Intuition and Reflection in Arithmetic: Bob Hale

INTUITION AND REFLECTION INARITHMETIC

Michael Potter and Bob Hale

II—Bob Hale

ARITHMETIC REFLECTION WITHOUT INTUITION

ABSTRACT Michael Potter considers several versions of the view that the truthsof arithmetic are analytic and finds difficulties with all of them. There is, I think,no gainsaying his claim that arithmetic cannot be analytic in Kant’s sense. How-ever, his pessimistic assessment of the view that what is now widely called Hume’sprinciple can serve as an analytic foundation for arithmetic seems to me un-justified. I consider and offer some answers to the objections he brings against it.

I

Michael Potter begins with two questions: ‘Can we give anaccount of arithmetic which does not make it depend for

its truth on the way the world is? And if so, what constrains theworld to conform to arithmetic?’ If, as I think we may, we take‘depends for its truth on the way the world is’ as an ellipsis for‘depends for its truth on the way the world is, in respects in whichit might have been otherwise’, then his first question asks whetherthe truths of arithmetic are in some sense necessary. Asking what,if so, constrains the world to conform to arithmetic is then likeasking what constrains the world to conform to the laws oflogic—an odd question, if taken at face value; but I think Pot-ter’s second question is really just a picturesque way of askinghow, if its truths are necessary, the applicability of arithmetic isto be explained.

If Potter’s first question is thus understood, then amongstthose who have favoured an affirmative answer must be reckonedKant, with whose position Potter’s paper manifests a degree ofsympathy, without being very clear on the precise extent of it.His primary concern, however, is with those who have agreedwith Kant that the truths of arithmetic are necessary but who,in contrast with Kant, have been drawn to the idea—as Potter

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expresses it—that arithmetic is ‘in some sense devoid of content’.In the absence of closer circumscription of the utterly vaguenotion of content, there is no sense in disputing this descriptionof their position. Obviously there is a sense of ‘content’ in whichanyone (Kant included) who holds arithmetic to be necessarymust deny that its truths have content—they must not assertanything about worldly matters which might have been other-wise. Equally obviously, there is a sense in which they can—and I think should—insist that arithmetical statements do havecontent: everyone except a very extreme kind of formalist willhold that they are meaningful, and have truth-conditions.

Potter discusses three positions which he takes to involve deny-ing that arithmetic has content: (i) it is analytic in ‘somethinglike Kant’s sense’, (ii) it is analytic in some wider sense thanKant’s and (iii) it is tautologous in the sense of the Tractatus. Heasserts that these positions involve denial of content in differentsenses of ‘content’—I cannot myself see how this claim is justi-fied, in any sense that makes it interesting. As yet, the only clearsense in which a proponent of an affirmative answer to Potter’sfirst question must deny that arithmetic has content is: assertingclaims about worldly matters which might have been otherwise.Where the three positions differ, on the face of it, is not over thesense in which they deny content to arithmetic, but in the positiveaccounts they offer of how it gets to be necessary.

Pending some clearer identification of a notion of content inwhich those who maintain that arithmetic is, in some sense, ana-lytic must deny content to it—in contrast with those who denythat it is analytic (either because they hold, with Kant, that it issynthetic or because, with Quine et al., they deny that there isany philosophically useful notion of analyticity), it seems to meat best unhelpful, and probably misleading, to pose the issues inthese terms. One might, to be sure, hold that Kant treats arith-metic as having content in the sense that it says (or is anywaytrue in virtue of) something about the structure of time andyorspace—and so depends for its truth on the way the world (or atleast our world, the world as it presents itself to us) is; but if thissuggestion is not to have the rather bizarre consequence that anyphilosophy of arithmetic opposed to Kant’s eo ipso involvesdenying content, there had better be some characterisation—inless parochial terms—of what having content amounts to (under

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which it can be seen that being about the structure or time andyor space is one way of having content). So far, it remains quiteunclear how that might go. One might, instead, simply equatehaving content with not being analytic, i.e. being synthetic. Potterwould then be right to say that (i) and (ii) (and (iii), if beingtautologous is a way of being analytic distinct from (i) and (ii))involve denying content in different senses. If this were all hemeant, I would have no quarrel with the substance of it. I would,however, want to register some disgruntlement at his way ofexpressing it, which gives the impression that dispute over theanalyticity of arithmetic may illuminatingly be characterised asderiving from disagreement over whether it has content—whereas if ‘having content’ just means ‘not being analytic’, itcomes to the same thing whether we depict Kant’s opponents asdenying that arithmetic has content or as asserting its analyticity.Any appearance that the nature of the dispute had been clarifiedby bringing in the notion of content would be wholly illusory.

