THE CONSISTENCY OF THE NAÏVE

7/27/2019 THE CONSISTENCY OF THE NAVE

1/27

THE CONSISTENCY OF THE NAVETHEORY OF PROPERTIES

B H F

If properties are to play a useful role in semantics, it is hard to avoid assuming the nave theory ofproperties: for any predicate(x), there is a property such that an object o has it if and only if(o).Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic isweakened appropriately; the main difficulty is finding a semantics that can handle a conditionalobeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued,with the values only partially ordered. Can the solution be adapted to nave set theory? Probably not,but limiting nave comprehension in set theory is perfectly satisfactory, whereas this is not so in aproperty theory used for semantics.

I. INTRODUCTION

According to the nave theory of properties, for every predicate (x) there isa corresponding property x(x). Moreover, this property x(x) is instan-tiated by an object o if and only if (o). More generally, the nave theoryinvolves the following nave comprehension schema:

NC. u1 ...uny[Property (y) x(xinstantiatesy(x, u1 ... un))].

This nave theory of properties has many virtues, but it seems to have been

shattered by (the property version of) Russells paradox.Seems to have been shattered? There is no doubt that it wasshattered, ifwe presuppose full classical logic. Let us use the symbol to mean instan-tiates. The Russell paradox involves the Russell property Rcorrespondingto the predicate does not instantiate itself. So according to the navetheory, x[xR (xx)]. Therefore in particular,

(*) RR (RR).

But (*) is classically inconsistent.

There are two solution routes (routes for modifying the nave theory)within classical logic. The first says that for certain predicates, such as does

The Philosophical Quarterly, Vol. ,No. January ISSN

The Editors ofThe Philosophical Quarterly, . Published by Blackwell Publishing, Garsington Road, Oxford , UK,and Main Street, Malden, , USA.


2/27

not instantiate itself, there is no corresponding property. The second saysthat there is one, but it is not instantiated by what you might think: there areeither (i) cases where an object o has the property x(x) even though (o),

or (ii) cases where an object o does not have the property x(x) even though(o). In particular, when (x) is does not instantiate itself, the Russellproperty is either of sort (i) or of sort (ii). This second solution route sub-divides into three variants. One variant commits itself to a solution of type(i): the Russell property instantiates itself, but none the less has the propertyof not instantiating itself. A second offers a solution of type (ii): the Russellproperty does not instantiate itself, but none the less fails to have the proper-ty of not instantiating itself. A third variant hedges: it says that the Russellproperty is either of sort (i) or of sort (ii), but refuses to say which.

These four classical theories the three variants that admit the existenceof the Russell property and the one that denies it all seem to me proble-matic. (In the prima facie analogous case of sets, I take the approach thatdenies the existence of the Russell set to be quite unproblematic. But I takeproperties to be very different from sets in this regard, for reasons to bediscussed in the final section.) In my view we need a different sort of solutionroute, and it must inevitably involve a weakening of classical logic. It is theaim of this paper to provide one.

The idea of weakening logic to avoid the Russell paradox is not new, but

the proposal presented here is unlike many in that it saves the full navecomprehension schema in the form stated above: it saves this not only fromthe Russell paradox (which is relatively easy) but from far more virulentforms of paradox (such as the Curry paradox and its many extensions). Iknow of no other ways of saving nave comprehension in as strong or asnatural a logic.

II. BACKGROUND

If we are going to weaken classical logic to get around the Russell paradox(along with others), it is useful to look at how it is that (*) leads to contra-diction in classical logic; then we shall know which steps in the argumentfor contradiction might be denied. Actually, one well known approachaccepts contradictions, in the sense of assertions of the form A A, andwhile I do not favour it, I want my initial discussion to recognize it as analternative. For that reason, I shall stipulate that a theory is to be calledinconsistentif it implies not just a contradiction in the above sense, but any-

thing at all: the existence of Santa Claus, the omniscience of George Bushabout matters of quantum field theory, you name it. So even those who

THE CONSISTENCY OF THE NAVE THEORY OF PROPERTIES

The Editors ofThe Philosophical Quarterly,


3/27

accept contradictions will not want their theory to be inconsistent, in theway I am now using these terms. With this terminology in mind, here arethe main steps in an obvious argument to prove that (*) is inconsistent:

. (*) and RRtogether imply the contradiction(**) (RR) (RR)

since the first conjunct is one of the premises and the second conjunctfollows by modus ponens

. Analogously, (*) and (RR) together imply (**). So by disjunction elimination, (*) and (RR) (RR) together imply

the contradiction (**). But (R R) (R R) is a logical truth (law of excluded middle), so

(*) all by itselfimplies the contradiction (**). Anything that implies a contradiction implies anything whatever, and

hence is inconsistent in the most obviously odious sense of the term.

