythe interpretation of the lower terms k1and some theory k .

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2.2 The analyticsynthetic distinction

On the basis of a bipartitioned vocabulary, Carnap (1958, 245246; cf. 1963, 24.D)suggests a distinction between the synthetic (empirical) and analytic componentof a theory as follows: Assume that is a finite conjunction of sentences that

describe the theory, and assume that = (O1, . . . , Om , T1, . . . ,Tn), that is, contains only the -termsO1, . . . , Omand the -termsT1, . . . ,Tn .12 Then

R() := X1 . . .Xn(O1, . . . , Om ,X1, . . . ,Xn) (2)

is sRamsey sentence, which entails the same -sentences as itself. sCarnapsentenceis given by

C() := R() . (3)

According toCarnap(1963), the Ramsey sentence and the Carnap sentencefulfill the conditions of adequacy for any distinction between the analytic andsynthetic component of a theory. To spell out these conditions,Carnap(1963,963) defines the observational content of any sentenceSas follows:

Definition 1. Theobservational contentor O-content ofS=Dfthe class of allnon-L-true[not logically true]sentences inL

Owhich are implied byS.

LO

refers to the logically extended observation language. On this basis, Car-nap(1963,963) suggests

Definition 2. S is O-equivalent(observationally equivalent) to S=Df S is a

sentence inLO

and S has the same O-content asS.13

bestimmte Raum-Zeit-Stellen bezogene Stze abzuleiten [ . . . ]. Aus diesen konkreten Stzen sindmit Hilfe anderer Gesetze und logisch-mathematischer Schluregeln weitere konkrete Stze abzu-leiten, bis man zu Stzen kommt, die man im gerade vorliegenden Fall anerkennen will. Dabei ist

es Sache des Entschlusses, welche Stze man jeweils als derartige Endpunkte der Zurckfhrung[ . . . ]verwenden will. Sobald man will, etwa wenn Zweifel auftreten oder wenn man die wis-senschaftlichen Thesen sicherer zu fundieren wnscht, kann man die zunchst als Endpunktegenommenen Stze ihrerseits wieder auf andere zurckfhren und jetzt diese durch Beschlu zuEndpunkten erklren.[E]s gibt keine absoluten Anfangsstzefr den Aufbau der Wissenschaft.

11In this particular exposition,Carnap(1939)does not even distinguish between observationaland theoretical terms, but rather between elementary and abstract ones. And he notes, for instance,that if iron is not accepted as sufficiently elementary, the rules can be stated for more elementaryterms (Carnap 1939,207).

12Note that an unadorned T is a set of sentences (a subset of), while a subscripted Ti is atheoretical term. This notation is traditional, albeit confusing.

13Note that Carnaps definition is asymmetric.

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Taking R()and C()as the first and second components of, respec-

tively,Carnap (1963, 965) states:14

The two components satisfy the following conditions:

(a) The two components together are L-equivalent toT C[:= TC].

(b) The first component is O-equivalent toT C.

(c) The second component contains theoretical terms; but its O-content is null, since its Ramsey-sentence is L-true inL

O.

These results show, in my opinion, that this method supplies an ade-quate explication for the distinction between those postulates whichrepresent factual relations between completely given meanings, andthose which merely represent meaning relations.

By definition2, Carnaps condition (b) entails that the first component does notcontain -terms, so that the conditions on a theorysanalytic componentAn()and itssynthetic componentSyn() can be formulated as follows:

Definition 3. An()is anadequate analytic componentof and Syn()is anadequate synthetic componentof if and only if

1. An() Syn() is L-equivalent to ,

2. Syn() has the same O-content as ,3. Syn() contains no theoretical terms, and

4. the O-content of An() is the empty set.

As Carnap points out, a sentences O-content is the empty set if and only ifits Ramsey sentence is logically true. This result follows from

Lemma 1. Letand be sentences. Then s O-content is the empty set if and onlyif R(), andand have the same O-content if and only ifR() R().

Proof. sO-contentistheemptysetifandonlyifallthe-sentences of any order

that it entails are tautologies (since LO contains sentences of any order). Since

R()is an -sentence, R(), and R()entails all-sentences that areentailed by (Rozeboom 1962, 291293), this is the case if and only if R().

14These conditions are equivalent to earlier ones thatCarnap (1958,245246) uses to argue forthe adequacy of the Ramsey sentence and the Carnap sentence:

(2) Jeder Satz ohneT-Terme, der ausT Cfolgt, folgt auch ausR.[ . . . ]

(3) a)Die KonjunktionR AT istL-quivalent mitT C.b)Jeder Satz ohneT-Terme, der ausATfolgt, istL-wahr.

In these conditions, Carnap speaks of sentences without theoretical terms, which, though con-taining only observational terms, are unrestricted in their logical apparatus.

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Two sentences have the same O-content if and only if they entail the same -

sentences of any order. A sentence has the same O-content as its Ramsey sentence,so two sentences have the same O-content if and only if they have equivalentRamsey sentences.

Carnaps conditions of adequacy can thus be rephrased:

Corollary 2. An() is an adequate analytic component of andSyn() is anadequate synthetic component of if and only if

1. An() Syn() ,

2. RSyn()

R(),

3. Syn() contains no theoretical terms, and

4. RAn()

.

The Ramsey sentence thus allows for a compact description of central em-piricist concepts. More importantly, together with the Carnap sentence it fulfillsthe central need of the logical empiricists philosophy for an analytic-syntheticdistinction.

While the syntactic analysis of scientific theories had at this point in the his-torical development of the Received View reached a very high level, the ReceivedViews semantics was a mere impressionistic story, not yet described with any for-

mal rigor. It was Marian Przeecki who first developed and analyzed the seman-tics of the Received View as rigorously as Carnap did with its syntactic aspect.

3 Przeeckis semantics for the Received View

In his monograph, Przeecki develops a semantics based on two central assump-tions. For one,Przeecki (1969,105106) explicitly sets aside the matter of thedevelopmentof theories. Instead he focuses his analysis on a theory at a specificpoint in timethe theorys cross-sectionand considers the development of atheory a series of such cross-sections.15 This perspective fits nicely with Carnapsand Hempels analyses, which, although typically referring to a fixed set of sen-

tences of a theory, often assumed that those sentences might change in the future.The best example of this is probably their reliance on conditional rather thanexplicit definitions in order to allow for the definitions to be strengthened later(Carnap 1936,449450;Hempel 1952, 680681).

More substantially,Przeecki (1969,2930) also assumes that the domain ofa theory is fixed in advance in the theorys metalanguage.Przeecki (1974a,405)

15Przeecki(1969,106) provides a nice analogy: The logical technique resembles here a biolog-ical one. Logical reconstruction of a scientific theory is like making slices of a living organism.This certainly distorts our original object of inquiry. But only then can it be put under a logicalmicroscope.