IIPotter claims that arithmetic cannot be analytic in Kant’s sense,on the ground that ‘the extent of the Kantian analytic coincideswith what can be proved in [Aristotelian] logic’ which ‘recognizesno means of distinguishing between objects independent of theconcepts they are supposed to satisfy’, with the result that inAristotelian logic ‘we cannot even make sense of the idea ofcounting’. I shall not quibble with Potter over the details here,since he is surely right that Kant’s sense of analytic, in so far asit is clear, is too restrictive to support the thesis that arithmeticis analytic. In my view, he would be right, too, to think Kant’snotion too restrictive simpliciter. For not only arithmetic, butquantification logic too, will fail to be analytic in Kant’s sense.Actually, it is not clear that Potter would agree that this marksa defect in Kant’s conception of analyticity. What he declares tobe implausible is making quantification logic ‘depend ... on thespatio-temporal structure of the world as we experience it’—hedoes not expressly say that quantification logic ought to comeout as analytic, in some more reasonable sense. But I don’t knowwhat else he would say, and that is what I think he ought to say.If that is right, we have a strong independent reason to lookbeyond Kant’s restricted notion of analyticity, and need not betroubled over the fact that arithmetic fails to be Kant-analytic.

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III

As is well-known, we can define numerically definite quantifiersin first-order logic with identity, by setting:

∃0xFx Gdf ∀x™Fx

and ∃nC1xFxGdf ∃x(Fx∧∃ny(Fy∧x≠y)).

Potter puts this, somewhat unusually, in a higher-order setting—we can define constant second-level predicates 0, 1, ... and anoperation s by:

0(F )Gdf ™∃xFx

s(n)(F )Gdf ∃y(Fy · n(λx(Fx∧x≠y)))

where s is a function from second-level properties to second-levelproperties. Moving up into third-order, we can define:

m is a number Gdf ∀F((F(0)∧∀n(Fn→F(s(n)))→Fm)

On this account, ‘ξ is a number’ is a third-level predicate andnumbers themselves are not objects, as Potter observes, butsecond-level properties. As he also observes, the account hassome more or less obvious drawbacks. It provides for countingobjects, but not for counting things other than objects; thus itwill not allow us to count concepts (properties), and so—since ittakes numbers to be properties rather than objects—it will notallow us to count numbers. Nor will it lead to arithmetic as stan-dardly practised without the independent assumption that thereare infinitely many objects.

Frege, of course, considered essentially the same account (atGrundlagen §55) and found it wanting—not (or at least notexplicitly) for the reasons Potter gives, but for the ostensiblyquite different reason that it does not suffice to establish numbersas objects. This does not impress Potter as a drawback because,he says, ‘we have not yet been given any strong reason to thinkthat numbers are objects’. If he means merely that he has nothimself, up to this point in his paper, considered any strongreason to think numbers objects, I would agree; if he is makinga less autobiographical claim, I would dispute it.

In fact, Frege’s reasons for dismissing the account as in-adequate are much closer to Potter’s than what I have said makes


apparent. For one thing, Frege frequently insists that numbersthemselves are among the things which can be counted, and it isessential to his project that this should be so. Why it should beso, and why it was so crucial for him that numbers be recognisedas objects, becomes clear when we consider how he proposes toprove that every finite number is immediately followed byanother (and hence, in the presence of others of the Dedekind–Peano axioms which Frege can easily establish, that the sequenceof finite numbers is infinite). As is well-known, the idea of hisproof is to show that for each finite n, the number belonging tothe concept: finite number ancestrally preceding or equal to n itselfdirectly follows after n. To prove this, in the manner sketched inGrundlagen §§82–3, requires applying Frege’s criterion of identityfor cardinal numbers to first-level concepts of the type just indi-cated—that is, concepts under which numbers, and indeed onlynumbers, fall—and thus requires numbers to be objects.