That is the argument. I have been a bit sloppy about use and mention, sinceI have defined Rto be a property, but appear to have spoken of a sentence(*) that contains it. There are several ways in which this could be maderight. One is to work in a language where we have a property-abstractionoperator, so that we could name R in the language; then that name wouldbe used in (*). A second is to replace R in (*) with a free variabley; then the

argument in the text goes over to an argument that formulae of the formy y(yy) imply contradictions, so their existential generalizations dotoo, and (NC) implies such an existential generalization. A third involves theintroduction of a convention of parameterized formulae, pairs of formulaeand assignments of objects to their free variables. Then (*) is simply aconvenient notation for the pair of yy(yy) and an assignment ofRto y, and what appears in the text is a literally correct derivation involvingparameterized formulae. Do things however you like.

Obviously there are several different ways of restricting classical logic so

as to evade the above argument for the inconsistency of (*). (I take it that theargument that (NC) implies (*) involves nothing in the least controversial, sothat it is the argument that (*) leads to inconsistency that must be chal-lenged.) I shall simply state my preferred approach, without arguing that it isbest: in my view, the most appealing way to weaken classical logic so as toevade the argument that (*) leads to inconsistency is to restrict the law ofexcluded middle, thereby undermining step (). Disjunction elimination canbe retained (even in the strong sense used in step (), i.e., allowing sideformulae). So can the odiousness of contradictions assumed in step ().

Of course, if you can evade the argument in a logic 1 that contains theodiousness of contradictions ruleAAB, you can equally evade it in a

HARTRY FIELD



4/27

logic 2 which is just like 1 but which is paraconsistent in that the ruleAABis dropped. But since classical laws like excluded middle that areabsent from 1 will be absent from 2 as well, this has no evident advan-

tages. What might have advantages, if it could be achieved, would be to savenave property theory in a paraconsistent logic in which we retain lawswhich are absent from 1, such as excluded middle, but I know of nointeresting way to do this.1

Unfortunately, restricting excluded middle falls far short of giving anadequate theory. In the first place, though restricting the law of excludedmiddle blocks the above argumentfor the inconsistency ofRR(RR), itis by no means obvious that there is a satisfactory logic without unrestrictedexcluded middle in which that biconditional can be maintained. In the

second place, it is still less obvious that there is a satisfactory such logic inwhich the full nave theory of properties can be maintained.To elaborate, the most obvious ways to deal with the paradoxes in logics

without excluded middle (e.g., the property-theoretic adaptation of theKleene version of Kripkes fixed-point approach to the semantic para-doxes) do not vindicate (NC), nor do they even vindicate its weak con-sequence (*).2 The reason is that they do not contain an appropriateconditional (or biconditional).

Indeed, the main issues involved in showing the consistency of the nave

theory centre on the problem of finding an adequate treatment of, andhence of: I shall assume that ABmeans (AB) (BA). Even ifour goal were limited to the consistent assertion of the biconditional (*), thatwould rule out our definingABin terms of the other connectives in themanner familiar from classical logic, viz A B. For on that materialconditional reading of, (*) amounts to

[(RR) (RR)] [(RR) (RR)].

Assuming distributivity and a few other simple laws, this is equivalent to adisjunction of the classical inconsistencies (RR) (RR) and (RR) (RR). If we assume double-negation elimination, that is in effect justthe simple contradiction (R R) (R R); and even if double-negationelimination is not assumed, a disjunction of contradictions seems just as



1 For a discussion of some obstacles to doing it in an interesting way, see my Is the LiarSentence Both True and False?, in J. Beall and B. Armour-Garb (eds), Deflationism and Paradox(Oxford UP, forthcoming).

2 See S. Kripke, Outline of a Theory of Truth,Journal of Philosophy, (), pp. .A property-theoretic adaptation of the Kleene variant of Kripkes approach is in effect givenin P. Maddy, Proper Classes, Journal of Symbolic Logic, (), pp. , as a theory ofproper classes. I shall discuss in the final section the use of the theory to be given in this paper

in connection with proper classes. In my opinion, not only is the presence of (NC) needed forproperty theory generally, it also makes for a more adequate theory of proper classes.