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later calls this assumption unfounded [and] involv[ing]certain undue restric-

tions, but he is too harsh in judging his assumption unfounded, since in hismonograph he refers toKemeny (1956, 17), who argues that this

restriction is motivated by the idea that a formal system is the for-malization of the abstract structure of a given set of individuals. Wecan see this in examples from [ . . . ]Science. [ . . . ]Sociology dealswith human beings[and]a sociologist would allow only the set ofall human beings (past, present, and future)[.]

Przeecki(1974b, 405) notes another defense in connection with an argument byWinnie(1967), which I will discuss in the following section. In spite of these twopossible defenses,Przeecki (1973)provides two modifications of his semantics

that lift the restriction, which I will discuss in 3.3.

3.1 Ostension, vagueness, and approximation

In a central discussion of his monograph, Przeecki (1969, 2430) argues thatpurely verbal means (i. e., the exclusive use of sentences) are insufficient for givinga theory an empirical interpretation. This is justified by by

Claim 3. A set of sentences cannot determine the domains of its models.

Proof (cf.Przeecki 1969, 3031). Let A = |A|,R1, . . . ,Rr,f1, . . .fs , c1, . . . , ct bea model of and let Bbe any set with the same cardinality as |A|. Now de-

fine, for anyk -tuplex1, . . .xk of objects and any function gwith domain|A|,g Rg(x1), . . . ,g(xk )

R i (x1, . . . ,xk ). Then any bijection g :|A| Bis an

isomorphism from A toB= B,g R1, . . . ,g Rr,g f1, . . . ,g fs ,g(c1), . . . ,g(ct),and thus B .

Hence, if all intended interpretations of are over some domainA, willalways also be true of interpretations over some completely unrelated domainB.Specifically, cannot identify any objects in its domain.16 Even under Przeeckisrestriction of all models of to the same domain,gmay still be any permutationon the domain; hence any object of the domain can be exchanged for any otherobject of the domain, always leading to another model of (Przeecki 1969, 3031). Relying on the same formal result,Putnam(1989,353) puts this point in a

discussion of his famous model theoretic argument against realism (Putnam 1977)as follows:

[I]f there is such a thing as an ideal theory[I], then that theory cannever implicitly define its own intended reference relation. In fact,there are always many different reference relations that makeI true,ifIis a consistent theory which postulates the existence of more thanone object.

16This point is closely related to Newmans objection to Russells theory of perception (New-man 1928,2).

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For this reason, at leastsometerms of have to be interpreted directly, and along-

side Carnap, Przeecki assumes that these are the -terms.As to the means of direct interpretation,Przeecki(1969,3637) suggestsos-tension, the presentation of paradigmatic cases (thepositive standards) and paradig-matic non-cases (thenegative standards) of a predicate. By apsychologicalprocess,a student who is presented with the positive and negative standards can then learnto identify reliably other objects that the teacher considers to fall under the pred-icate. Importantly,Przeecki (1969, 37) points out:

If one chooses the right class [i. e. the one intended by the teacher]as the denotation of the predicate, one is not compelled to this choiceby purely logical reasons. This conclusion is arrived at in some pro-cess of abstraction whose analysis presents a problem for a psycholo-gist rather than for a logician.

Hence the efficacy of ostension for observational terms is an empirical prob-lem, which lies outside the philosophy of science.17 Because-terms are onlyostensively interpreted there are no analytic -sentences; otherwise, accordingtoPrzeecki(1969, 37), their interpretation would be partly determined verbally,and thus not by ostension.

A direct interpretation of the -terms and a fixed domain determine an-structure N := |N|, O

N1 , . . . , O

Nm . The interpretation of the -terms

within is determined only byN and , just as assumed by Carnap. Thisis the Thesis of Semantic Empiricism (Przeecki 1974b, 402, 405; cf.Rozeboom

1962). But only a portion of contributes to the interpretation of the-terms,namely s analytic component An(). s synthetic component Syn() rathercontains s empirical claims. Hence any expansion ofN to a model N ofAn() (N| = N N An()) provides a possible interpretation of the-terms (Przeecki 1969,4850). Such an expansion is unique for any Nif andonly if An() entails a definition for every-term(Beth 1953), and so N andAn() typically determine a set of structures for rather than a single structure.In an earlier paper, Przeecki (1964a) considers the resulting sets of structuresfor a special kind of analytic sentence: conditional definitions of-terms. As hepoints out, a conditional definition ofTi leads to avague denotation, for someobjects of the domain|N|will be in the extension ofTi in some expansions ofN, but not in others. More precisely, a k -place predicateTi tripartitions |N|kinto a setT+

i ofk -tuples of objects that are always inTi s extension (thepositive

extensionofTi ), a setT

i ofk -tuples that are never inTi s extension (thenegative

extension), and a set ofk -tuples that are only sometimes inTi s extension, whichI will call T (the neutral extension). In a generalization,Przeecki (1976,375)

17Incidentally, because of this psychological process of abstraction,Przeecki (1974b,403404)contends that ostension is less restricted thanTuomela (1972,2) makes it out to be in his criticismof Przeecki. As to Tuomelas rejoinder that he attributed to Przeecki somewhat too strict a viewon ostension, if[Przeeckis]reply really states what he said in his monograph(Tuomela 1974,407): See pages 3637.

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also discusses the denotation of a function symbol Fthat is vague over |N|,

which does not assign a single element b |N| to ak -tuple a1, . . . ,ak |N|k ,but rather a set F+(a1, . . . ,ak ) =B |N|(call this the non-negative extensionofF).18 Bcan be seen as the set of possible values of the function named by Ffor the argumentsa1, . . . ,ak . The denotation of a constant symbol c(consideredas a 0-place function symbol) that is vague over |N| is thus a setc

+ |N|.In a keen move,Przeecki (1969, ch. 5) draws on the multiplicity of possible

interpretations to formally describe the semantics of vague observationalterms.Because the interpretation of any-term is ostensive and some objects will beneither like its positive nor like its negative standards, these objects will be in theterms neutral extension(Przeecki 1969,3839). Like Carnap,Przeecki (1969,34) distinguishes between the observable objectsOand the unobservable objects

Uin the domainO Uof. Unlike Carnap,Przeecki (1969, 38) suggests thefollowing delineation of observable objects:

We shall call an object observable, if the possibility of its being ob-served is guaranteed by some natural law. In other words, xis observ-able ifxhas a propertyPsuch that the following statement: whoever(in suitable conditions) looks at an object possessing propertyPwillperceive the objectis a statement of a natural law. This loose expli-cation is not meant to serve as a definition of observability. It is onlyintended to point out some of its characteristic features.

Whether an object is observable or not is thus an empirical question. This doesnot mean that itself determines the observable objects, since can be one the-ory among many. The theories containing the natural laws about observabilitymay be different from .