It may, with some justice, be claimed that this gives Frege amotive to treat numbers as objects, but not a justification fordoing so. Some writers, including Michael Dummett, have seenFrege’s criticism (in §56) of the inductive definitions proposed in§55 as an attempt to prove that numbers are objects, and havepointed out that, considered as such, it is hopelessly question-begging. This reading gains some plausibility from the fact that§§55–6 lie at the beginning of a subsection of Part IV headedEvery individual number is a self-subsistent object. But it is hardto believe Frege guilty of so gross and transparent a circularity,and a more sympathetic reading is possible. In objecting to thedefinitions of §55, he appears to be assuming—rather than tryingto show—that adequate definitions of 0, 1, etc., should definethem as objects. In so far as he provides a defence of this assump-tion, it comes partly before, and partly after, §§55–6. Earlier inGrundlagen, he emphasises the use of number words as propernames as a reason to taking numbers to be objects; in the sectionsimmediately following his criticism of his first attempt at context-ual definition of numbers (i.e. §§57–61), he re-iterates this reason(‘I have already drawn attention... to the fact that we speak of‘‘the number 1’’, where the definite article serves to class it as anobject. In arithmetic, this self-subsistence comes out at everyturn, as for example in the identity 1C1G2’.) and goes on, withsome assistance from his famous Context principle, to demolish

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various bad reasons for refusing to recognise number words asproper names, standing for objects (the impossibility of forming‘ideas’ of their referents, their lack of spatial position). I thinkthese sections contain the materials from which a strong primafacie case for admitting numbers as objects may be constructed.

We start from two ideas. First: objects, as distinct from entitiesof other types (properties, relations or, more generally, functionsof different types and levels), just are what (actual or possible)singular terms refer to. Second: no more is to be required, forsub-sentential expressions of any given logical type to have refer-ence, than that such expressions should occur—functioning asexpressions of that type—in true statements. In particular, ifcertain expressions function as singular terms in various truestatements, there can be no further question but that thoseexpressions have reference, and, since they are singular terms,refer to objects. The underlying thought is that—from a semanticpoint of view—a singular term just is an expression whose func-tion is to effect reference to an object, and that a statement con-taining such terms cannot be true unless those terms successfullydischarge their referential function. Provided, then, (as certainlyappears to be the case) there are true statements featuringnumber words as singular terms, there are objects—numbers—to which they make reference.

Various qualifications are, more or less obviously, required,which I cannot go into in detail here. Firstly, if an argument ofthis kind is to be of any use at all, it must be possible to identifyexpressions as functioning as singular terms independently of theassumption that those expression refer—or purport reference—to objects. It must be possible to discern, from independentlyaccessible features of their use, which expressions function—successfully or not—as singular terms. I hold that this can bedone, employing criteria relating to inferential role of the kindproposed by Dummett1—though I believe those he actuallyproposes require some refinement.2 Second, argument of this

1. See Michael Dummett Frege: Philosophy of Language London: Duckworth, 1973ch. 4.

2. See my ‘Singular terms (1)’ in Frege: Importance and Legacy, pp. 438–457, ed.Matthias Schirn (Berlin and New York: Walter de Gruyter, 1996) and ‘Singularterms’ in The Philosophy of Michael Dummett, pp. 17–44, eds. Brian McGuiness andGianluigi Oliveri (Dordrecht, Boston and London: Kluwer, 1994) for an attempt tosupply the needed refinements.


kind does not—and I think, could not—conclusively establish theexistence of objects of the sort it concerns. There will obviouslybe various ways of resisting. An opponent might agree that thestatements to which appeal is made have, superficially, the rightlogical form, but deny that this is their real logical form. Perhapsmore radically, she may deny that any of the relevant statementsare—taken literally and at face-value—true. It might be con-ceded that the argument makes a prima facie case, but arguedthat the relevant statements cannot really be seen as involvingreference to numbers, on the ground that there are (allegedly)insuperable obstacles in the way of making sense of the idea thatwe are able to engage in identifying reference to or thought about(abstract) objects, to which we stand in no spatial, causal or othernatural relations, however remote or indirect. A related objectionhas it that if mathematical statements really were about abstractobjects such as numbers or sets, our lack of causal or othernatural relations to such objects would render mathematicalknowledge, or justified mathematical belief, impossible and con-cludes that since we do have such knowledge or justified belief,the supposition that mathematical statements involve abstractreference must be rejected. A proponent of the argument shouldconcede that his case could, in principle, be undermined in anyof these ways and perhaps others, and that he must thereforeseek to answer such challenges. But none of this shows that theargument fails of its primary purpose—to create a presumptionin favour of the existence of numbers, as a species of objects, inlight of which the onus lies firmly with those who would deny itto show that that presumption is defeated.

