Speaking of Everything Author(s): Richard L. Cartwright Source: Noûs, Vol. 28, No. 1 (Mar., 1994), pp. 1-20 Published by: Blackwell Publishing Stable URL: http://www.jstor.org/stable/2215917 Accessed: 12/03/2009 03:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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NOUS28:1 (1994) 1-20

Speaking of Everything RICHARD L. CARTWRIGHT

MassachusettsInstituteof Technology I

To speak of everything, as I shall speak of it, is to asserta propositionthatcan be expressedin a quantifiedsentenceof a first-orderlanguagethe variablesof which range over everything there is. That we can thus speak of everything seems plainly true. We can say that everythingis self-identical: (x)x = x;

that everything is mortal if human: (x)(Hx -- Mx);

that nothing is a unicorn: (x)>Ux.

In each case the variable 'x' ranges over everythingthere is. Not that the variables of a first-orderlanguage always range so widely. Their values may be limited to the naturalnumbers,or the pure sets, or the trees in the garden. But no such limitationneed be imposed:the naturalnumbers,the pure sets, and the trees in the garden-all of them, along with any other objects there are-can simultaneously be the values of the variables of a first-orderlanguage. So it seems, at any rate, and so it sometimes appearsin logical writings. In expounding the system of set theory known as New Foundations, Quine remarked that "the variables are to be regardedas taking as values any objects whatever";' and the variables of the system of his MathematicalLogic were similarly to be understoodas unrestrictedin range.2Earlier,in The Principles of Mathematics, Russell not only permittedunrestrictedvariablesof quantification ?D1994 Basil Blackwell, Inc., 238 Main Street, Cambridge,MA 02142, USA, and 108 Cowley Road, Oxford OX4 lJF, UK.

2 NOLS

but also saw them as alreadypresent, if only implicitly, in the propositions of mathematics:"in every propositionof pure mathematics,when fully stated, the variableshave an absolutelyunrestrictedfield."3 It is truethatFrege's individual variables, as they are now called, ranged only over so-called objects, exclusive of functions; but they ranged over them all. Judged by some recent pronouncements,however, logicians who go in for unrestrictedquantificationare out of date, or at least out of step. Peter Geach says that "few modem logicians wholly agree with Frege and Quine on this matter."4In the first edition of his book on Frege's philosophy of language, Michael Dummett went further:"the one thing we may confidently say that no modem logician believes in is wholly unrestrictedquantification."5In the second edition, Dummettretreated:unwilling to count Quine a pre-modern,he changed 'no' to 'hardly any';6 and in still anotherplace he is content to say that "the overwhelming majority of logicians.. .do not think it possible intelligibly to quantify over all objects whatever."'7 Can an overwhelmingmajorityof modernlogicians be wrong?Not when they speak ex cathedra, one naturallythinks;and if in the presentinstancethey do not so speak, to cite them is irrelevant.Dummett'sremarksthus hint at some logical mistake involved in the use of variablesunrestrictedin range;and the impression that he thinks unrestrictedquantificationhas been demonstratedto be somehow illegitimate is confirmedby his saying that "the one lesson of the set-theoretical paradoxeswhich seems quite certainis that we cannot interpretindividualvariables in Frege's way, as ranging simultaneouslyover the totality of all objects which could meaningfullybe referredto or quantifiedover."8 A lesson need not be a logical consequence; yet we are apparentlyto understandthat the settheoretic paradoxes somehow show that the variables of a first-orderlanguage cannot range over all objects whatever. I intend to speak on behalf of unrestrictedquantification. Objections to it come from a variety of quarters,and I cannot addressthem all. I will be especially concerned to respond to allegations that use of variables unrestrictedin range involves some logical or mathematicalmistake-some demonstrableerror that modem mathematicallogicians have finally exposed. But first some necessary explanations, distinctions, and qualifications. II The disputedproposition, the one I propose to defend, is this: any objects there are can simultaneouslybe the values of the variables of a first-orderlanguage. The modalitymakes for a certainvagueness:can-that is to say, withoutthe dire consequences, whateverprecisely they are, alludedto by those who say that the variables of a first-orderlanguage must be decently confined in range. It is generally agreed that the members of any nonempty set can simultaneously be the values of the variables of a first-orderlanguage. But the dis-

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puted proposition extends to cases in which the objects in question do not constitute a set. If x is any set, then the objects not in x are the values of the variables of some first-orderlanguage;the pure sets-that is, the sets that have only sets in their transitive closures-are the values of the variables of some first-orderlanguage;so are the ordinalnumbers,and the cardinalnumbers.Here I assume certain familiar limitations on sethood. Some such limitations are inevitable. There is no set that has as membersall and only those things that are not members of themselves. But the things that are not membersof themselves can simultaneouslybe the values of the variablesof a first-orderlanguage;so at any rate I claim. I do not exclude so-called ultimate, or proper, classes. But neitherdo I assume them. I do not need the assumption,for in orderthat certain objects be the values of the variables of some first-orderlanguage, it is on my view not necessarythattherebe some one object of which they are the members. Anyhow, the assumptioncould not be invoked across the board:there is nothing the members of which are all and only those things that are not members of themselves. It is consistent with the view I want to defend to speak of the universe of discourse of a language. Thus we may say that the universe of discourse of certain first-orderlanguages comprises the pure sets, or the ordinalnumbers,or everythingthere is. It is a convenientway of speaking, and one there is no need to forego. But it does involve a certainrisk, the risk of being understoodto imply that the universe of discourse is an object-a set, or class, or collection-of which the values of the variablesof the language are the members. The implication must simply be disavowed: to say that the universe of discourse of a language comprises the ordinal numbers is to say no more than that the ordinal numbers are the values of the variablesof the language. It is sometimes possible to list the objects in the universe of discourse of a language: they may be the numbers 1 through 10, or the people listed in the Boston telephone directory.But giving a list is for one or anotherreason usually out of the question, and we say insteadthat the universeof discourse consists of (say) the naturalnumbers,or the ordinalnumbers,or the pure sets, or the things there are. In any such case we are at the mercy of the world:which objects are the values of our variables will depend not only on what we say but also on what there is-what naturalnumbers, what ordinal numbers, what pure sets, what objects. If I tell you that the values of the variablesare trees, all the trees there are, I do not therebyensure that elms are among the values of the variables;the world must cooperate. If I tell you that the values of the variables are the pure sets, you do not yet have it that there is an infinite set among the values of the variables;for that to be true there must be an infinite pure set. A theory can for presentpurposesbe taken to be a set of sentences of a firstorderlanguage. Now the universeof discourseof a theory is simply the universe of discourse of the language of the theory. But a differentnotion is that of the