Since an unobservable object is similar to neither positive nor negative stan-dards of any -term,Przeecki (1969, 4041) argues that all -terms are com-pletely vague overU, that is, every object in U is in the neutral or non-negativeextension of every-term. This is markedly different from Carnaps approach,in which unobservable objects have to be in the negative extension of all -termssimply because unobservable objects are numbers.Przeecki (1969, 4041), onthe other hand, intends unobservable objects to be physical (cf.Przeecki 1974b,405), which provides a second justification of his assumption of a fixed domainO U. For a fixed domain of the theory blocks a proof byWinnie(1967), ac-cording to which there is for any unobservable object a a model of in whichany other unobservable object (e. g. a number) is exchanged fora. Przeecki thuschooses a fixed domain over the danger of an antirealist interpretation of unob-servable objects. And this may be the biggest difference between Carnaps andPrzeeckis approaches: While Przeecki aims to be a realist about unobservableobjects, Carnap explicitly allows for an antirealist stance towards them.

18This is a slight generalization of Przeeckis account, who assumes thatBis an interval of reals,which would therefore have to be in |N|.

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Incidentally, one can now also see a central restriction of Przeeckis seman-

tics in his monograph, which is explicitly acknowledged byPrzeecki (1969, 103104): It does not contain mathematical objects, even though could containaxiomatizations of mathematics. In subsequent discussions, however, Przeecki(1974a, 347348;1976, 376) drops this restriction and allows terms of the theoryto be directly and uniquely interpreted by mathematical objects. In other words,mathematical terms are interpreted like (non-vague) -terms. AsBalzer and Re-iter(1989)show, such a formalization is possible in a sorted first order languagewithout the loss of completeness.

In Przeeckis semantics the existence of unobservable objects, and in generalthe direct interpretation of-terms by ostension leads to a non-trivial classNof intended -structures, and thesethen determine, together with An(), the

classN:= {N :N| NN An()} (4)

of intended -structures. The effect of this is that onceNis determined, -termsand -terms can be treated in the same way (as interpreted non-uniquely byclasses of intended structures), even though -terms are directly interpreted, and-terms are indirectly interpreted.

After a cogent discussion of different possibilities for defining truth and fal-sity in sets of structures (rather than single structures),Przeecki(1969,22) optsfor conditional definitions: is true in N ifN for all N N, and isfalse inNifN for all N N. AsPrzeecki (1976, 375, n. 3) points out, theconditionals together with their converses make truth and falsity intosupertruthandsuperfalsity, respectively (cf.Fine 1975), which I will assume for ease of ex-position.Przeecki (1976, 378379) also suggests that thesubtruthof (cf.Hyde1997), with N forsomeN N, can serve as an explication of the notion ofapproximate truth.Przeecki(1976,379, my notation) states:

The notion of approximate truth makes it possible to dispense withcertain idealizing assumptions in the semantics of empirical theories.The requirement that a physical theorybe approximately true al-lows to treat theory as a theory of real, not idealized, objects. Theuniverse of the structures in Nmay be thought of as a set of realthings which are close enough to the alleged ideal entities. Thus, in

the case of particle mechanics, its universe will be composed not ofideal point-masses, but of actual things, such as planets, projectiles,and the like. The theory is approximately true of themin the sensebeing here considered. That is to say, among proper structures inNthere[are]some in which the theory is (strictly) true. This amountsto the fact that the values of the theorys functions for those objects,as stated by the theory, fall into the intervals determined by the rele-vant measurement procedures.

For example, when measuring a magnitudeFwith imperfect precision, an objectb (e. g., a body of condensed matter, liquid, or gas, or a whole system thereof)

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is assigned a range of possible values, and the pair consisting ofband its range

of values is in the non-negative extension F+ ofF. The measurements and ob-servations for all objects in the theorys domain and all relations and functionsinthen determine, together with the domain itself, the class Nof intended-structures. If contains only-terms, is approximately true if and only ifit is subtrue inN, and thus, because of its definition(4), also subtrue in N.

In general, of course, the definition of approximate truth inNhinges on theas-of-yet undefined notion of An(). But under the assumption that An()(discussed below), it is trivial to show that is approximately true if and only ifthere is some N Nthat can be expanded to a model of. Delineating theanalytic component ofis therefore not necessary for determining whetheris approximately true. But it is necessary for determining whether is true in a

vague language. For in that case, has to be true inevery N N, and there maybe intended structures that are not models of since typically An() .

3.2 The analytic-synthetic distinction

Towards a delineation of An(), Przeecki (1969,5859) assumes that con-tains a distinguished set of postulatesPthat determine the interpretation of thetheorys theoretical terms. Since Pcan have empirical content, Przeecki allowsP and I will assume this in the sequel.Przeecki(1969,5051) then argues asfollows:

The language of any empirical theory always seems to be treatedby the scientist as an interpreted, meaningful language, and not asa mere formal, meaningless calculus. And it seems to be treated so in-dependently of any empirical findings. Experience may decide onlywhether an empirical theory is true or false, not whether it is mean-ingful or meaningless.

Since, first, an interpretation of the-terms is given by an expansion of someintended structure in Nto a model of An(), and second, Nhas to be de-termined exclusively by empirical means,Przeecki(1969,55, my notation) canconclude that An() must be sufficiently weak to fulfil the semantic conditionof non-creativity.

Definition 4. An() issemantically non-creativeif and only if

AB .B|=AB An() . (5)

Any -structure can be expanded to a model of An() if it is semanti-cally non-creative. Thus An() is weak enough that it does not restrict the-structures.

On the other hand, Przeecki (1969, 55) argues, it must be sufficientlystrong to include all of the meaning postulates contained in . To express this

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demand formally, Przeecki paraphrases a demand byWjcicki(1963):19

Definition 5. An() includes all of the meaning postulatescontained in if andonly if

A . B[B|=A|B ] [A An() A ] . (6)

The conditions of adequacy for the analytic component of are thereforegiven by

Definition 6. An()is a semantically adequate analytic componentof if andonly if it is semantically non-creative and includes all of the meaning postulatescontained in .

Incidentally, Przeecki does not assume (and does not need to assume) that, An(), and Syn() are single sentences. Hence I will treat them as sets in thefollowing.

With Przeeckis conditions of adequacy, one can now establish the alreadyannounced

Claim 4. IfAn()is a semantically adequate analytic component of, then istrue (false) in the intended structuresNif and only if every (no) intended-structurecan be expanded to a model of.

Proof. : Assume that is true inN. Then is true in every N Nand thusin every expansion of every intended-structure to a model of An(). Since

An() is semantically non-creative, is thus true in at least one such expansionand, since An() contains all meaning postulates of, in all of them.

Now assume that some intended-structure can be expanded to a model of. Then it can be expanded to a model of An()because An()contains allmeaning postulates of. Thus is not false inN.

: Assume thatevery N Ncan be expanded to a model of. There-fore, since An() includes all of the meaning postulates contained in , an expan-sion ofNis a model of An() only if it is a model of. Hence everyN Nisa model of.

Now assume that no N N can be expanded to a modelN of. SinceAn() is semantically non-creative, no N Nis a model of.