5/27

inconsistent as a single contradiction. So if we put aside the paraconsistentapproaches mentioned above, it is clear that we cannot in general interpretABas AB if we want to retain even (*). And on the paraconsistent

approaches, the material conditional reading of seems inappropriate ona different ground: this reading invalidates modus ponens. (Although it isimportant not to interpret as the material conditional, the theory that Ishall advocate does posit a close relation between the two: while (AB) (A B) is not a logical truth, it is a logical consequence of the premisesA A andBB. In other words, it is only in the context of a breakdownin the law of excluded middle that the divergence between and thematerial conditional emerges.)

The first problem about getting a decent conditional, then, is licensing

the assertion of (*). But there are plenty of logics of that solve thatproblem while still being inadequate to the nave theory, for the fullcomprehension schema (NC) is not consistently assertable in them. Indeed,many of these logics fail to handle a close analogue of Russells paradox dueto Curry. The problem is this: (NC) implies the existence of a Curryproperty K, for which x[xK (xx)], where is any absurdityyou like. So KK (KK); that is,

(i) KK (KK)

and(ii) (KK) KK.

But in many logics of we have the contraction ruleA (AB) AB,on which (i) implies

(i*) KK.

But this with (ii) leads to KKby modus ponens; and another application ofmodus ponensleads from that and (i*) to .

Unless we restrict modus ponens(and it turns out that a very drastic restric-tion of it would be required), we need to restrict the contraction rule. Thisrequires further restrictions on the logic as well. For instance, given thatwe are keeping modus ponens in the form A, A B B, we certainly haveA, A (A B) B simply by using modus ponens twice; so to preventcontraction, we cannot have the generalized -introduction meta-rule thatallows passage from ,ABto AB. Indeed, even the weaker versionwhich allows the inference only when is empty should be given up: it is theobvious culprit in an alternative derivation of the Curry paradox.

It turns out, though, that the difficulty in finding an adequate treatment of is not insuperable, and that the nave comprehension principle (NC) can

HARTRY FIELD



6/27

be maintained, indeed, can be maintained in a logic that, though not con-taining excluded middle or the contraction rule, is not altogether unnaturalor hopelessly weak.3 The aim of this paper is to show this.4 Whether the

theory should still count as nave when the logic is weakened in this way is aquestion I leave to the reader.It is worth emphasizing that though the law of excluded middle will need

restriction, there is no need to give it up entirely: it can be retained invarious restricted circumstances. For instance, the notion of property isnormally employed in connection with a base language Lthat does not talkof properties; we then expand Lto a language L+ that allows for properties,including but not limited to properties of things talked about in L. (It is notlimited to properties of things talked about in Lbecause it will also include

properties of properties: indeed, it is some of these that give rise to theapparent paradoxes.) It is within the ground language Lthat most of mathe-matics, physics and so forth takes place; and the theory advocated here doesnot require any limitation of excluded middle in these domains, because aslong as we restrict our quantifiers to the domain of the ground language, wecan retain full classical logic. We can also retain full classical logic in con-nection with those special (rank ) properties that are explicitly limited so asto apply to non-properties; and to those special (rank ) properties that areexplicitly limited so as to apply to non-properties and rank properties; and

so on. Where excluded middle cannot be assumed is only in connection withcertain properties that do not appear anywhere in such a rank hierarchy,like the Russell property and the Curry property (though for other suchproperties, e.g., those whose complement appears in the rank hierarchy, ex-cluded middle is also unproblematic). Even for the problematic properties,there is no need to give up excluded middle for claims about propertyidentity; it is only when it comes to claims about instantiation of problematicproperties that excluded middle cannot be assumed in general.

I do not know if the theory here can be adapted to a theory of nave sets,

by adding an axiom or rule of extensionality; I shall discuss this in the finalsection, and also why the matter is much less pressing for sets than for pro-perties. But if it is possible to develop a theory of nave sets, it seems unlikelythat we would be able to maintain excluded middle for identities betweennave sets (e.g., between the empty set and {x|x=xKK}, where Kis the



3 For instance, the conditional obeys contraposition in the strong form (AB) (B A). Also, when A B and CB results from CA by substitutingB for one or moreoccurrences ofA, then CA CB; so (NC) yields that y x(x) is everywhere inter-substitutable with (y), even within the scope of a conditional.

4 The approach I shall be giving is an adaptation of the approach to the semantic para-

doxes developed in my A Revenge-Immune Solution to the Semantic Paradoxes, Journal ofPhilosophical Logic, (), pp. .


7/27

Curry set, defined by analogy with the Curry property). Because of this, anave set theory, if possible at all, would have an importantly differentcharacter from the nave property theory about to be developed.

III. THE GOAL

I have said I want a consistent nave theory of properties, but actually what Iwant is stronger than mere consistency. It is time to be more precise.