The conclusion Potter draws from the inadequacy of his envis-aged account of numbers as second-level properties is: ‘If we wishto extend arithmetic any further, we must appeal to somethingelse to provide it with the necessary content. But our search... isheavily constrained by our insistence that we should aim toexplain the applicability of arithmetic to the world’. To the extentthat Potter’s first sentence goes beyond the obvious—that werequire a better way of introducing numbers, which allows num-bers themselves to be counted and, crucially, allows us to estab-lish the existence of infinitely many finite numbers withoutrelying upon any extraneous assumption to the effect that thereare infinitely many objects of some other kind—I find it rather

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opaque. That it is meant to go beyond the obvious is, perhaps,suggested by his talk of extra content being needed. That mightmean no more than that we need to go beyond the narrowKantian notion of analyticity—in which case, I certainly agree—but I suspect he has more in mind. If so, it eludes me. The con-clusion I would myself draw, anyway, is the one Frege drew, i.e.the obvious one just stated, that a better explanation of the con-cept of number is required.

IV

Guided by his belief that ‘our search for a source of the contentof arithmetic is heavily constrained by our insistence that weshould aim to explain the applicability of arithmetic to theworld’, and sinking that explanatory task into the ‘wider puzzleof explaining the link between experience, language, thought andthe world’, Potter discerns four types of account which might‘supply the content we require: those that involve the structureof our experience; those that explicitly involve our grasp of a‘‘third realm’’ of abstract objects...; those that appeal to some-thing non-physical that is nevertheless an aspect of reality in har-mony with which the physical aspect is configured; and thosethat involve only our grasp of language’ (pp. x). I do not havespace either to trace and evaluate the intriguing sequence of con-siderations by which he seeks to make it plausible that the‘additional content’ needed ‘cannot be derived directly from thestructure of any of our four sources—experience, thought, theworld, and language’, or to assess his concluding speculativeelaboration of the idea that a full grasp of arithmetic requires‘appeal to higher order concepts’ (in the sense indicated in thepassage from Godel he quotes). Instead, I shall confine my dis-cussion to the second of the four types of account Potter con-siders. I shall further restrict attention, as he does, to oneparticular account of this type—Frege’s—and a variant of itdefended, principally, by Crispin Wright with occasional assist-ance from the present author. Naturally, I believe that there ismuch more to be said for an account of this sort than Potterallows and that his criticisms of it are a good deal less thancompelling.


V

One principle known to suffice for the derivation of arithmetic—in the sense that its adjunction to second-order logic yields asystem of which the Dedekind-Peano axioms are theorems—isthe principle which expresses Frege’s criterion of identity for car-dinal numbers, which Potter calls the numerical equivalence andwhich is also commonly labelled:

Hume’s principle: ∀F∀G(NxFxGNxGx↔F≈G )

where the right hand side abbreviates the second-order conditionfor there to be a 1-1 correspondence between the Fs and the Gs.Hume’s principle provides for a straightforward explanation ofempirical applications of arithmetic. To calculate the number ofnuts and bolts, for example, establish—by counting, if you like—that nut≈finite number ancestrally preceding n, for appropriate n,and infer by the relevant instance of Hume that Nx nut(x)GNxfinite number ancestrally preceding n(x). Do the same for bolt andfinite number ancestrally preceding m. Calculate nCm. Infer,assuming that no nut is a bolt, that Nx(nut(x)∨bolt(x))GnCm.Thus provided that Hume’s principle can be regarded as analytic,we have a competitive account of the second of the three sortsPotter distinguishes in his opening paragraph, on which arith-metic is analytic in some wider sense than Kant’s. What doesPotter have against it? He writes

[Frege] hoped to show that this [Hume’s principle] was a purelylogical principle. He thought this could be done indirectly via anexplicit definition of numbers as extensions of concepts, but thatdepended on Basic Law V, which cannot be a logical law since itis contradictory. To do it directly instead we should have to estab-lish a notion of content for which the left hand side of the numeri-cal equivalence could be seen as doing no more than recarving thecontent expressed by the right hand side. Frege did not supplysuch a notion of content, and the difficulty involved in doing so issevere since we know that the syntactically similar Basic Law Vcannot be viewed in terms of such a recarving.

Frege’s appeal to the notion of content at this point indicatesthat he had failed to justify the numerical equivalence as purelylogical. (pp. 67)

There are two main claims here: (i) Hume’s principle cannotbe shown indirectly to be a purely logical one by deriving it from

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Frege’s explicit definition of numbers, (ii) nor can it be showndirectly to be so by arguing that (instances of) its left hand side‘carve up the content’ of (corresponding instances of) the righthand side in a new way. Claim (i) is, on the face of it, indisput-able since, as Potter observes, a derivation of Hume from Frege’sexplicit definition of numbers would appear to have to appeal toBasic Law V, which cannot be a logical law because it isinconsistent.3 In support of claim (ii), Potter makes two further