4 NOUS

ontological commitmentsof a theory,the notion of what accordingto the theory there is. And between the two notions there is no firm relationship.9 How are we to determinethe ontological commitmentsof a theory? A suggested answer is that to show a theory to be committed to a given object, or objects of a given class, we must show that the theory would be false if that object did not exist, or if that class were empty; hence that the theory requires that object, or membersof that class, in order for all its sentences to be true.10 The suggestion is that questions of the form: What according to T is there? can be recast as correspondingquestions of the form: What must exist for the sentences in T to be true? It is an initially plausible suggestion, but it will not do. Considera theorythatconsists of a single sentence to the effect thatthere is at least one tree. Whatmust exist in orderfor its single sentence to be true?At least one tree, of course. But also, according to some philosophers, the attributeof being a tree; and numbers, according to those philosophers who think them necessary existents; God, according to Kilmer. These answers need not be endorsed, but neithercan they be dismissed as perversedistortionsof the question: if it cannot be true that there is at least one tree unless there is at least one tree, and if there cannot be at least one tree unless God exists, then it cannot be true thatthereis at least one tree unless God exists. But it would be perverse, even for a theist, to say that God is something there is according to any theory that contains a sentence to the effect that there is at least one tree. The same for numbers, I suppose. Treehood, the attributeof being a tree, is doubtless a more delicate matter. But even here we can distinguish two questions: whether,on the one hand, theremust be such a thing as treehoodif it is to be truethat there is at least one tree and whether,on the otherhand, theories that contain a sentence to the effect that there is at least one tree are therebytheories accordingto which there is such a thing as treehood. It may be considerationsof this kind that led Quine to suggest a refinement. How are we to show that a theoryrequiresthis or thatobject?Answer: "Toshow that some given object is requiredin a theory,what we have to show is no more nor less than that that object is required,for the truthof the theory,to be among the values over which the bound variables range."" Again: "The ontology to which an (interpreted)theory is committed comprises all and only the objects over which the bound variablesof the theory have to be construedas ranging in orderthat the statementsaffirmedin the theory be true."'12But thus understood, the notion of the ontology of a theoryis of doubtfulutility.If treehood, numbers, and God need not be amongthe values of the variablesof our tree theory,in order

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5

for its single sentence to be true, neitherdoes anythingelse: any tree will do, as just now noticed.'3 In spite of its commitmentto therebeing at least one tree, our tree theory has an empty ontology. A circumstanceby no means unusual; for theoriesoften requirefor theirtruththattherebe some Fs, or even exactly one F, without there being one or more particularFs that must be among the values of the variables if the theories are to consist of none but true sentences. The ontological commitmentsof a theory thus typically outrunthe theory's ontology. Indeed, they sometimes outrun the theory's universe of discourse. There is no tree in the garden, we may suppose, and yet some theories say there is. There is no set of all those sets that are not members of themselves, even though naive set theory is said to say there is. The ontological commitmentsof a theory are presumablyto be found somehow in its existentially quantifiedsentences. But how exactly? In the narrowest sense, a theory is committed to there being just such things as its existentially quantifiedsentences say there are. It will be suggested that these are the explicit ontological commitmentsof the theory, its implicit commitmentsbeing the explicit commitmentsof its closure underthe relationof first-orderlogical consequence. But it may be doubtedthatthis extension goes far enough. A theory that contains an instance of the schema: (1)

(x)>Fxx & (x)(y)(z)(Fxy & Fyz -> Fxz) & (x)(3y)Fxy

may fairly be said to be committedto there being infinitely many objects. But it is not easy to see how simple inspectionof the existentiallyquantifiedsentences among its first-orderconsequences could reveal the commitment. The example suggests that in orderto discernthe ontological commitmentsof a theory, examination of the existential sentences in its deductive closure be supplementedby looking to common featuresof its models. No doubt. But we must look with an educatedeye, takingcare not to mistakeartifactsof the models for propertiesof the theory.It is truethat a theorythatcontains an instanceof (1) is committedto therebeing infinitely many objects, and it is also truethat we can recognize this by noticing that any model of the theory must have an infinite domain. But the domainwill in each case be an infiniteset, and mere presenceof an instance of (1) is not enough to saddle a theory with commitment to the existence of an infinite set, or indeed of any set at all: it all depends on what specific predicatereplaces the schematic 'F'. Perhapsno instance of (1) can be true unless thereexists an infinite set, just as it is alleged thatunless God exists it cannot be true that there is at least one tree; that is a matter to be addressed furtheron. Still, the theory need not be committedto there being an infinite set, even though it is committedto therebeing the infinitely many objects that would constitute the set. To be quite specific, consider the effect of simply dropping the axiom of infinity from Zermelo-Fraenkelset theory.The resultingtheoryremainscommit-

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ted to there being infinitely many sets-for instance, 0, {O}, {{O}},and so on. Hence every model of the theory has an infinite domain. But we surely do not want to conclude from this feature of its models that the theory is after all committedto there being an infinite set. And if the example somehow fails to be convincing, consider set-theoreticarithmetic,which is like ZF save for containing the negation of the axiom of infinity.14 When we talk of the ontological commitmentsof a theory,we are in uncertain territory.It nonethelessseems clear that if it is said thatsuch-and-suchobjects are the values of the variables of a first-order language, nothing-or next to nothing-is thereby implied as to the ontological commitmentsof theories expressible in the language. A theory in the language that contains one or more existentially quantifiedsentences is committedto there being at least one suchand-such, for an existentially quantified sentence '3xFx' says that there is at least one such-and-suchthat is an F. But specification of the universe of discourse of a language determines nothing more with respect to the ontological commitmentsof theoriesexpressiblein it. Withinthe language, thereis plenty of room for disagreementas to what there is.