Let Lbe any first-order language with identity. Since I shall not want toidentifyABwith AB, it is necessary to assume that is a primitiveconnective, along with , and/or , and and/or . And to avoidannoying complications about how to extend function symbols when we add

to the ontology, I shall assume that Lcontains no function symbols (exceptperhaps for zero-place symbols, i.e., individual constants). Lcan be taken tobe a language for mathematics, or physics, or whatever you like other thanproperties. (So it should not contain the terms Property or in the senses tobe introduced. If it contains these terms in other senses, e.g., for member-ship among the iterative sets of standard set theory, then imagine thesereplaced by other terms.)

Let L+ result from L by adding a new -place predicate Property and anew -place predicate meaning instantiates. For any formula A ofL, let

ALbe the formula ofL+ obtained fromA by restricting all bound occurrencesof any variable zby the condition Property(z). Let Tbe any theory in thelanguage L. Nave property theory over T is the theory T+ that consists ofthe following non-logical axioms:

I. AL, for anyA that is a closure of a formula that follows from TII. xy[xyProperty(y)]III. u1 ... uny[Property (y) x(xy(x, u1 ... un))],

where (x, u1 ... un) is any formula ofL+ in whichy is not free.

(III) is just (NC). Then a minimal goal is to show that in a suitable logic, thetheory T+ consisting of (I)(III) is always consistent as long as T itself isconsistent. (IfTis itself a classical theory, i.e., is closed under classical con-sequence, then nave property theory over T effectively keeps classical logicamong sentences of form AL, even though its official logic is weaker: for ifA1 ...An are formulae ofLthat classically entailB, thenA1 A2 ...AnBisin T, so [(u1 ... uk)(A1A2 ... AnB)]L is in T+, and this is the same as(u1 ... uk)[Property(u1) ... Property(uk) (A1LA2L ...AnLBL)].)

The minimal goal is to show that T+ is consistent whenever T is, but I

actually want something slightly stronger: I want to introduce a kind of multi-valued model for L+ (infinite-valued, in fact), and then prove

HARTRY FIELD



8/27

G. For each classical model MofL, there is at least one model M+ ofL+

that validates (II) and (III) and hasMas its reduct

where to say that M is the reduct of M+

means roughly that whenyou restrict the domain ofM+ to the things that do not satisfy Property(and forget about the assignments to Property and to ) then what youare left with is just M. Since the connectives of L+ will reduce to theirclassical counterparts on the reduct, the fact that M is the reduct ofM+

will guarantee the validity of axiom schema (I); so ifMsatisfies T,M+ satis-fies T+.5

There is good reason why (G) says at least one rather than exactly one:we should expect that most or all models of T+ can be extended to models

that contain new properties but leave the propertyless reduct unchanged.The proof that I shall give yields the minimal M+ for a given M, but ex-tensions of the model with the same reduct could easily be given.

I shall prove (G) in a classical set-theoretic meta-language, so anyone whois willing to accept classical set theory should be able to accept the co-herence of the non-classical property theory to be introduced.

IV. THE SEMANTIC FRAMEWORK

The goal just enunciated calls for developing a model-theoretic semanticsfor L+ in a classical set-theoretic meta-language. The semantics will be multi-valued: in addition to (analogues of ) the usual two truth-values there will beothers, infinitely many in fact.

IV.. The space of values

My approach to achieving the goal is an extension of the Kripke-styleapproach previously mentioned, but it needs to be substantially more com-

plicated because of the need for a reasonable conditional.One complication has to do with method of proof: the new conditional isnot monotonic in the sense of Kripke, which means that we cannotmake do merely with the sort of fixed-point argument that is central to hisapproach (though such a fixed-point argument will play an important role inmy approach too).



5 The reason for the roughly in the definition of reduct is that M is a classical model,whereasM+ will be multi-valued; so its reduct will have to assign objects that live in the largerspace of values. The larger space of values will contain two rather special ones, to be denoted1 and 0, and we can take A has value 1 inM+ and A has value 0 inM+ to correspond to A is

true inM and A is false inM, whenA is in L. The reduct ofM+ will not strictly beM, but itwill be the {0,1}-valued model that corresponds toMin the obvious way.


9/27

The other complication is that the semantic framework itself should begeneralized: whereas Kripke uses a -valued semantics, I shall use a modeltheory in which sentences take on values in a subspace W of the set F of

functions from Pred() to {,,}, where is an initial ordinal (ordinal withno predecessor of the same cardinality) that is greater than , and wherePred() is the set of its predecessors.6 (I do not fix on a particular value ofat this point, because I shall later impose further minimum size require-ments on it.)