3. Actually, the situation is a little more complicated. If we say that a concept F isa subconcept of a concept G iff ∀x(Fx→Gx), that F goes into G iff F is equinumerouswith a subconcept of G and finally that a concept F is Small just in case the universalconcept ξGξ does not go into F, then we may, following George Boolos, formulatea (plausibly) consistent replacement for Basic Law V—called New V by Boolos—which runs: ∀F∀G(*FG*g↔ ((Small (F )∨(Small (G ))→∀x(Fx↔Gx))). By New V,there will be an object *F—which Boolos dubs the subtension of F—correspondingto each concept F. If, but only if, F is Small, *F will be a set, satisfying the usualaxiom of extensionality. The obvious question is then: Can we derive Hume’s prin-ciple from Frege’s explicit definition of number with the aid of New V? There is anobstacle to doing so, if the explicit definition is taken to be the definition given inGrundlagen §68, which identifies the number of Fs–NxFx–with the extension of theconcept equal to the concept F (i.e. in effect, with the class of concepts equinumerouswith FA{H uH≈F}). The snag is that New V, as formulated, generates subtensionsonly for first-level concepts, and so does not yield subtensions corresponding to suchsecond-level concepts as ≈F and ≈G. However, this obstacle is easily circumvented.One way round it would be to appeal to a higher-order version of New V–NewV2A∀Ξ∀Ψ(*ΞG*Ψ↔ ((Small(Ξ)∨Small(Ψ))→∀φ(Ξφ↔Ψφ))) [Here Ξ, Ψ range oversecond-level concepts and φ ranges over first-level concepts]. Hume’s principle can bederived from Frege’s Grundlagen explicit definition of numbers with the aid of NewV2. Note that the Smallness condition in New V2 needs to be understood in termsof a second-level concept under which every first-level concept falls, just as Smallnessin New V is understood in terms of a first-level concept (self-identity) under whichevery object falls. The obvious choice is self-co-extensiveness—i.e. the second-levelconcept expressed by ∀x(φx↔φx). A second-level concept Ξ is thus Small iff∀x(φx↔φx) does not go into Ξ. Under Frege’s explicit definition adapted to this newsetting, NxFx will be identified with *≈F, so that to prove Hume’s principle, whatneeds to be established is that *≈FG*≈G↔F≈G. An instance of New V2 is: *≈FG*≈G↔ ((Small (≈F )∨Small (≈G ))→∀φ (φ≈F↔φG )). Note that for any first-level con-cept F, Small (F ) iff Small (≈F ). For Hume left-right note that if neither F nor G isSmall, then neither ≈F nor ≈G is Small, so that ∀x(φx↔φx) goes into both, and sinceboth obviously go into ∀x(φx↔φx), they go into one another, so that F≈G asrequired, while if either F or G is Small, one of ≈F and ≈G is Small and by New V2

∀φ (φ≈F↔φ≈G ), whence F≈G easily follows. Hume right-left is also easily proved. Asimpler and perhaps preferable alternative would be to amend the explicit definitionof number so that numbers are identified with subtensions of suitable first-level con-cepts. Roughly, instead of defining NxFx to be the class of concepts equal to F, wecould define it as the class of classes determined by concepts equal to F. This is, ineffect, the course Frege actually takes in Grundgesetze, where he shows how he candefine the number of Fs as the course-of-values of a certain first-level function (SeeGrundgesetze §40—since the function in question takes us from objects to truth-values, it is a concept, so that numbers are identified with extensions of certain first-level concepts, rather than with extensions of certain second-level concepts as inGrundlagen). If this is done, we can get home with New V as it is.

One might feel queasy about the first alternative in the absence of a proof that


claims: (iii) Frege failed to explain a suitable notion of contentunder which the left and right sides of Hume might be held tohave the same content, and (iv) the obvious syntactic parallelbetween Hume and Basic Law V gives a strong reason to thinkthat there is no suitable such notion of sameness of content. Inany case, he adds, (v) the very fact that the direct route appealsto the notion of content means that it cannot justify Hume aspurely logical.

The first point to be made here is that, by demanding thatHume be established as a purely logical principle, Potter has sur-reptitiously moved the goalposts. It is true, near enough, thatthis is what Frege sought to accomplish. But the question is—orshould be—not whether Hume is a purely logical principle butwhether it is analytic in some wider sense than Kant’s. Claims (i)and (v) may therefore be conceded without prejudice to Hume’scapacity to play a key role in a case for the (non-Kantian) ana-lyticity of arithmetic; and the same goes for claim (ii) as stated,since what matters is whether Hume can be regarded as analyticin some reasonable sense. This last point does not remove theneed for a response to claims (iii) and (iv); but there is an obviousprior question: What sense of ‘analytic’?