III In saying that any objects there are can simultaneously be the values of the variablesof a first-orderlanguage, I do not mean to imply that there is some one first-orderlanguage in which everythingthere is to be said can be said. In order that the thesis be true, it is enough that there be a first-orderlanguage the variables of which have an unrestrictedrange and the lexicon of which contains, besides the usual apparatusof connectives and quantifiers, only a symbol for identity.Obviously, such a languageis extremelylimited in expressivepower. In general, the expressive power of a language depends not only upon its universe of discourse but also upon its lexical resources; and I do not for a moment suppose that there is a first-orderlanguage, or for that matter any language, which is in this second respect without limitations. Closely connected with limitation in expressive power is a limitation on specifiability.Wherex is a value of the variablesof a first-orderlanguage L, x is specifiable in L just in case L contains an open sentence satisfied by x and x alone. Obviously, not all the things over which the variablesof L range need be specifiable in L. Indeed, in typical cases the universe of discourse far outruns what can be specified within the language. Nothing to the contraryis implied by the thesis thatthe universeof discoursemay embraceeverythingthere is. To be a value of a variable of L is one thing; to be specifiable in L is another. IV It is sometimes assumedthat we can speak of all so-and-so's only if the so-andso's constitutesome one set-like object. Such an assumptionseems to lie behind

SPEAKINGOF EVERYTHING 7

FrankDrake'sdissatisfactionwith the impredicativeextension of ZF proposedby A. P. Morse'5 and expoundedin a more familiarform by J. L. Kelley.17 Drake writes: This impredicativeextension has an unsatisfactorynaturefrom the point of view of the cumulative type structure.If we consider V to be the universe of all sets, then classes are subcollections of things from V; if we quantifyover classes, this implies thatwe have the collection of all classes to talk about, andthe collection of all classes would be exactly the thing we should take as the next level, following all the levels used to make up V. In other words, talking about all classes is tantamountto saying that we have not taken all levels, with no end, but we have anotherone (the level of classes) which we have not used for making sets. From this point of view it is more naturalto regardclasses as not forminga completedcollection, so that we should not quantify over classes.17

The pertinentremarkis that "if we quantify over classes, this implies that we have the collection of all classes to talk about."The general principleappearsto be that to quantifyover certainobjects is to presupposethat those objects constitute a "collection,"or a "completedcollection"-some one thing of which those objects are the members. I call this the All-in-One Principle. It would be more accurate to speak of a battery of principles, varying in strength. According to one, the values of the variablesof a first-orderlanguage must constitutea set; anotherrequiresonly a class, perhapsultimate;still another, designed to accommodatetalk of all classes, requiresonly a hyper-class;'8 and so on. But a common idea runs throughthem all: the values of the variables must be in, or belong to, some one thing. The All-in-One Principle threatenssimultaneousquantificationover certain objects only if those objects do not togetherconstitutesome one set-like object, only if there is no one thing of which they are the members. It is thus of no consequence to those who would embrace the furtherprinciple that, as Benson Mates puts it, "Any thing or things whateverconstitutethe entire membershipof a class; in other words, for any things there are, there is exactly one class having just those things as members."'9But this second principleis false. There surely are objects thatdo not togetherconstitute,as its members,some one object:some objects are not members of themselves, and there is nothing the members of which are precisely the things that are not members of themselves. But then according to the All-in-One Principle those objects cannot be the values of the variables of a first-orderlanguage. It is a short step to the conclusion that absolutely unrestrictedquantificationis illegitimate:if certainobjects can simultaneously be the values of the variablesof a first-orderlanguage, then so as well can some of them. In risky language:any sub-universeof a legitimateuniverseof discourse is a legitimate universe of discourse.20 It may be protestedthat the reasoningrequiresindulgence in the very quantification its conclusion would prohibit. Appeal is made to the truthof:

Edited by Foxit Reader Copyright(C) by Foxit Software Company,2005-2008 For Evaluation Only. 8 NOUS -(3y)(x)(xEy <-

xEx),

wherein the variables have an unrestrictedrange. But perhapsit can be replied that the appeal is ratherto the provabilityof the schema: -(3y)(x)(Fxy <-

Fxx).

In any case, dialecticalmoves aside, the importantissue concernsthe All-in-One Principle itself: is there any reason to think it true? There would appear to be every reason to think it false. Consider what it implies: that we cannotspeakof the cookies in thejar unless they constitutea set; thatwe cannotspeak of the naturalnumbersunless thereis a set of which they are the members;that we cannot speak of all pure sets unless there is a class having them as members. I do not mean to imply that there is no set the members of which are the cookies in the jar, nor thatthe naturalnumbersdo not constitutea set, nor even that there is no class comprisingthe pure sets. The point is rather that the needs of quantificationare already served by there being simply the cookies in the jar, the naturalnumbers, the pure sets; no additionalobjects are required. It is one thing for there to be certainobjects;it is anotherfor there to be a set, or set-like object, of which those objects are the members.21Russell intendedto respect the distinction, I think, in his talk of the class as many and the class as one; it was no triviality,and in fact was soon held to be doubtful, that "the class as one is to be found whereverthereis a class as many."22But the terminologyis badly chosen. For a class as many is inevitablytaken to be a class as many, an object distinct from the many it comprises; and the very distinction intended is therebythreatened,or allowed to degenerateinto an obscuredistinctionbetween ways or modes in which one thing, the class per se, may be regarded. Cantor may have intendedto respect the distinctionin his talk of multiplicities,some of which are "inconsistent"in the sense that "the assumptionthat all their elements 'are together' leads to a contradiction."23For he said that an inconsistentmultiplicity "cannotbe understoodas one whole and thus cannotbe understoodas one thing."24If thatwas his intention,the terminologyis again ill chosen: witness the not uncommoninterpretationof Cantoras anticipatingtalk of ultimateclasses.25 V The All-in-One Principlemay be thoughtto derive supportfrom currentmodeltheoretic accounts of first-orderlogical truthand consequence. I think in fact it does not, but the matterneeds brief discussion. It will be enough to consider logical truth.For this purpose let p be a closed sentence of a first-orderlanguage L, and assume for convenience that the only nonlogical constantsof L arepredicates.With each predicatethatoccurs in p, we may pair a variableof L that has no occurrencein p, and this in such a way that