Which subset ofF do I choose as my W? If is a non-zero ordinal lessthan , call a memberfofF-cyclicif for all and for which +


10/27

Also, iff() is and fW, thenfassumes the value arbitrarily late,viz at all right-multiples of f. (By f, I mean the smallest for which f is-cyclic.) Also, for anyfandgin W, there are


11/27

independent of, since fand gare (-cyclic and hence) -cyclic. Similarlyfor the -independence of the condition for (fg)(+)=. Case : =.We need that (fg)() = for all . The reason is that for any


12/27

||p(t1 ... tn)||s is p(dens(t1) ... dens(tn)), which in the future I shall also writeasp*dens(t1) ... dens(tn)

||A||s is (||A||s)*||AB||s is {||A||s, ||B||s}||AB||s is {||A||s, ||B||s}||xA||s is {||A||s|s differs from sexcept perhaps in what is assigned to

the variable x}||xA||s is {||A||s |s differs from sexcept perhaps in what is assigned to

the variable x}||AB||s is (||A||s, ||B||s).

For the quantifier clause to make sense in general, it is essential that the

domain of quantification has lower cardinality than . But this is no real re-striction: it is simply that if you want to consider models of large cardinalityyou have to choose a large value of. (Recall that the goal (G) is to establisha strong form of consistency in which for any classical starting model Mforthe base language L, there is a non-classical modelM+ in L+ that hasMas itsreduct. There is no reason why the space of values used for M+ cannotdepend on the cardinality ofM.) So I shall take my non-classical model M+

to be a W-model for some initial ordinal of cardinality greater than thatofM(as well as being greater than ).M+ will have a cardinality that is the

maximum of the cardinalities ofMand of, so this restriction will suffice forthe quantifier clause to be well defined.

I have written the valuation rules for ordinary formulae, but in the futureI shall adopt the convention of using parameterized formulae in which wecombine the effect of the formula and the assignment function in our nota-tion by plugging a meta-linguistic name for an object assigned to a variablein for free occurrences of the variable in the displayed formula; that willallow me to drop the subscript s, and simplify the appearance of otherclauses. For instance, the clauses for atomic formulae and universal quanti-

fications become

||p(o1, ... on)|| is the functionfp, o1 ...on that takes any intop(o1 ... on)

||xA|| is the function {||A(o)|||oD}.

(Sometimes I shall make the parameters explicit, e.g.,

For all o1 ... on, ||xA(x, o1 ... on)|| is the function {||A(o, o1 ... on)|||oD}

but the absence of explicit parameters should not be taken to imply that

there are no parameters in the formula.)




13/27

V. A MODEL FOR NAVE PROPERTY THEORY

V.. The basics

The next step is to specify the particular model to be used for nave propertytheory. I am imagining that we are given a model Mfor the base languageL. We can assume without real loss of generality that |M| (the domain ofM)does not contain formulae ofL+, or n-tuples that include such formulae; forif the domain does contain such things, we can replace it with an isomorphiccopy that does not. With this done, let E0 be |M|. For each natural number k,we define a set Ek+1 ofersatz properties of level k+. A member ofEk+1 is a triple

consisting of a formula ofL+

, a distinguished variable ofL+

, and a functionthat assigns a member of{Ej|jk} to each free variable ofL+ other thanthe distinguished one, meeting the condition that if k> then at least oneelement ofEk is assigned.8 If(x, u1 ... un) is the formula, xthe distinguishedvariable and o1 ... on the objects assigned to u1 ... un respectively, I shall usethe notation x(x, o1 ... on) for the ersatz property. Let Ebe the union of allthe Ek for k, and let |M+| be |M|E. (The cardinality of |M+| is thus thesame as that of|M| when |M| is infinite, and is 0 whenMis finite.) The onlyterms ofL+ besides variables are the individual constants ofL; they get the

same values inM+

as inM.I hope it is clear that the fact that I am taking the items in the domain tobe constructed out of linguistic items does not commit me to viewing pro-perties as linguistic constructions; the point of the model is simply to give astrong form of consistency proof, i.e., to satisfy goal (G), and this is the mostconvenient way to do it.

Putting aside the unimportant issue of the nature of the entities in thedomain, the domain does have a very special feature: all the properties inthe model are ultimately generated (in an obvious sense I shall not bother to

make precise) from the entities in the ground model by the vocabulary of theground model; so the model contains the minimal number of properties thatare possible, given the ground model. It is useful to consider such a specialmodel for doing the consistency proof for nave property theory, but notall models of nave property theory will have this form (as is obvious simplyfrom the fact that if we were to add new predicates to the ground modelbefore starting the construction, we would generate new properties).