New V2 is consistent. The consistency of New V, however, is scarcely open to seriousquestion—Boolos shows that it is consistent provided that second-order arithmeticis (see ‘Saving Frege from contradiction’ Proceedings of the Aristotelian Society 87(1986y87), pp. 137–51). In any case, even New V is an appreciably stronger compre-hension principle than is actually needed to derive Hume’s principle from Frege’s(original) explicit definition. As Dummett observed long ago (in his article on Fregein the Edwards’ Encyclopaedia, reprinted as ‘Frege’s Philosophy’ in Dummett’s Truthand other enigmas (London: Duckworth, 1978), pp. 87–115; see p. 114), no strongerform of comprehension is needed than one which guarantees the existence, for eachSmall concept F (i.e. in the sense of New V) of its equivalence class under the relationof (cardinal) equality. I take it that no one thinks that even that much comprehensionis inconsistent. There is thus not much to be said for the idea that Hume’s principlecannot be shown to be a logical truth by deriving it from Frege’s explicit definitionbecause such a derivation must appeal to inconsistent principles governing the notionof extension. A much more plausible objection is that the additional principle(s)needed to carry out the derivation cannot be logically true, even if they are consistent.Boolos was adamant that neither New V nor Hume is a logical truth, and would, Iam sure, have taken the same view of New V2 (and even of the weakest comprehen-sion principle needed to ensure the existence of sufficiently many cardinal equivalenceclasses) for much the same reasons—reasons having, mostly, to do with their existen-tial implications. Since I think the more important issue to be whether Hume’s prin-ciple can be regarded as analytic, rather than whether it is a logical truth, I shall notpursue the matter further now, save to make the obvious point that since the statusof New V and New V2 as logical truths is certainly no less open to doubt than thatof Hume’s principle, deriving the latter from the former is hardly a promising methodof persuading anyone that it is a logical truth.

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One wider sense is Frege’s: a statement is analytic if and onlyif provable using only general logical laws and definitions.4 Humeis not analytic in this sense—there is, in the background as itwere, no definition of the numerical operator which would enableus either to view Hume as a definitional abbreviation of a logicallaw or to derive it from logical laws; and Hume cannot itself beregarded as a definition in any strict sense, because it does notprovide for the elimination of the numerical operator in all con-texts. However, there is some reason to think that Frege’s notion,whilst broader in some respects than Kant’s, is still undulyrestrictive. For one thing, there are many statements which areplausibly viewed as conceptual truths, true in virtue of the mean-ings of their ingredient words, which do not qualify as analyticunder Frege’s definition, because they are not themselves logicaltruths and cannot be reduced to logical truths by means of defi-nitions. Plausible examples are ‘Whatever is yellow is coloured’and ‘If one event precedes another, which in turn precedes athird, then the first precedes the third’. Anyone who understandsthese statements is in a position to recognise them as being true,independently of any experience beyond what is needed toacquire an understanding of the words involved, and it seemsclear enough that their ability to do so derives entirely from theirknowledge of meanings. But although the key words involved—‘yellow’, ‘coloured’, ‘precedes’—can of course be taught andlearned, they are not capable of being verbally defined. There is,for example, no correct definition of ‘yellow’ of the form ‘x isyellow iff x is φ and x is coloured’, by means of which the firststatement could be transformed into a logical truth of the form‘∀x(x is φ∧x is coloured→x is coloured)’. Particular examples—even these examples—might, perhaps, be disputed. But it seemsto me that there are bound to be statements of the kind I have

4. A definition very similar to Frege’s is envisaged—but not, of course, endorsed—by Quine in ‘Two Dogmas of Empiricism’ From a logical point of view (CambridgeMa.: Harvard University Press, 1953) pp. 20–46, where he distinguishes two classesof statements commonly regarded as analytic. The first class consists of those state-ments which are logically true; the second, supposedly wider, class comprises inaddition those statements transformable into statements of the first class by ‘puttingsynonyms for synonyms’ or, equivalently, replacing defined expressions by theirdefinientia. Of course, even if Quine had not had other reasons for regarding theproposed definition as unsatisfactory, he would not have regarded it as equivalentto Frege’s, since he—unlike Frege—counts only first-order logical truths as logicaltruths.


sought to illustrate. For it is no accident that our language con-tains expressions whose meanings cannot be fully captured byverbal definitions—if there were not some words whose mean-ings could be grasped without their being defined using otherwords, the whole process of definition would be either viciouslycircular or infinitely regressive. And whatever words they are,whose meanings have to be learned otherwise than via verbaldefinitions, there will be correct and incorrect ways of applyingthem, and so meaning relations between them, such as thatbetween ‘yellow’ and ‘coloured’, which will give rise to analytictruths of the sort in question.