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no two predicatesof p are paired with the same variable. From this pairing we may derive another,which correlateseach atomic open sentence Rx, ... .Xn

that occurs in p with an (abbreviationof an) open sentence of the language of set theory, namely, Ez y,

wherey is the variableoriginallypairedwith the predicateR. If in p each atomic open sentence is replaced with its correlatedset-theoreticformula, the result is what may be called, following Quine, a set-theoreticanalogue of p.26 Now it is temptingto say that a first-orderclosed sentence is a logical truthjust in case its set-theoreticanalogues are satisfied by every sequence of sets. If to a set-theoretic analogue q of a closed first-ordersentence we prefix universal quantifiersanswering to each of the free variables of q, we obtain a closed sentence of the language of set theory, a universal closure of q. Satisfaction of q by all sequences of sets comes to truthof all universal closures of q. Hence the temptingdefinition of logical truthmay be alternativelyphrasedthus: a closed sentencep of a first-orderlanguageL is a logical truthif and only if the universalclosures of its set-theoreticanaloguesare true-or, equivalently,if and only if some universalclosure of some one of its set-theoreticanalogues is true. A naturalthought underliesthe definition, namely, that a sentence of a firstorder language is a logical truthjust in case it is true on any assignment of extensions to its predicates-any assignment, that is, which respects the polyadicity of the predicates. In spite of its naturalness,however, the thought is mistaken. A universal closure of a set-theoreticanalogue of (2)

-(x)Gx

is

(3)

(y) -(x)(x~zy).

Evidently (2) is not a logical truth;but (3) is true, for it says that no set has every set as a member.27 The example is only one of infinitely many, but it suffices to reveal what is wrong. What faults (2) is that (3), though it expresses a truth, does not do so on all relativizations-that is, all resultsof restrictingits variable 'x' to membersof a nonemptyset U and its variable'y' to subsetsof U. In general, a sentencep of a first-orderlanguage is a logical truthif and only if the universal closures of its set-theoretic analogues are true on all relativizations-all results, that is, of

10 NOUS

restricting those of its variables that occur in p to a nonempty set U and its remainingvariablesto subsets of U, or sets of orderedpairs of membersof U, or sets of ordered triples of members of U, and so on, as increasing polyadicity requires. The amendeddefinitionmay encouragethe idea thatamong the relativizations of the universalclosures of the set-theoreticanaloguesof the first-ordersentence p there must be one that expresses the intendedinterpretationof p, or at least a propositionequivalentin some strongsense to the intendedinterpretationof p. It might be thoughtthatotherwisetruthof the relativizationswould give no guarantee even of the plain truth of p. But the thought is mistaken. It is a fact, if a somewhat surprisingfact, that even if the universeof discourseof L is not a set, a closed sentencep of L is true if the universal closures of its set-theoreticanalogues are true on all relativizations to nonempty sets. The proof, if that is the right word, is well known: obviously, every provable sentence is true; by the completeness theorem, a sentence is provable if the universal closures of its set-theoretic analogues are true on all relativizations;the conclusion follows.28 VI Russell statedhis paradoxsubstantiallyas follows. Let w be the class of all those classes which are not membersof themselves. Then, whateverclass x may be, x is a memberof w if andonly if x is not a memberof x. Hence, in particular,w is a member of w if and only if w is not a member of w.29 There is a straightforwardresponse. (i) The reasoningis impeccable. (ii) The conclusion to be drawn is that there is no such class as w. (iii) This conclusion does not involve essentially the concepts class and membership.Indeed, it is a logical truth, one that can be representedas an instance of the valid schema: -(3y)(x)(Fxy

*->

-Fxx).

(iv) The conclusion surprisedthose who, like Frege,30had assumed that every instance of the schema: (3y)(x)(xEy

*->

Fx)

is true. But the paradoxshows that some instances of the schema are false. (v) Though the paradoxis in Quine's sense veridical,3' we are left with what James Thomsoncalled the task of clearingup afterwards.32Whatclasses can we consistently say there are? One can detect the straightforwardresponse, or something close to it, among Russell's various early reactionsto his paradox.In his letter to Frege of 16 June 1902,33 Russell concluded from the paradoxthat "thereis no class (as a totality) of those classes which, each taken as a totality,do not belong to themselves"and

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that "undercertaincircumstancesa definablecollection does not form a totality." In The Principles of Mathematicshe wrote: The reason that a contradictionemerges... is that we have taken it as an axiom that any propositional function containing only one variable is equivalent to asserting membershipof a class definedby the propositionalfunction. Eitherthis axiom, or the principle that every class can be taken as one term, is plainly false....34

And in a paper of late 1905 Russell stated that "what is demonstratedby the contradictions"(i.e., his own and those of Cantorandof Burali-Forti)"is broadly this: 'A propositional function of one variable does not always determine a class'."35He did not cite a first-orderschema an instanceof which is the conclusion to be drawn from the paradox.36But he saw that the concepts class and membershipare not involved essentially: he noticed that analogous reasoning leads to the conclusion thatthereis no propertyexemplifiedby all and only those propertiesthat do not exemplify themselves and no propositionalfunction satisfied by all and only those propositional functions that do not satisfy themselves.37 But Russell's adherenceto the straightforwardresponsewas never firm,38and already in 1906 he appearsdefinitely to have rejected it; for in a paper of that year39he announcedhis agreementwith Poincar' thatthe paradoxes,semanticas well as set-theoretic,"springfrom a vicious circle."40 Thenceforwardthe attitude is that in each paradoxapparentlysound reasoningleads to a contradictoryconclusion. So there is in each case a mistake-the same mistake, as Russell now thought. To "solve" the paradoxes is in part to expose the "precise fallacy" involved and in part to construct a system of mathematicallogic secured in advance against the fallacy.4' Russell's paradoxinvolves a fallacy? The suggestion may well seem off the wall. It surely cannot be denied that from: (4) (3 y)(x)(xCyx

xEx)

there follows: (5)

(3y)(y~y <-

yEy).