To complete the specification ofM+ we must specify an appropriate ,and then assign to each n-place atomic predicate p a W-extension: a

HARTRY FIELD


8 The exception for k= is needed only for formulae that contain no free variables beyondthe distinguished one.


14/27


15/27

In fact there is an easy way to find out what function fKK is. First,fKK()cannot be ; for the only function in W that has value at is 1, and sofK K() would have to be ; but fK K() can only be iffK K() is . By a

similar argument, fK K() cannot be . It follows that fK K() must be ,and from that it is easy to obtain all the other values successively. (The valueis at and all limit ordinals, at odd ordinals, and at even successors.)

Other cases will not be so simple. Consider a more general class ofCurry-like properties, the properties of the form x(xxA(x; o1 ... on)).LettingQbe the property for a specific choice ofA and ofo1 ... on, we want|QQ| to have the same value as QQA(Q; o1 ... on). But A can be aformula of arbitrary complexity, itself containing and , and the ois canthemselves be odd properties of various sorts. It is not obvious how the

reasoning that works for the simple Curry sentence will work moregenerally.In many specific cases, actually, it is also easy to come up with a

consistent value for the sentences involved often a unique one, though incases like the parameterized sentence x(x x) x(x x) it is far fromunique unless further constraints are added. But it is one thing to figure outwhat the value would have to be in a lot of individual cases, another to comeup with a general proof that values can always be consistently assigned. Andit is still another thing to specify a method that determines a unique value

for any formula relative to any assignment function. How are we to dothese further things? The reasoning about the valuation of K K suggeststhat for > we might be able to figure out the function Z that assigns toeach parameterized formula B the value ||B||(), if only we had the func-tions Z for


16/27

to construct the assignment to by approximations when the assignmentto is given. And we need to do these two approximation processestogether somehow; this is where most of the difficulties arise.

V.. Constructing the valuation of: first steps

Now to business. The construction will assign values in the set {,,} toformulae relative to two ordinal parameters and (as well as to an assignment ofvalues to the variables in the formula). will initially be unrestricted; willbe restricted to being no greater than , the initial ordinal of the cardinalitythat immediately succeeds that of |M+|. (Forget about for now; we shallultimately take it to be at least , but it is not yet in the picture.) We order

pairs lexicographically, that is, iffeither


17/27

form x(x, b1 ... bn) for some specific formula and objects b1 ... bn. Inthat case, |o1o2|, is

if for some


18/27

from Pred() to {,,}. To do better, we need to explore what happens athigher and higher values of. That is the goal of the next subsection.

Before proceding to that, I note a substitutivity result:

FP-Cor. . IfA is any parameterized formula, and A results from it byreplacing an occurrence ofy x(x, o1 ... on) by an occurrence of(y, o1 ... on), then for each , |A|=|A|.

It is worth emphasizing that this holds even when the substitution is insidethe scope of . The proof (whose details I leave to the reader) is aninduction on complexity, with a subinduction on to handle the condi-tionals and the identity claims. (It is essential that the assignment of values toconditionals for = did not give conditionals different values when they

differ by such a substitution; but it clearly did not do that, since it gave allconditionals the value .)

V.. The fundamental theorem

Is there a way to get from our single-bar semantic values relative to levels to double-bar semantic values in a space W? A nave thought might be todefine ||o1o2|| as the function that maps each into |o1o2|. But it shouldbe obvious that this does not work: it does not meet the regularity conditionthat we need. (It does work in a few simple cases, like fK K, but not in

general.) The fact that all conditionals have value at = is the mostobvious indication of this.

But something like it will work: I shall show that there are certain ordinals, which I shall call acceptable ordinals, with some useful properties. It turnsout that if is any acceptable ordinal and is any sufficiently larger accept-able ordinal that is also initial (so that it is equal to +), then we can usethis for our value space W, and we can define ||o1o2|| as the functionthat maps each


19/27

that is equivalent to . I shall make use of an obvious lemma which thereader can easily prove by induction on :

Lemma. If is equivalent to then for any , +is equivalent to +.

Now let FINAL be the set of functions v that are represented arbitrarilylate, i.e., are such that ()()(v=H).

Prop. . FINAL .

Proof: if it were empty, then for each function vfrom SENT (the set of sen-tences) to {,,}, there would be an v such that (v)(vH). Let bethe supremum of all the v. Then for each function vfrom SENT to {,,},vH. Since H itself is such a function, this is a contradiction. QED.

Call an ordinal ultimate if it represents some v in FINAL; that is, if()()(H=H).