A second point is that Frege’s definition makes the analyticityof fundamental logical truths simply a matter of stipulation. Heis free, of course, to do that; but it leaves wholly unexplained oursense that his choice of logical laws as his base class is not arbi-trary, but (very nearly) right, given that his aim is ‘not... to assigna new sense [to ‘analytic’], but only to state accurately what earl-ier writers, KANT in particular, have meant by [it]’ (Grundlagenp. 3, fn. 1). Why choose logical laws as the base class, rather than,say, the laws of physics, or—if, like Frege in another context, Imay be permitted a rather crude example—the truths enunciatedin Mrs. Beaton’s Manual of Cookery and Household Manage-ment? Since ‘provable’ in Frege’s definition means ‘provable inlogic’, he has a compelling motive for holding that logical infer-ence preserves analyticity, and so for regarding logical truths asanalytic. But once again, a motive is not the same thing as ajustification. What justifies him, if it is not that statements like‘If A and B, then A’ are, as we might put it, analytic of the basiclogical concepts involved? And now why shouldn’t Hume’s prin-ciple, though not analytic in either Frege’s sense or Kant’s, beheld to be analytic of the concept of (cardinal) number?

VI

I turn now to claims (iii) and (iv). For an obvious reason, I take(iv) first. The thought behind it would seem to be as follows. Inview of the fact that Hume’s principle and Basic Law V bothexemplify the same syntactic pattern, instances of the left handside of the former can be viewed as recarving the content of cor-responding instances of its right hand side only if the same is

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true of instances of the left and right hand sides of the latter. Butif instances of the latter’s left hand side merely recarved, and sopossessed the same content as, instances of its right hand side,Basic Law V would be true. Since it is inconsistent and thereforefalse, this cannot be so. Hence it cannot be the case that instancesof the left hand side of Hume recarve the same content as corre-sponding instances of its right hand side. What is questionablein this modus tollens is its major premiss, which rests upon theassumption that whether instances of the left hand side of anabstraction principle have the same content as correspondinginstances of its right hand side depends solely upon its syntacticshape. But why make that assumption? However, precisely, theappropriate notion of identity in content is to be explained, onemight, on the contrary, expect that it would not be a purely syn-tactic matter. The assumption is tantamount to the claim thateither all abstraction principles—that is, near enough, all prin-ciples of the form:

§αG§β↔Eq(α,β) where Eq is an equivalence relation onentities of the type of α,β... and § is afunction from entities of that type to objects

—are true, or none are. Why should a Fregean abstractionistbuy that, rather than hold that some abstraction principles areacceptable, others not? In his view, a good abstraction furnishesa means of introducing or explaining a (sortal) concept—number,in the case of Hume’s principle. The fact that some would-beexplanations of this style are unacceptable because they lead tocontradiction does not suffice to discredit the whole genre, anymore than the fact that some would-be explanations of predicatesin terms of their satisfaction conditions—e.g. w is heterologicalif and only if w is not true of itself—generate contradictionshows that all such explanations are inherently defective.5 Theinvidious parallel with Basic Law V at best enjoins the Fregeanabstractionist to point to differences in virtue of which Hume isan acceptable explanation but Basic Law V not—but that he caneasily do: the latter is provably inconsistent, while the former isnot and is, in fact, provably consistent, relative to the assumption

5. cf. Crispin Wright ‘On the Philosophical Significance of Frege’s Theorem’, inLogic and Language: Essays for Michael Dummett, pp. 201–44, ed. Richard Heck(Oxford: Oxford University Press); see sections II and V.


(which few would seriously question) that second-order arith-metic is.6

VII

Defending the proposal7 to define number contextually by meansof Hume’s principle, Frege invites us to view the statement thatthe number of Fs is identical with the number of Gs as the resultof carving up in a new way the content of the statement that theFs and the Gs are in 1–1 correspondence.8 Evidently this requiresan account of when two statements are to be regarded as sharingtheir content in the relevant sense. Frege, as Potter complains,provided none. Trawling the pages of Grundlagen and earlierwritings for clues in Frege’s other uses of ‘Inhalt’ and ‘enthalten’yields little to the purpose—he assigns content both to names