Neither can it be denied that (5) is false. It follows that (4) is false. Where is the fallacy? Russell's paradox would be no paradox, veridical or otherwise, were it not that the schema: (6)

(3y)(x)(Fxy

*->

implies the schema:

-Fxx)

12 NOUS (7)

(3y)(Fyy <-

Fyy)9

and were it not that (7), and hence (6), is invalid. His paradoxrequires,too, that at least in some minimal syntactical sense (4) and (5) be instances of, respectively, (6) and (7). Still, there is room for fallacy: (4) and (5) may be meaningless, and hence only in a syntacticalsense instances of (6) and (7). And that is precisely the tack Russell took. Explanationis wanted, of course. If (4) is a meaningfulinstanceof (6), then it is logically false. But meaninglessness of (4) hardly follows; after all, some instancesof (6) are meaningful.How is (4) any differentfrom 'Thereis a barber who shaves all and only those barberswho do not shave themselves'? Russell's answer is double-barreled.42Meaninglessness of (4) is alleged to resultfrom meaninglessnessof its containedopen sentence 'xEx'. But, independently of that, (4) purportsto be what we are told more thanonce no "significant statement"can be, namely, a statementabout "all classes." Thus, in Russell's version of the paradox, the defining condition of the purportedclass w is: (8)

(x)(xEw

<->

-xEx).

Contradictionensues from taking the variable 'x' here to range over all classes; for then there is no protectionagainst inference by instantiationto: (9)

wEw

<->

-wEw.

Russell concludes thatthere is "no total"of "all classes," which is to say that (8) and its ilk must be condemned as meaningless.43 But the reasoninglimps at its last step. If 'x' in (8) rangesover all classes, and if there is such a class as w, then instantiationto (9) is correct.Why not conclude simply thatthere is no such class as w? Why conclude ratherthatit is illegitimate to speak of all classes? After all, significant statementscan be made about all barbers. Russell's explanationis obscure: In this case the class w is defined by referenceto 'all classes', and then turnsout to be one among classes. If we seek help by deciding that no class is a memberof itself, then w becomes the class of all classes, and we have to decide that this is not a memberof itself, ie., is not a class. This is only possible if thereis no such thing as the class of all classes in the sense requiredby the paradox. That there is no such class results from the fact that, if we suppose there is, the supposition immediately gives rise (as in the above contradiction)to new classes lying outside the supposed total of all classes.44

But why in the face of the paradoxwould anyone "seek help"by deciding thatno class is a memberof itself, given that at no point in the reasoning is it assumed

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that-anyclass is a memberof itself? In what sense does the paradox"require"that there be such a thing as the class of all classes? How is it relevantto the paradox to point out that there is no such class? It is likely thatRussell intendedto call attentionto an instanceof what he says "all our contradictionshave in common," namely, "the assumptionof a totality such that, if it were legitimate, it would at once be enlarged by new members defined in terms of itself."45 In this instance the "totality"is the class of all classes: if it were "legitimate," it would at once "give rise to" classes lying outside it. The collection of classes thus "has no total." To get a handle on what Russell may have meant, consider again (8), the defining conditionof the purportedclass w. Russell appearsto argue:if 'x' in (8) ranges over the membersof some "legitimatetotality"U, then w is not in U; if in particularU is the totalityof all classes, then w will both be in U and not be in U. Hence the totality of all classes is illegitimate. Thus stated, the reasoningappearsto requirethe existence of w. But it can be restated so as to avoid that assumption, and so as to constitute a persuasive argumentfor the nonexistence of a class of all classes. Whateverclass U is the range of its variable 'x', (8) defines a subclass w of U; but on pain of contradiction, w is not itself in U. Hence U is no class of all classes. It is indeed a persuasiveargument.But Russell needs somethingmorethan, or perhapsotherthan, its conclusion. Therewas to be "no total"of all classes, in the sense that no significant statementcan be made about all classes. Suspicion is aroused that the All-in-One Principleis at work.46 But there is a more charitableinterpretation.Suppose that quantificationover all classes is permitted,and suppose that the usual laws of quantificationallogic are not to be disturbed.Then meaningfulnessof '-xEx' cannot be denied. For, from an innocent formula such as (3x)(y) -yEx there will follow (3x) -xEx. Given the same suppositions, (8) will count as meaningful.Perhaps,then, it was Russell's conviction that if (8) is meaningful, it must define a class-that, in general, meaningfulinstances of the schema: (3y)(x)(xEy <->Fx) must be true. That they are so accordingto the theory of types helps to explain Russell's belief that that theory has "a certain consonance with common sense which makes it inherentlycredible."47

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VII to be regardedas an opponentof unreevidently is Dummett Michael Although over all sets, or all ordinals, or quantification of strictedquantification-indeed, grounds for it must be deand his his opposition of nature cardinals-the all scribed with some care. The reason is in part that, though he more than once declares such quantificationunintelligible, he seems to allow that we can speak of everything. The view "thatthere are no quantifiedstatementswhich can be understood,or at least which can be recognized as true, when the variables are thoughtof as rangingover all objects, or all sets, or all ordinalnumbers"is, he says, "manifestlyabsurd."48We may say, and say truly,that everythingis identical with something: (x)(3y)x = y; that there is something not identical with something: (3x)(3y)