Prop. . If is ultimate and then is ultimate.

Proof: if, then for some , =+. Suppose is ultimate. Then forany , there is an which is equivalent to . But then , i.e., +, isequivalent to + by the Lemma, and +; so is ultimate. QED.

Call a parameterized formula A ultimately good if for every ultimate ,|A| = ; ultimately badif for every ultimate , |A|=; and ultimately indeterminate

if it is neither ultimately good nor ultimately bad. If is a class of para-meterized formulae, call an ordinal correct forif

ULT. For anyA, |A|= iffA is ultimately good, and |A|= iffA isultimately bad.

(It follows that |A|= iffA is ultimately indeterminate. Also, ifis closedunder negation then the clause for follows from the clause for .) And callan ordinal acceptableif it is universally correct, that is, correct for the set of allparameterized formulae. (So if two ordinals are acceptable, they are equi-

valent, i.e., they assign the same values to every parameterized formula.)Prop. . If is ultimate, then the following suffices for it to be correct for :

for allA, ifA is ultimately indeterminate then |A|= .

Proof: since is ultimate, anything that is ultimately good or ultimatelybad has the right value at , so only the ultimately indeterminate A have achance of being treated incorrectly. QED.

I now proceed to show that there are acceptable ordinals; indeed, arbi-trarily large ones. Start with any ultimate ordinal , however large. Then

every member of FINAL is represented by some ordinal ; and sinceFINAL is a set rather than a proper class, and is ultimate, there must be a

HARTRY FIELD



20/27

such that + is equivalent to and every member of FINAL is repre-sented in the interval [, +). Finally, let be +. I shall show that isacceptable.

Prop. . For any n, every member of FINAL is represented in the interval[+n, +(n+)).

Proof: from the fact that + is equivalent to , a trivial induction yieldsthat for any finite n, +n is equivalent to ; so for any finite n and any < , + n + is equivalent to + . So anything represented in theinterval [, +) is represented in [+n, +(n+)). QED.

Prop. . is correct with respect to all conditionals.

Proof: since is ultimate, any ultimately good A has value at , and anyultimately badA has value at . It remains to prove the converses for thecase whereA is a conditional.

Suppose |B C| = . Then for some


21/27

IfA is a conditional, then by the valuation rules |A|,0 is |A|,, i.e., |A|,which (whenA is ultimately indeterminate) is by Prop. ().

The other cases use the claim that (***) holds for simpler sentences, and

are fairly routine. E.g., ifA is xB, then ifA is ultimately indeterminate,there is a t0 such thatB(t0/x) is ultimately indeterminate and for no tisB(t/x)ultimately bad. But for any t for which B(t/x) is ultimately indeterminate,includingt0, the induction hypothesis gives that |B(t0/x)|,0= ; and for anytfor whichB(t/x) is ultimately good, |B(t/x)|, is and so |B(t/x)|,0 {,}.So by the valuation rules for , |xB|,0= . QED.

V.. The valuation of concluded

We are now ready to choose the value of for our space W, and to choose

a W

-extension for .The acceptable ordinal just constructed was chosen to be bigger thanan arbitrarily big; so the fundamental theorem gives that acceptable ord-inals occur arbitrarily late. Let 0 be the first acceptable ordinal and 0+be the second; then an ordinal is acceptable iffit is of the form 0+.

If I had not already imposed stringent requirements on the space ofsemantic values (so as to be able to develop the semantics generally with aslittle bother as possible), I could now simply let o1o2 be the function thatmaps each


22/27

The last thing that must be shown, to show that (E) does in fact succeedin assigning a W-extension to for the recently chosen, is that each||o1 o2|| is regular. But that is clear: if it maps into either or , then

|o1 o2|0 is or , so by acceptability, o1 o2 is either ultimately bad orultimately good, and so |o1 o2|0 + is either for all or for all ; so||o1 o2|| is either 0 or 1.

So we have a W-model. All that now remains for the consistency proofis to verify that the model validates axiom schema (III). This requires thefollowing:

Theorem. For each parameterized formula A, ||A|| is the function thatassigns to each ordinal


23/27


24/27

certain laws involving it to hold in full conditional form or only in the formof rules. But the main problem seems to be independent of these issues, for itdoes not involve identity: the issue is how we can secure the rule

Set(x) Set(y) w(wxwy) z(xzyz)and preferably also the reverse negated rule

z(xzyz) w(wxwy)

without any weakening of the nave comprehension schema (III). Thenatural way to try to secure these rules is to modify the treatment of, sothat what the fixed-point construction ensures is not the (FP) of V., butrather, (FP) only for the special case =, supplemented with

FP-Mod. For all > and all o and all and all b1 ... bn,|ox(x, b1 ... bn)|=|x[xo(x, b1 ... bn)]|

where xy abbreviates [Set(x) x=y] [Set(x) Set(y) z(zxzy)]. (|xy| depends only on the single-bar values of membership claimsfor


25/27

This does not seem to me enough to count as nave set theory. I do not ruleit out that we might be able to do better by a cleverer construction, but itdoes not look easy.