Even without having determined a theorys analytic component, Przeeckissemantics already provides the notion of approximate truth under the reasonableassumption that a theory entails its analytic component (which follows fromclaim5below). Claim4now shows that for truth in a vague language it is notnecessary to determine a theorys analytic component either. Hence, while it isnecessary to know how to delineate an analytic component of in principle (byway of its conditions of adequacy), it is notnecessary to do so in any specific case.

19I am very grateful to Krystian Jobczyk for giving me an overview and partial translation ofWjcickis article.

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Also based on work byWjcicki(1963),Przeecki and Wjcicki (1969,376)

give conditions of adequacy for the synthetic component of, based onDefinition 7. An -structure A isadmittedby if and only if

B .B|=AB . (7)

Two sets of sentences that admit the same -structures place the same restric-tions on them, and thus can be considered semantically empirically equivalent:

Definition 8. Syn() and aresemantically empirically equivalentif and only ifthey admit the same -structures.

Przeecki and Wjcicki (1969,387) demand that Syn()admit the same -structures as , so Syn()and must be semantically empirically equivalent.They demand further that Syn() provide no empirical interpretation of.

Definition 9. Syn()provides no empirical interpretationof if and only if

AB .A|=B| [A Syn() B Syn()] (8)

Together, these demands lead to

Definition 10. Syn() is asemantically adequate synthetic componentof if andonly if Syn() is semantically empirically equivalent to and provides no empir-

ical interpretation of.The conditions of adequacy for An()and Syn()can be combined as fol-

lows:

Claim 5. An() is a semantically adequate analytic component ofandSyn() isa semantically adequate synthetic component of if and only if

1. An() Syn() ,

2. Syn() is semantically empirically equivalent to .

3. Syn()provides no empirical interpretation of, and

4. An() is semantically non-creative.

Proof. :Przeecki and Wjcicki (1969, theorem 7) note that condition1fol-lows from the conditions of adequacy; that the others follow is trivial.

: Syn()is trivially a semantically adequate synthetic component of,and An()is trivially semantically conservative. It remains to be shown thatAn() contains all of the meaning postulates in . This is the case if and only if

A . B[B|=A|B ] A A An() , (9)

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and

A . B[B|=A|B ] A An() A . (10)Conditional (9) is equivalent to An() and follows trivially from condition1.Conditional(10)follows from conditions1,2,and3:Because of the first conjunctof the antecedent and condition2,B[B| =A|B Syn()]. Hence, bycondition3, A Syn(). A then follows from the second conjunct of theantecedent and condition1.

Although the conditions of adequacy in claim5are arguably more intuitivethan those given by Przeecki and Wjcicki, their conditions have an indisputableadvantage: Unlike claim5,Przeecki and Wjcickis conditions can be applied toa purported analytic component without knowing the synthetic component and

vice versa. This will be convenient in 4.3below.The conditions of claim5seem very similar to Carnaps (definition3), and

indeed, they are direct generalizations of Carnaps conditions to sets of sentences.The connection is given by the paraphrase of Carnaps conditions in terms ofRamsey sentences (claim2) and

Lemma 6. Acan be expanded to a model of the sentence if and only ifA R().

Proof. Let be the result of substituting each-term in by a correspondingvariable.

: Since A R(), there is a relationVifor every relation symbolPiin, a functionGjfor every function symbolFjin , and a constantdkfor everyconstant symbolckin such that {Vi , Gj, dk } satisfies

in A. Define C so thatPC

i = Vi for eachVi ,F

Cj

= Gjfor eachGj,cCk

= dkfor everydk , and C|= A.Induction on the complexity of shows that C .

: Induction shows that {PCi ,FC

j , cC

k}satisfies in A, so A iXi jYj

kxk .

Lemma6immediately entails

Corollary 7. Let,Syn(), andAn() be single sentences. ThenSyn() is seman-tically equivalent to if and only ifRSyn() R()andAn() is semanti-cally non-creative if and only if R

An()

.

Furthermore the following holds:

Lemma 8. LetSyn() be a single sentence. Then Syn() provides no empiricalinterpretation ofif and only ifSyn() R

Syn()

.

Proof. : Since RSyn()

is an existential generalization of Syn(), it is clear

that Syn() RSyn()

. For the converse, assume that A R

Syn()

. By

lemma6,there is a B Syn() with B|=A|. Therefore A .: Immediate.

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Corollary7and lemma8furthermore entail

Claim 9. Let andSyn() be single sentences. ThenSyn() is a semanticallyadequate synthetic component of if and only ifSyn() R().

Proof. If Syn() is a semantically adequate synthetic component of , thenR() R

Syn()

and R

Syn()

Syn(), and thus Syn() R().

The converse holds because RR()

R().

Because of claim9,a semantically adequate synthetic component ofmustbe equivalent to a (singleton) set of-sentences. Hence claim2,corollary7,corol-lary8, and claim9together entail

Claim 10. Let,An(), andSyn() be single sentences. ThenAn() is a seman-

tically adequate analytic component of andSyn() is a semantically adequatesynthetic component of if and only ifAn() is an adequate analytic component ofandSyn() is, up to equivalent reformulation, an adequate synthetic componentofaccording to definition3.

Thus for single sentences, Wjcicki and Przeeckis conditions of adequacyare essentially equivalent to Carnaps and can be phrased in a very compact wayin terms of of Ramsey sentences.

Claim9shows that Przeecki and Wjcickis (and hence Carnaps) conditionsof adequacy uniquely determine the synthetic component of up to logicalequivalence when -sentences can be of any order. In contradistinction, An() isnot in general uniquely determined by their conditions of adequacy, asPrzeeckiand Wjcicki(1969,391) point out. In particular, the Carnap sentence is just theweakest of a variety of adequate analytic components of. Przeecki (1969, 7.III)provides a nice example for reduction sentences

x[(x) T1x] x[(x) T1x] , (11)

where and are -formulas andT1is a -term. Then

C() x[(x) (x)] x[(x) T1x] x[(x) T1x] (12)

is an adequate analytic component of. However, the logically stronger sentence

x[(x) (x) T1x] x[(x) (x) T

1x] (13)

is also an adequate analytic component of. This solution has general advan-tages (Przeecki 1961b) and can, for example, be used to salvage ethical terms thatwere introduced by reduction sentences with empirically false implications (Lutz2010).Winnie(1970, 294296) andDemopoulos(2007,V) argue that the possibil-ity of choosing the analytic component of a theory is a problem for the analytic-synthetic distinction, and they suggest an additional condition of adequacy thatestablishes a theorys Carnap sentence as its sole adequate analytic component.However, neither their argument nor their suggested condition of adequacy areconvincing (Caulton 2012;Lutz 2012a,12.1).