-x

=

y;

that something is a naturalsatellite of the Earth: (3x)x is a naturalsatellite of the Earth. "Likewise," Dummett goes on to say, "since the statements'For some a, a has infinitely many members'and 'For some ordinalw, w has no immediatepredecessor' are true for some interpretationsof the variablesas rangingover specific domains, they must be acknowledged as true when the variables are taken as ranging over all sets and over all ordinalsrespectively."49 It is true that Dummett's acknowledgementof the intelligibility of the statementsjust mentionedis guarded:they are statements"whichcan be understood, or at least which can be recognizedas true." But ratherthan speculate as to the distinctionhere intended,let us look at Dummett'sexplicit "gloss"on his contention that "it is impossible coherentlyto understandindividualvariables as ranging over all objects, or even over all sets, all ordinal numbers."50 What is meant...is that it is not possible to suppose that, by specifying the range of some style of individual variables as being over 'all objects', or 'all sets', or 'all ordinals', we have thereby conferred a determinatetruth-valueon all statements containing quantifiersbinding such variables (even given that the other symbols occurringin these statementshave been assigned a determinatesense). Any attempt to stipulatesenses for the predicates,relationalexpressions and functionaloperators thatwe shall want to use relativeto such a domainwill either lead to contradictionor will prompt us to concede that we are not, after all, using the bound variables to range over absolutely everythingthat we could intuitively acknowledge as being an object, a set, or an ordinal number.5'

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But this explanationis not easily understood.The variablesof a first-orderlanguage range over only what there is; so if they range over all objects, or sets, or ordinals, then, whateverwe take on intuitive groundsto be an object, or set, or ordinalwill alreadybe one of the values of our variables-or else our intuitionis just wrong. Let me put the difficulty of understandinganother way, limiting attention simply to the case of sets. Is anythingthat "we could intuitively acknowledge as being a set" in fact a set? If so, and if the variables of a certain set-theoretic language L range over all sets, then they range over everything that we could intuitively acknowledge to be a set; the second horn of Dummett's dilemma disappears,and we are left with the contentionthatno assignmentof truthvalues to the sentences of L can both respect their intended meanings and also avoid assigning truthto some contradictorysentence. If not-if, thatis, not everything "we could intuitively acknowledge as being a set" is in fact a set, and if the variables of L range over all sets, then so much the worse for our intuitive acknowledgements;again the second horn disappears. Indeed, in some of Dummett'sglosses the second horn does not appear.Thus we are told that"whenthe domainis too large, for instance, when it is the totality of all sets, in the intuitive sense of 'set', or when it is regardedas being.. .the totality of all objects whatsoever,it is.. .the prime lesson of the discovery of the set-theoreticalparadoxesthatquantificationover thatdomaincannotbe regarded as yielding, in all cases, a sentence with a determinatetruth-value."52It may seem that here the first horn is absent as well. But I think not, for I take Dummett's 'cannot' to allude to it.53 The paradoxesof Cantorand Burali-Fortiare not strictly logical. Because of its relative simplicity, consider just the former. Its crucial sentence asserts the existence of a universal set: (10)

(3y)(x)xEy.

We have already seen good reason to assign (10) falsity. The argumentwas in effect that its negation follows from: (11)

(z)(3y)(x)(xEy

<-> xEz

& -xEx),

one of the so-called axioms of separation.Though (11) is not a logical truth,the principle on which it is based is surely hardto deny: if z is a set and if p is any condition meaningfulfor the membersof z, then thereexists the set of members of z that satisfy p. Where is the threatof contradiction? Now Dummett himself subscribes to the general principle just enunciated, takes (11) to be an instance, and infers the negation of (10).54 Curiously,however, he does so in the interests of deploring unrestrictedquantification,even quantificationover all sets. Why not say ratherthatsettlingthe truthvalue of (10)

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removes any threatCantor'sparadoxmay have posed to a consistent interpretation of the language of set theory? However the question is to be answered, it is clear that Dummettthinks that the negation of (10), or some propositionclosely relatedto it, bears somehow on the question of the limits of quantification. Here is a representativepassage: We neither need nor can follow Frege in supposing that one single all-embracing domainwill serve for all uses of individualvariables:for the most directlesson of the set-theoreticparadoxesis that, at least when we are concernedwith abstractobjects, there is no one domain which includes as a subset every domain over which we can legitimately quantify:we cannot give a coherent interpretationof a language, such thatevery sentence of the languagecan be taken as having a determinatetruth-value, by taking the individualvariablesto range over everythingthat answers to the intuitive notion of a set, or that of a cardinalnumberor that of an ordinal.55

I am uncertainas to the relationsof logical dependencyDummettintends. But it is plausible to take him to be assertingamong other things that "the most direct lesson of the set-theoreticparadoxes"is the proposition: (12) There is no one domain which includes every domainover which we can legitimately quantify;56 and that consequently we cannot "follow Frege in supposingthat one single allembracingdomain will serve for all uses of individualvariables." Elsewhere Dummett puts the alleged lesson of the set-theoretic paradoxes somewhat differently: "the one lesson of the set-theoreticalparadoxes which seems quite certain is that we cannot interpretindividual variables in Frege's way, as ranging simultaneously over the totality of all objects which could meaningfully be referredto or quantifiedover." Presumablythe reason is that there is no such totality: no totality has in it all the things that could be meaningfully referredto or quantified over.57 If we can take 'totality' to mean the same as 'domain', then we have: (13) There is no one domain that has in it everythingthat can legitimately be taken to be a value of a variable. How are (12) and (13) related?I suspect Dummetttakes them to be equivalent.58 But are they? It seems clear that (13) follows from (12): if there is a domain U that has in it everythingthatcan legitimatelybe takento be a value of a variable, and if D is a domain over which we can legitimately quantify,then surely D is included in U; thus if (13) is false, so is (12). It is not clear, however, that (12) follows from (13). One might try to arguefrom (13) to (12) as follows: if D is an arbitrarydomain, then by (13) there are objects not in D that can legitimately be

SPEAKING OF EVERYTHING

17

taken to be values of variables; where E is the domain constituted of those objects, E is not includedin D; and hence D does not include every domain over which one can legitimately quantify.But the argumentassumes: (14) Any objects that can be taken to be the values of the variables of a first-orderlanguage constitute a domain. Withoutthe assumptionthere remainsthe possibility that some domain includes every domain over which we can legitimately quantify but is nevertheless not exhaustiveof all the things thatcan be values of variables,thereremainingthings which can be values of variables but which together constitute no domain. Perhaps the possibility is remote. I mention it only to highlight the role of (14). For from neither (12) nor (13) alone does it follow that unrestrictedquantification is in any way illegitimate. But conjoin either with (14) and the illegitimacy follows. It is thus (14) that does the work. And (14) is the All-in-One Principle. Dummett'sargumentseems to be simply that since there is no universal "domain,"and since the All-in-One Principleis true, unrestrictedquantification is illegitimate. Perhapsthe first premise can reasonablybe described as a lesson of the set-theoreticparadoxes;the second cannot.59 Notes 1W. V. Quine, From A Logical Point Of View (Cambridge, Mass.: HarvardUniversity Press,