But why do we need a nave theory of sets (or other extensional relations)anyway? We have a very neat non-nave theory of sets, namely, theZermeloFraenkel theory; and this can be extended to a nave theory ofextensional relations either artificially, by defining extensional relationswithin it by the usual trick, or by a notationally messy but conceptuallyobvious generalization of ZF that treats multi-place extensional relationsautonomously. (Formulations of ZF in terms of a relation of having nogreater rank than greatly facilitate this generalization.)

It is true that the absence of proper classes in ZF is sometimes awkward.

It is also true that adding proper classes in the usual ways (either predicativeclasses as in GdelBernays, or impredicative ones as in MorseKelley) isconceptually unsettling: in each case (and especially in the more convenientMorseKelley case) they look too much like just another level of sets, andthe fact that there is no entity that captures the extension of predicates trueof proper classes suggests the introduction of still further entities (super-classes that can have proper classes as members), and so on ad infinitum. Butonce we have properties (and non-extensional relations more generally), thisdifficulty is overcome: properties can serve the function that proper classes

have traditionally served. The rules they obey are so different from the rulesfor iterative sets (for instance, they can apply to themselves) that there is nodanger of their appearing as just another level of sets. And since everypredicate of properties itself has a corresponding property, there is no fearthat the arguments for the introduction of properties will also support theintroduction of further entities (super-properties).12

Of course, in standard proper class theories, proper classes are exten-sional, whereas properties are not. Does this show that the properties willnot serve the purposes that proper classes have been used for? No. I doubt

that extensionality among proper classes plays much of a role anyway, butwithout getting into that, one could always use the surrogate as a pseudo-identity among properties that is bound to be adequate in all traditionalapplications of proper classes; and an extensionality law stated in terms ofrather than = is trivially true. Of course, is very bad at imitating identityamong properties generally: if it were not, the problem of getting an exten-sional analogue of nave property theory would be easy. But when weconfine our attention to those properties that correspond to the proper

HARTRY FIELD


12 The general philosophical view here is quite similar to that in Maddys paper Proper

Classes, though the theory of properties on offer here is much stronger because of thepresence of a serious conditional.


26/27


27/27

anything like nave comprehension: it is central to the idea of naturalproperties that it is up to science to tell us which natural properties thereare. (It is also doubtful that we want natural properties of natural properties.

Even if we do, it seems likely that we should adopt a picture which isZF-like in that each natural property has a rank and applies only to non-properties and to properties of lower rank. But there is no need to decidethese issues here.) But in addition to the notion of natural property, there isalso a conception of property that is useful in semantics. And it is the raisondtreof such semantically conceived properties (sc-properties for short) thatevery meaningful open sentence (in a given context) corresponds to one.15

(Open sentences in the language of sc-properties are themselves meaningful,so they must correspond to sc-properties too.) Again a ZF-like solution in

which the existence of the properties is denied goes against the whole pointof the notion.In a theory of semantically conceived properties, then, it is unsatisfactory

to say that for a meaningful formula (x), there is no such thing as x(x). Italso seems unsatisfactory to say that though x(x) exists, the things thatinstantiate it are not the o for which (o). In classical logic, those are the onlytwo options, but what I have shown in this paper is how to develop a thirdoption in which we weaken classical logic. If we do that, then we can retainthe nave theory of (sc-)properties, and that has an important payoffwhich

has no analogue in the case of sets. At the very least, the value of a nave settheory is unobvious; but the value of a nave theory of satisfaction is over-whelmingly clear, and it is almost as clear that we ought to want a navetheory of sc-properties if we are going to posit sc-properties at all.

You may still want a nave theory of sets, for whatever reason; but whatyou need is a nave theory of properties and a nave theory of satisfaction. Isuspect that you cannot get what you want; but you get what you need.

New York University

HARTRY FIELD

15 Or rather, every meaningful open sentence with a distinguished free variable corresponds

to a sc-property relative to any assignment of entities, possibly including sc-properties, to theother free variables.

THE CONSISTENCY OF THE NAÏVE

Documents

Transcript of THE CONSISTENCY OF THE NAÏVE