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3.3 Przeeckis modifications of his semantics

Przeecki (1969,10.I) generalizes his semantics in a significant way already inhis monograph. He introduces a hierarchy of languages , 1, 2, . . . that beginswith a setof observational terms and continues with a series of sets of the-oretical terms. The intended-structures and some theory 1 determine theintended structures for 1, which in turn determines with some theory 2the intended structures for 1 2, and so forth. The intended -structuresdo not interpret the vocabulary of the theory n ,n> 1, so the class of intendedstructures for nis the reduct of the intended structures for 1 nton1 n . Indeed, there need not even be a well-determined class of-terms; itmay simply be convenient within the formalism to postulate this starting pointof the hierarchy(Przeecki 1969, 10.II). Such a hierarchy of vocabularies wassuggested at about the same time byRozeboom(1970, 202), and Przeeckis for-malism is a plausible explication of the hierarchies of vocabularies discussed byCarnap. For convenience, nand n1can be renamed to and , respectively,keeping in mind that does not have to be observational in any specific sense. Infact,Carnap(1931,437438) suggested that observation reports are formulatedmore expediently in such a vocabulary.

The new-terms can thus be considered to refer to concepts that are notthemselves under investigation, as was suggested explicitly byReichenbach(1951,49),Lewis(1970,428), and Carnap himself. Since the -terms are unproblematicinthissense,therecanbeaset of analytic -sentences. Accordingly, Przeecki(1969,9899) generalizes the conditions of adequacy for a theorys analytic com-ponent:

Definition 11. An()is asemantically adequate analytic component of givenanalytic-sentences if and only if

A.A B .B|=AB An() (14)

and

A .A B[B|=A|B ] [A An() A ] . (15)

The first condition (14) generalizes semantic non-creativity to semantic non-creativity relative to-sentences 20 and the second condition generalizes thedemand that An() include all of the meaning postulates of.

In a later work,Przeecki (1973) discusses the possibilities for avoiding theassumption of a fixed domain. The reason is simple: Fixing the set Uof unob-servable objects in the metalanguage is impossible if the interpretation of the-terms is to be determined solely by the interpretation of the-terms and by. Therefore the assumption of a fixed domain O Uis incompatible with the

20This can be further generalized to semantic non-creativity relative to -sentences (Lutz 2012a,definition 6.6, cf. 6.11.1).

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thesis of semantic empiricism (Przeecki 1974b, 405).Przeecki(1973,287) avoids

the assumption in two alternative modifications of his semantics. Given the in-tended-structureN, the intended-structures can be given by the class ofall expansions of elementary extensions ofN or the class of all expansions ofextensions ofN. If there is no single intended observational structureNbut aclass Nthereof, the class of intended structures may be defined in three differentways:

N:=A :A An() B N.B=A|

(16)

N:=A :A An() B N.BA|

(17)

N:=A :A An() B N.BA|

(18)

The first definition(16) paraphrases Przeeckis initial definition (4). The twomodifications (17) and (18) avoid the assumption of a fixed domain and are com-patible with the thesis of semantic empiricism, but are also subject to Winniesproof: In both semantics, an intended -structure can always be extended to con-tain mathematical objects if it can be extended at all.

Przeecki (1973,289) modifies the condition of non-creativity accordingly,assuming the possibility of analytic -sentences. When elementary extensions ofthe intended -structures are allowed, this leads to

Definition 12. An() isup to elementary extensions semantically non-creative rel-ative to -sentences if and only if

A.A

B .AB|B

An() . (19)As Przeecki points out, definition 4 has no syntactic formulation in first

order logic, while definition12corresponds to syntactic non-creativity:

Definition 13. An()is syntactically non-creative relative to-sentences ifand only if for all -sentences , An() only if.

Claim 11. An() is up to elementary extensions semantically non-creative relativeto-sentencesif and only ifAn() is syntactically non-creative relative to-

sentences .

The difference between semantic and syntactic non-creativity lies at the core

of the critique ofDemopoulos and Friedman (1985) by Ketland (2004, 297299).

21

Demopouloss response turns on the equivalence of syntactic non-creativity andsemantic non-creativity up to elementary extension(Demopoulos 2011, 4).

In the case where the semantics allows for any kind of extensions of the -structures,Przeecki(1973,289) modifies semantic non-creativity as follows:

Definition 14. An()isup to extensions semantically non-creative relative to-sentences if and only if

A.A B .AB|B An() . (20)21AsPrzeecki and Wjcicki(1971,9495) show, this difference holds even if= .

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Przeecki(1973, 289) notes

Claim 12. An() is up to extensions semantically non-creative relative to -sentences if and only if for all purely universally -sentences,An()only if.

Note that in the semantics that allows any kind of extensions of intended-structures, the truth-value of a first order -sentence can be different inN andN, which is impossible in the previous two.

It is clear that Przeeckis modifications of his semantics lead to natural modifi-cations of definition 6 for an adequate analyticcomponent and of definition 10 foran adequate synthetic component of a theory. Przeecki demonstrates this withhis modifications of non-creativity (definitions12and14); the general methodcan be read off of these: To take previously accepted analytic sentences intoaccount, all that is needed is a restriction of the quantifiers over the class ofstructures to the class of-models of. To take the change from simple expan-sions to expansions of elementary extensions or expansions of extensions intoaccount, all that is needed is a systematic substitution of = between structuresby or , respectively.

4 After the Received View

In the preceding section, I have argued that Przeeckis semantics can be seen as

an explication of Carnaps informal description of the semantics of scientific the-ories. I now want to argue that with his semantics, Przeecki both influenced andanticipated later developments in the philosophy of science. AlthoughPrzeecki(1975,284) considered himself positivistically-minded and his monograph anintroduction to the Received View, he had the greatest influence on the ReceivedViews main opponentthe so-called Semantic Viewwhich describes theoriesas classes of model theoretic structures rather than sets of sentences. Probablydue to Przeeckis heavy use of model theoretic methods,da Costa and French(1990,249) consider his monograph a precursor of the Semantic View andVolpe(1995,566) even lists it as an early workwithinthe Semantic View.22 It is notablethat, even though Przeeckis monograph does not seem to have had any specific

influence on the article byda Costa and French(1990),Przeecki (1969;1976)anticipates and even generalizes their central concepts of partial structures andquasi-truth in his concepts of vague terms and approximate truth (Lutz 2012a,4.3.2).

22This suggests that the differences between the Received View and the Semantic View are notas large as some proponents of the latter have claimed (cf. Lutz 2012a, 4.1).Przeecki(1974c)alsooutlines how to bridge the gap to Sneeds Structuralist View, which is sometimes considered onevariety of the Semantic View.

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4.1 Constructive empiricism

Przeecki explicitly influenced van Fraassen, who famously declared that the Re-ceived View is in principle unable to capture the relation between theory andphenomena(van Fraassen 1980,3.6). Within his alternative, constructive em-piricism,van Fraassen(1980, 64) states that

[t]o present a theory is to specify a family of structures, its models;and secondly, to specify certain parts of those models (theempirical

substructures) as candidates for the direct representation of observablephenomena.