1953), p. 81. 2W. V. Quine, MathematicalLogic, rev. edn. (Cambridge, Mass.: HarvardUniversity Press, 1951), p. 69. 3BertrandRussell, The Principles of Mathematics (Cambridge:Cambridge University Press, 1903), p. 7. 4Peter Thomas Geach, Reference and Generality, 3rd edn. (Ithaca: Cornell University Press, 1980), p. 174. 5Michael Dummett, Frege: Philosophy of Language, 1st edn. (Cambridge, Mass.: Harvard University Press, 1973), p. 567. 6Michael Dummett, Frege: Philosophy of Language, 2nd edn. (Cambridge, Mass.: Harvard University Press, 1981), p. 567. 7MichaelDummett, The Interpretationof Frege's Philosophy (Cambridge,Mass.: HarvardUniversity Press, 1981), p. 229. 8Frege, 2nd edn., p. 567. 91tis sometimes said that"theontologicalcommitmentsof a theoryare the values of its variables" (StewartShapiro, "Second-orderLanguagesand MathematicalPractice,"Journal of SymbolicLogic 59 (1985): 714-741, at p. 737). See also Michael Dummett, Frege, 2nd edn., pp. 476-477. In the paragraphsthat follow I argue against such a view. 10SeeW.V. Quine, Ontological Relativityand Other Essays (New York: Columbia University Press, 1969), p. 93. 1'Ontological Relativity, p. 94. 12"Ontologyand Ideology,"Philosophical Studies 2 (1951): 11-15. The quotedremarkis from p. 11, with italics added. See also: "OnWhat There Is," in FromA Logical Point Of View (Cambridge, Mass.: HarvardUniversity Press, 1953), pp. 1-19, especially pp. 13-14; "On Carnap'sViews on Ontology,"in The Waysof Paradox, 2nd edn. (Cambridge,Mass.: HarvardUniversity Press, 1976), pp. 126-134, especially p. 128; "Semanticsand AbstractObjects," Proceedings of the American Academy of Arts and Sciences 80-1 (1951): 90-96, especially p. 93. It should be remarkedthat Quine sometimes uses 'ontology' synonymouslywith 'universeof discourse':see, e.g., "Replies,"in

18 NOUS Donald Davidson and JaakkoHintikka, eds., Wordsand Objections:Essays on the Workof W. V. Quine (Dordrecht:Reidel, 1969), pp. 292-352, at p. 315. B3AsQuine agrees, for he writes that"thereis more to be said of a theory,ontologically, thanjust saying what objects, if any, the theory requires;we can also ask what various universes would be severally sufficient. The specific objects required, if any, are the objects common to all those universes." (Ontological Relativityand Other Essays, p. 96.) 141t should be pointed out that, on its intended interpretation,the language of set-theoretic arithmeticis presumablyunlike that of ZF in containingno infinite sets in its universeof discourse. I take the term 'set-theoreticarithmetic'from C. C. Chang and H. J. Keisler,Model Theory(Amsterdam: North-Holland, 1973), p. 509. '5A. P. Morse, A Theoryof Sets (New York:Academic Press, 1965). 16J. L. Kelley, General Topology(New York:Van Nostrand, 1955), pp. 250-281. '7FrankR. Drake, Set Theory:An Introductionto Large Cardinals (Amsterdam:North-Holland, 1974), p. 17. 18J take the term from Azriel Levy, "The Role of Classes in Set Theory,"in Gert H. Muller, ed., Sets and Classes (Amsterdam:North-Holland, 1976), pp. 173-216. Original publicationin A. A. Fraenkel,Y. Bar-Hillel,and A. Levy, Foundationsof Set Theory(Amsterdam:NorthHolland, 1973), pp. 119-153. '9Benson Mates, SkepticalEssays (Chicago: University of Chicago Press, 1981), p. 43. George Boolos called the quotationto my attention. 20CompareRussell, "MathematicalLogic as Based on the Theory of Types", in RobertCharles Marsh, ed., Logic and Knowledge (New York: Macmillan, 1956), pp. 59-102, especially p. 73. 2'When Aquinassaid, cagily, thatthe angels exist in exceeding greatnumber(SummaTheologiae 1, 50, 3), he did not mean thatthereis a set-like object of large cardinalityof which the membersare the angels. He meantjust what he said. 22ThePrinciples of Mathematics, p. 104. 23Letterto Dedekind, 28 July 1899. English translationin Jean van Heijenoort,ed., FromFrege to Godel (Cambridge,Mass.: HarvardUniversity Press, 1967), pp. 113-117. 24Letterto Jourdain,9 July 1904, in I. Grattan-Guiness,"The Correspondencebetween Georg 73 (1971): 111Cantorand Philip Jourdain,"Jahresberichtder deutschenMathematiker-Vereinigung 130. The quotation is taken from Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford: ClarendonPress, 1984), p. 286. 25See, e.g., Hao Wang, "The Concept of Set," in Paul Benacerrafand Hilary Putnam, eds., Philosophy of Mathematics,2nd edn. (Cambridge:CambridgeUniversityPress, 1983), pp. 530-570, at p. 539; Azriel Levy, Basic Set Theory(Berlin:Springer-Verlag,1979), p. 11; Paul Postal and D. T. Langendoen,The Vastnessof Language (Oxford:Blackwell, 1984), p. 19, n. 10. For doubtsas to the correct interpretationof Cantor,see W. V. Quine, Set Theoryand its Logic, 2nd edn. (Cambridge, Mass.: HarvardUniversity Press, 1969), p. 20. 26W.V. Quine, Philosophy of Logic (Cambridge,Mass.: HarvardUniversityPress, 1986), p. 51. 271 here take for grantedthat there is no set of which every set is a member. For discussion, see below. 28Theargumentappearsto have its origin in Georg Kreisel, "InformalRigour and Completeness Proofs," in Imre Lakatos, ed., Problems in the Philosophy of Mathematics (Amsterdam:NorthHolland, 1967), pp. 138-157, at pp. 153-155. The presentversion derives from Quine, Philosophy of Logic, pp. 54-55. 29See "MathematicalLogic as Based on the Theory of Types," Logic and Knowledge, p. 59; Alfred NorthWhiteheadand BertrandRussell, PrincipiaMathematica,2nd edn., vol. 1 (Cambridge: CambridgeUniversity Press, 1925), p. 60. 30See his Letter to Russell, 22 June 1902. English translationin van Heijenoort, From Frege to Godel, pp. 127-128. Frege wrote: "Yourdiscovery of the contradictioncaused me the greatest surpriseand, I would almost say, consternation,since it has shakenthe basis on which I intendedto build arithmetic." 31See The Waysof Paradox, p. 3. 32JamesF. Thomson, "On Some Paradoxes",in R. J. Butler,ed., AnalyticalPhilosophy (Oxford: Blackwell, 1962), pp. 104-119. The remarkoccurs on p. 116. 331n van Heijenoort,From Frege to Godel, pp. 124-125. 34pp. 102-103

SPEAKINGOF EVERYTHING 19 35"On Some Difficulties in the Theory of TransfiniteNumbersand OrderTypes,"Proceedingsof the London MathematicalSociety, series 2, 4 (1906): 29-53. Reprintedin Douglas Lackey, ed., Essays in Analysis (New York:Braziller, 1973), pp. 135-164. The quotationis from pp. 144-145. 36Hehad come close to doing so in ThePrinciples of Mathematics,at-p. 102. Thus: "LetR be a relation, and consider the class w of terms which do not have the relationR to themselves. Then it is impossible that there should be any terma to which all of them and no other terms have the relation R. For, if there were such a term, the propositionalfunction "x does not have the relationR to x" would be equivalent to "x has the relation R to a." Substitutinga for x throughout...we find a contradiction." 37"On Some Difficulties in the Theory of TransfiniteNumbers and Order Types", Essays in Analysis, pp. 137-138. 381ndeed,in Appendix B of The Principles of Mathematics he proposes, albeit tentatively, a version of the simple theory of types "as a possible solution of the contradiction"(p. 522). 39"On 'Insolubilia' and their Solution by Symbolic Logic," in Essays in Analysis, pp. 190-214. English version of "Les Paradoxesde la Logique",Revue de Metaphysiqueet de Morale 14 (1906): 627-650. 40Essaysin Analysis, p. 196. 41See Principia Mathematica,vol. 1, p. 1, where the authorssay thattheir system of mathematical logic "is specially framedto solve the paradoxeswhich, in recentyears, have troubledstudentsof symbolic logic and the theoryof aggregates;it is believed thatthe theoryof types, as set forthin what follows, leads both to the avoidance of contradictions,and to the detection of the precise fallacy which has given rise to them." 42CompareW. V. Quine, Set Theoryand its Logic, p. 249. 431n "MathematicalLogic as Based on the Theory of Types"Russell wrote: "When I say that a collection has no total, I mean that statements about all its members are nonsense." (Logic and Knowledge, p. 63, n.). In Principia Mathematica:"By saying that a set has 'no total', we mean, primarily,that no significant statementcan be made about 'all its members'."(p. 37) 44"MathematicalLogic as Based on the Theory of Types",Logic and Knowledge, p. 62. 45"MathematicalLogic as Based on the Theory of Types",Logic and Knowledge, p. 63. 46CompareR. M. Sainsbury,Russell (London:Routledge and Kegan Paul, 1979), p. 328. 47"MathematicalLogic as Based on the Theory of Types",Logic and Knowledge, p. 59. 48Frege, 2nd edn., p. 568. 49Frege,2nd ed., p. 568. My 'a' and 'w' are substitutesfor the symbols used in Dummett'sbook. 50Frege,2nd edn., p. 568. 51Frege,2nd edn., pp. 568-569. See also pp. 476, 530. 52Frege,2nd edn., p. 516. 53Thuswe read furtheron that "we cannot take quantificationover the totality of all objects as a sentence-formingoperationwhich will always generatea sentence with a determinatetruth-value;we cannot, in other words, interpretit classically as infinitaryconjunctionor disjunction.If we attempt to do so, we shall be led into contradiction."(p. 530) Here referenceto the second horn is explicit. It is possible, of course, that Dummett'stalk of indeterminacyof truthvalue alludes ratherto propositions, such as the continuumhypothesis, thatare undecidablerelativeto the axioms of ZF (or ZF plus an axiom of choice). I cannothere discuss the possibility with the care it no doubtdeserves. Suffice it to say that the existence of such propositionsseems at best remotelyconnected with the set-theoretic paradoxes. 54Frege,2nd edn., pp. 530-531. In fact, he affirmsa principlethatsounds more general, namely, that if z is any totality and p any condition meaningful for the members of z, then there exists the "subset"of members of z that satisfy p. But I think the sound is deceptive. If at any rate ultimate classes are among "totalities,"the principleis plainly false. Identityis well defined for the members of the class of all sets; but of course no subset of that class comprises all self-identical sets, as Dummettwell knows. That Dummettuses 'subset' in formulatinghis principleseems to me to force the interpretationI have adopted in the text. 55Frege,2nd edn., p. 476 56HereI suppress'as a subset', on the groundthatits presenceis designed merely to guardagainst possible misunderstandingsof 'includes'. 57Frege,2nd edn., p. 567. 58Thathe does think them equivalentis stronglysuggested when he says that "theoverwhelming

20 NOUS majorityof logicians... do not think it possible intelligibly to quantifyover all objects whatever,that is, over a domain which includes every domainover which it is possible intelligibly to quantify"(The Interpretationof Frege's Philosophy, p. 229). 591 am gratefulto George Boolos and Helen Cartwrightfor numerousdiscussions of mattersdealt with in this paper, and to John Etchemendyfor helpful correspondence.