Furthermore the models of the theory are describable only up to structural iso-

morphism (van Fraassen 2008,238; cf.2002,22). Under the simplifying assump-tion that each model of the theory has exactly one empirical substructure, thiscan be phrased as follows:

Definition 15. A theory is a family {Tn}nNof structures (the models of thetheory) such that each of its members Tn has exactly one empirical substructureEn Tn . With each model, a theory also contains every isomorphic structurewith its corresponding empirical substructure.

Van Fraassen (1980, 64) strictly distinguishes between the setOof observableobjects and the unobservable objects and suggests describing observable phenom-ena by structures as well: The structures which can be described in experimental

and measurement reports we can callappearances (van Fraassen 2008, 286). Withthe simplifying assumption that all appearances can be included in a single struc-ture, this suggests

Definition 16. Theappearancesare given by a structureP with |P| = O.

Van Fraassen(1980,64) then defines a theory as empirically adequate if it hassome model such that all appearances are isomorphic to empirical substructuresof that model (cf.van Fraassen 1991, 12). With the simplifications given above,this results in

Definition 17. A theory{Tn}nN isempirically adequatefor the appearance P

if and only if there is some n Nsuch that En =P.

Van Fraassen(1980, 64;1989,227) traces his inspiration for relying on empir-ical substructures to Przeeckis monograph, but a much more direct connectionis given by Przeeckis second modification of his semantics:

Claim 13. Assume that{Tn : n N}= {B: B }, that contains only-terms, and that all and only expansions of extensions of intended-structures arein N. Then, ifN = {P}, {Tn}nN is empirically adequate if and only if is

approximately true inN.

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Proof. {Tn}nNis empirically adequate for P if and only ifP has an extension

to someTn , n N. This holds if and only ifTn N, that is, N for someN N, which is the approximate truth of.

The first assumption in claim13is a restriction on van Fraassens notion of atheory, because the language of may not be strong enough to describe {Tn}nNup to isomorphism. The second assumption is a restriction on Przeeckis notionof a theory, which allows theories to contain terms that do not occur in the de-scription of the appearances. The third assumption simply describes Przeeckissecond modification of his semantics. The condition N = {P} shows veryclearly that within the domain of observable objects, van Fraassen assumes per-fect information about the phenomena without any vagueness of terms or im-

precision of measurement.Van Fraassen (1989, 366, n. 5) is aware of this, as hesuggests introducing an approximate notion of empirical substructures to gener-alize empirical adequacy. In Przeeckis terminology, and in full agreement withPrzeeckis own position, the -terms are completely vague for the unobservableobjects.

Together with claim12,claim13has the nice

Corollary 14. Assume a language of first order logic. Assume further that{Tn :n N} = {B : B }, that contains only-terms, and that all and onlyexpansions of extensions of intended-structures are inN. Then ifP is described upto isomorphism by , {Tn}nN is empirically adequate if and only if is compatiblewith all purely universal-sentences entailed by .

Proof. Since describes P up to isomorphism, is approximately true inNifand only if is up to extensions semantically non-creative relative to , which isthe case if and only if for every purely universal -sentence, only if . Since is maximally consistent, this holds if and only if is compatiblewith all purely universal -sentences entailed by .

An analogue to corollary14for another semantic concept of van Fraassenswas later pointed out by Scott Weinstein in a defense of syntactic approacheslike the Received View (Friedman 1982, 277) . This would be the second timethat Przeecki anticipated a defense of the Received View.

4.2 Suppe on partial interpretations

A defense bySuppe(1971) was possibly the first that Przeecki anticipated. Al-though highly critical of the Received View (Suppe 1972), Suppe gives an eluci-dation of the Received Views notion of a partial interpretation with the aimof defending it against Putnams claim that the notion was completely broken-backed (Putnam 1962,241). Towards this goal, Suppe develops a number of sup-porting concepts. First, he assumes that the language of the theory is of firstorder and that the -terms are interpreted over a domain of concrete observableentities (Suppe 1972, 5859, my notation here and in the following). Suppe (1972,

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60) goes on to assume that a fixed set of rules of designation has been specified

for the-terms and calls the class of interpretations specified by these rules thepermissible interpretations S for. The result of adding a new property P to[S]and a new rule of designation that P designatesPis said to be apermissibleextensionof[S](Suppe 1971,63).Suppe(1971,65) adds that for a permissibleextension fromStoS, we do not require that Sand S have the same domain,but rather only that the domain ofScontain the domain ofS. This allows thepossibility that the domain ofSmay contain both theoretical entities and observ-able entities, etc.23 Finally, in effect treating the correspondence rulesC (fromT C= T C) like the postulates that determine the interpretation of inPrzeeckis account,24 Suppe(1971,65) concludes that

the assumed truth ofT Cwill impose restrictions upon the class oftrue permissible extensions to[].[ . . . ]This, then, suggests thatthe sense in which the interpretative system Csupplies[]with apartial interpretation is that it imposes restrictions on the class ofpermissible models for it.

The resulting semantics is that of Przeeckis second modification: The -termsare interpreted by an -structure S, and an expansion S of an extension is apermissible extension in Suppes sense if and only ifS S|.

25

Przeecki (1973)suggested his second modification at a conference in 1971(Bogdan and Niiniluoto 1973,v), in the year thatSuppe (1971) published his se-mantics. But it seems that the core idea of Suppes semantics is that of letting theinterpretation offix a structureNand consider any structure a possible inter-pretation of that in some sense includesN; and this account is already workedout with much precision and in a very general way in Przeeckis monograph. Ofcourse, the question of priority is basically moot since Suppe and Przeecki havedeveloped their accounts wholly independently. The important point is ratherthat their accounts are equivalent, for this provides another reason to considerPrzeeckis semantics a faithful formal account of the semantics of the ReceivedView, since such an account was an explicit goal of Suppes.

Unfortunately,Suppe(1971, 67) further argues that within the constraintsof the Received View, his semantics can be supplemented by a direct interpreta-tion of the-terms within some antecedently understood metalanguage, whichhe assumes to be a natural language. But this goes against the thesis of semanticempiricism and makes partial interpretation pointless, since there is no need to

23Suppes sequence of definitions takes a number of twists and turns that are sometimes difficultto follow, so that the relation betweenS and the permissible extensionsSmight be not as directas indicated here. This may be partly because of his puzzling use of some technical terms. ForexampleSuppe(1971,59, 60) repeatedly speaks of predicate and function variables although thelanguage is of first order. I hope to have done his semantics justice.

24As noted, I have instead used the whole of.25As noted above, Suppes outline of his semantics is not always clear, and I must restrict my

claim to the extent that I have been able to reconstruct Suppes semantics.

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construct an interpretation of the -terms with the help of if the -terms are

already directly interpreted.26 Furthermore, Suppes move only pushes questionsabout the interpretation of theoretical terms of the object language into the meta-language, as was already argued byCarnap(1939,204),Hempel (1963, 696), andRozeboom(1970,204205), and again pointed out byPrzeecki(1974b,402) in aresponse to a similar suggestion byTuomela (1972, 171). More generally, Suppes(and Tuomelas) suggestion seems to stem from a certain tendency of ascribingto natural language a mystical quality not inherent in formalized languages. Inthis wayBeth(1963,481) describes the view that natural languages can provide in-tended interpretations in a way that formalized languages cannot. Without such amystical view, supplementing a formal theory with natural language sentencesis just adding more theory, asPutnam(1980,477, emphasis removed) put it

most famously.

4.3 Andreas on the semantics of scientific theories

In a more recent article,Andreas(2010,530532) aims to give a formal seman-tics based on the indirect interpretation of theoretical terms outlined byCarnap(1939). Like Carnap and Przeecki, Andreas assumes a fixed domain. And likeCarnap, but unlike Przeecki, he assumes that there is a fixed bipartition of the do-main into observable objectsOand mathematical objectsU.Andreas(2010,529,532) further introduces the notion of an intended interpretation, specifically theintended interpretationNof the -terms; like Przeecki and Suppe, he assumes

that there is a subsetPof postulates of that determine the interpretation of thetheorys theoretical terms. Also like Przeecki, Andreas assumes that it is possibleand indeed preferable to letP= ,27 which I will suppose in the following. Withthese concepts,Andreas(2010, 533) suggests that the intended interpretations ofbe given by the admissible structures for the intended -structureNand thetheory .

Definition 18. Let O U = and let Nbe the intended-structure with|N| = O. Define two classes of-structures:

S1(N) ={A : |A| = O UA|O=NA } (21)

S2(N) ={A : |A| = O UA|O=N} (22)

The classS(N) ofadmissible-structuresfor and Nis defined as

S(N) := S1(N) ifS1(N) = , S(N) := S2(N) ifS1(N) = . (23)

A|Ohere stands for the relativized reductofA, the substructure ofAsreductA|that has domainO (i.e., A|O := A||O). It is a standard notionin model theory (cf.Hodges 1993, 5.1).

26The quotes from Hempel and Carnap that Suppe adduces to show that they accept directinterpretations of-terms in fact show the exact opposite (Lutz 2012b,9394).

27Personal communication from 10 January 2013.

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Given this semantics, Andreas defines truth in Nas supertruth inS. Since

S(N)|O := {S|O :S S(N)} = {N}, all-terms are completely preciseoverO . But as in Przeeckis semantics (and unlike in Carnaps semantics), most-terms are completely vague overU. This is because for any S S(N), anyfunctionfsuch that f |O= id andf |U= gfor some permutationgoverUleadsto another modelS of as described in the proof of claim 3. SinceS|O=N,S S(N). The only-terms that are not completely vague over Uare thosethat are invariant under any permutation; such terms may be called purely logicalover U (cf.Przeecki 1969,3031). Purely logical termsR occur only ifS1= and entails sentences likex.O x Rx orxy.Rx y x = y. Note thatthis exception does not hold for Przeeckis semantics, since there the extensionof the -terms overUis determined independently of.

Andreas (2010, 538) states that in his accountonly sentences qualifying as postulates are assumed to determine themeaning of theoretical terms. And the distinction between postu-lates and other theoretical sentences must clearly not be equated withthe analytic-synthetic distinction. Analyticity is therefore no require-ment for a sentence to determine the meaning of nonlogical symbols.

Although Andreas did not set out developing a semantics that allows a clearanalytic-synthetic distinction, I contend that he did. One way to show this isto follow Przeeckis suggestion of introducing a possibly fictitious new observa-tional vocabulary, and to treat Andreass -terms as theoretical terms. Andreass

semantics is then recovered by defining his-terms conditionally with the helpof the new observational terms (Lutz 2012a, 2.10.2). In this way, Andreass -terms are completely vague over all unobservable objects as a result of the con-ditional definitions, as was already suggested by Przeecki. A more direct wayleads via Przeeckis criteria of analyticity that take previously established ana-lytic sentences into account (definition11). The discussion in the following willbe much simplified by focusing not on the set of sentences describing a scien-tific theory, but rather the classT :={A : A }of its models. Similarly, onecan define the analytic component An(T) := {A : A An()}ofTand the classA :={A : A }of analytic-structures. As first suggested byCaulton(2012), I will go one step further and treat these these classes as primitive, thus

specifically not assuming that they have an axiomatization in first order logic.28Andreass semantics is intended forany -structure with domainO . Thus,

while he defines S(N)as the class of admissible structures for andN, itis of interest to determine which structures are admissible in principle, that is,independently of the empirically determined N.

Definition 19. The class ofadmissible structures for is given by

S :=

S(A) : |A| = O

(24)

28Caulton gives a semantic version ofCarnapsconditions of adequacy (definition3).

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This can be simplified by usingT|O := {T|O : T T}andA|Ofor the

claim that A|Ohas a relativized reduct with domainO:Claim 15. The class of admissible structures for is

S =A : |A| = O U A|O (A|O T|O A T)

(25)

Proof. First note that every-structureA with|A|= O is inS|O, either

because it can be extended and expanded to a model of with domainO U(which is then inS1), or because every such expansion makes false (and is thusinS2). Hence the following holds:

S ={A : |A| = O U A|OA T}

{A : |A| = O U A|O A|O T|O} (26)={A : |A| = O U A|O (A|O T|O A T)} (27)

Przeeckis definition11can be rephrased using classes of structures:

Definition 20. An(T)is a semantically adequate analytic component ofTgivenanalytic-structuresA if and only if

A.A A B .B|=AB An(T) (28)

and

A .A| A B[B|=A|B T] [A An(T) A T] . (29)

It is clear that forT = {A: A }, An(T) ={A: A An()}, andA ={A :A }, definitions11and20are equivalent. It is now possible to estab-lish the status ofS:

Claim 16. S is a semantically adequate analytic component ofT given analytic-structures

A:= A : |A| = O U A|O A|O T|OA T|. (30)

Proof. To show that equation (28) holds forS,T, andA, assume that A A.ThenA|O T|O or A T|. In the former case, there is no B T with|B|= O Usuch that B|O = A, and thus any extension and expansion ofA toO Uand is inS

. In the latter case, there is such a B T, and thus, byclaim15,B S.

To show that equation (29) holds, assume that A| A B[B|=A|B T]. If further A T, then A S by claim15.If on the other hand AS, then by claim15, A|O T|OA T. The first disjunct is false becauseB[B|=A|B T], so A T.

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Claim16hinges on the definition of the analytic -structuresA, and it is

clear thatS can always be an adequate analytic component ifA is chosen smallenough. For instance, ifA = , both conditions in definition20are triviallyfulfilled. But A is not too strict; more precisely, it does not place any restrictionson the -structures with domainO:

Claim 17.A|O = {A : |A| = O} (31)

Proof. Let|A|= O . For a proof by cases, assume that for some B, B|O =A |B| = O UB T. ThenB||O =Aand |B|| = O UB||OB| T|, and thus A A|O.

Now assume that there is no B with B|O = A |B| = O U B T.

Then choose any extension B ofA toO U. By assumption, Bhas noexpansion that is inT. Thus|B| = O U B|OB T|O, and thusA A|O.

The proof in the other direction is immediate.

Since T has an analytic component, it should also have a synthetic component.The analogue to definition20is given by

Definition 21. Syn(T)is asemantically adequate synthetic component ofT givenanalytic-structuresA if and only if

A.A A

B[B|=AB T] B[B|=AB Syn(T)]

(32)and

AB .A|=B| A| A [A Syn(T) B Syn(T)] . (33)

The first condition generalizes definition8, and the second condition gener-alizes definition9.WhileSleads to an analytic component ofT,S1leads toTssynthetic component. Define

S1:= {A : |A| = O U A|O S1(A|O) = } (34)

This can again be simplified:

S1 = {A : |A| = O U A|O A|O T|O} (35)

The following then holds:Claim 18. S1 is a semantically adequate synthetic component ofT given analytic-structuresA.

Proof. For a proof of equation (32), assume B| = A A and B S1. Then

by equation(35), A|O = B|O T|O. Hence, since A A, A T|,and thus there is some B withB|=AandB T. The other direction of theequivalence is immediate.

For a proof of equation(33), assume B| = A| A and A S1. Then by

equation(35),B|O=A|O T|Oand thus B S1.

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In summary, then, for each theoryT it is possible to find an analytic com-

ponent and a synthetic component under the assumption that there are alreadyanalytic -structures.This result might be unsurprising given theway Andreas sets up his semantics.

He assumes that any-structure with domainO can be an intended structure,and then allows any extension to the whole domain O U. Thus the empiricalinformation in an -structure with domainO Uis completely contained in itssubstructure with domainO. This means that the extensions of the -terms overUdo not provide empirical, but only analytic information. Thus any restrictionsof the extensions overUthat follow from are analytic, and when this is takeninto account, Przeeckis formalism for an analytic-synthetic distinction with pre-viously given analytic structures becomes applicable.

The overall result so far is this: Przeeckis semantics anticipates and generalizesconcepts and results from the partial structures approach, constructive empiri-cism and its critics, Suppes semantics, and Andreass semantics. Since Suppesand Andreass semantics are meant to be elaborations of Carnaps informal re-marks, this provides another argument for the adequacy of Przeeckis formalismas a semantics of the Received View.

5 Analyticity in vague languages

There is no question that most if not all terms are vague. This holds on empiricalgrounds for -terms and, due to Winnies argument, especially for -terms overU. Since Przeecki assumes that the intended -structuresN are determinedsolely by ostensive interpretation and that the negative extension of a term is de-termined analogously to the positive extension of a term (by comparison withthe negative standards), he can conclude that -terms are completely vague overthe unobservable objects. This is in contrast to Carnaps assumption that the-terms are precise overU, simply because their extension is disjoint fromU(andhence the permutation argument given in connection with Andreass semanticsdoes not apply). Whether Carnaps (and Suppes) or Przeeckis (and Andreass)

view on the interpretation of-terms is more apt depends on whether the nega-tive extension of a term canonlybe determined by comparison with the negativestandards. I at least doubt that this is in general the case, since, for example, Iexpect that the positive standards of blue are sufficient do definitely exclude akiss on the neck from its extension.29 In any case, nothing very important hingeson the decision: To move from Przeeckis view to Carnaps, every -term may

29While this is an empirical question in the case of observable objects, things are somewhatdifferent for unobservable objects since these cannot even be referred to by ostension. Hence onecannot ask whether this is red by pointing to it or, in the case of the kiss, performing it. I willhave to leave this line of thought for another time.

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be substituted by a new-term that is co-extensional except in that its negative

extension includesU. To move from Carnaps view to Przeeckis, every -termmay substituted by a new -term that is co-extensional except that its neutralextension includesU.

The move from Przeeckis view to Carnaps introduces analytic -sentences,which state for every -term that it does not apply to unobservable objects. Themove from Carnaps view to Przeeckis does not seem to introduce any suchstatements, since it only restricts the positive and negative extensions. Thereforethe new set of intended interpretationsNis a superset of the old set, and thusrenders at most as many sentences true and at most as many sentences false as theold. In that sense, Przeeckis view may seem preferable for allowing the assump-tion of-terms without analytic sentences. But it does not allow this; specifi-

cally, languages with vague -terms sometimes entail the existence of analytic-sentences.

Andreass semantics provides one instance of this phenomenon: Because theintended structures do not fix the extensions of the intended-structures toU,any such restriction by becomes non-empirical and thus analytic. In the discus-sion above, this is expressed by the claim thatA is not the set of all -structuresoverO U, so typically not all analytic sentences are tautological. More gener-ally, whenever := {:N }is not maximally consistent, there will be an-sentence that is analytic, since its truth value is not determined byN, which,by assumption, contains all empirical information there is. Only if a language istoo weak to distinguish between at least some of the elements ofNdo the vague

terms fail to lead to analytic sentences. This result has the interesting corollarythat Carnaps use of the Ramsey sentence as the synthetic component of a the-ory is generally justified only if all-terms are completely precise, or at least soprecise that even a higher order sentence cannot distinguish between their differ-ent extensions. Otherwise, some theories Ramsey sentences would have analyticcomponents, which is impossible by assumption. Thus Carnaps use of the Ram-sey sentence is incompatible with Andreass and Przeeckis semantics, which sug-gests that Carnap indeed assumed that unobservable objects are in the negativeextension of all -terms.

While the analytic sentences in Andreass semantic are easily determined, thegeneral case suggests an interesting and possibly deep puzzle.Nis assumed to be

determined by language and world together, and the argument just given showsthat mostNof vague languages lead to analytic -sentences. But sinceNis inpart determined by the world, there is no guarantee that the analytic -sentenceswill be the same for all worlds. Thus an-sentence may be clearly analytic, butthat it is clearly analytic may be an empirical fact.

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6 The semantics of scientific theories

Przeeckis formalism fulfills the central desiderata for a successful explicationof the semantics of the Received View. For one, it is broadly compatible withHempels and especially Carnaps elaborations of the Received View. This isshown by its equivalence or near-equivalence with Suppes and Andreass seman-tics, which are meant to explicate the Received View as well. More importantly,however, Przeeckis semantics provides a rigorous framework for many discus-sions among philosophers of science, including Putnam, French and DaCosta,Demopoulos, Ketland, and, in part, van Fraassen. Finally, and most significantly,Przeeckis semantics suggests the further investigation of vagueness, approxima-tion, Ramsey and Carnap sentences, analyticity, and much more that I have

not touched on in this discussion. Hence there are good reasons to considerPrzeeckis semantics the best we have for scientific theories, and, I would argue,for large parts of philosophy in general.

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