6. For details, see George Boolos ‘The Consistency of Frege’s Foundations of Arith-metic’ originally published in On being and saying: essays in honour of Richard Cart-wright, pp. 3–20, ed. Judith Jarvis Thomson (Cambridge: MIT Press, 1987) andreprinted in Frege’s Philosophy of Mathematics, pp. 211–33, ed. William Demopoulos(Cambridge: Harvard University Press, 1995). The relevant passage occurs atpp. 219–20 of the Demopoulos reprinting. After explaining how any proof in FA (i.e.Frege Arithmetic—the system obtained by adjoining Hume’s principle to second-order logic) Boolos asserts ‘It is therefore as certain as anything in mathematics that,if analysis [also known as second-order arithmetic] is consistent, so is FA.’ I shouldnote that whilst Boolos would not, so far as I know, have seriously questioned theconsistency of analysis, he did not regard the consistency of Hume’s principle, ortherefore that of analysis, as certain, beyond all possibility of doubt—cf. his ‘IsHume’s principle analytic?’ in Heck op. cit. pp. 245–61; see especially pp. 259–60.Boolos took this to be one reason for denying that Hume’s principle can be analytic,which it is, if you take analyticity to require indefeasible certainty—like CrispinWright, I reject that requirement (cf. Wright’s reply, also entitled ‘Is Hume’s principleanalytic?’, delivered at the recent memorial conference for Boolos in Notre Dameand forthcoming in a special issue of the Notre Dame Journal of Formal Logic).

7. cf. Grundlagen §63ff. Persuaded of the intractability of the Julius Caesar problem,he eventually rejects the proposal in favour of his explicit definition. Full defence ofthe claim that Hume’s principle can serve as an analytic basis for arithmetic obviouslyrequires showing that Caesar is not as intractable as Frege thought; but this is a largetask which must be deferred to another occasion.

8. Not in so many words, of course—Frege’s discussion throughout §§64–68 is con-ducted at one remove, in terms of a parallel contextual definition of direction bymeans of the equivalence: the direction of line aGthe direction of line b iff line ayyline b. Further, what he actually suggests is that the content of the symbol ‘yy’, ratherthan that of the whole sentence ‘ayyb’, is what is carved up—‘we replace the symbolyy by the more generic symbol G, by removing what is specific in the content of theformer and dividing it between a and b’. But it is hard to see how carving sub-sentential contents could fail to result in carving at the level of sentential content.See my ‘Grundlagen §64’ (Proceedings of the Aristotelian Society 1997, pp. 243–61),on which the discussion of this section draws heavily.

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and to judgements, sometimes in ways in which it is natural totake him to be speaking of what words refer to, sometimes inways which invite construal in terms of what he was later to calltheir sense. In a much discussed passage (Begriffsschrift §9) whichhas sometimes—wrongly in my view—been taken to provide themodel for his later talk of content-carving, Frege claims that dif-ferent ways of breaking up a statement into function and argu-ment involve no change in its ‘conceptual content’. This mightincline us to think of content, in the passage that concerns us, assomething sense-like; but no criterion of content-identity is inview. Our difficulties are, at first sight, compounded by the factthat, not much after Grundlagen, Frege rejected the notion of‘judgeable content’, replacing it with the contrasted notions ofthought and truth-value, as a special case of his sense-referencedistinction. In fact the shift is a help rather than a hindrance, asit forces us to ask what we should have asked anyway: Shouldthe content that is to be common to both sides of Hume’s prin-ciple be thought of as belonging to the realm of reference, orrather to the realm of sense?

The difficulties with taking the content of a sentence to be itsreference are obvious enough. In Frege’s view, sentences refer totruth-values and all materially equivalent sentences have thesame reference. But, first, it appears to make no sense at all tospeak, even figuratively, of carving up truth-values—the Trueand the False are indeed objects, on Frege’s eventual view, butthey seem not to be objects which might be carved up, like piecesof cheese. It is hard to see how they could be thought of as struc-tured, in ways that would give conceptual surgery something towork on, without depriving them of their capacity to serve as thereferents of all true and all false sentences respectively. In anycase, the view that they are simple objects appears to be imposedby Frege’s doctrine that truth and falsehood are indefinable.Second, it seems that a more intimate (equivalence) relationbetween sentences than mere identity in truth-value should berequired, for one to represent a recarving of the same content asthe other. One might think that taking states of affairs, conceivedof as structured entities of some kind, as the referents of sen-tences would alleviate these difficulties; but so long as the cri-terion of identity for states of affairs is purely extensional, aversion of the infamous Slingshot argument will show that we


are no better off than with the True and the False. This does notmean that the notion can play no useful role in a final account,since states of affairs might be more satisfactorily individuatedin non-extensional, or not purely extensional, terms. In particu-lar, it does not mean that we cannot think of the left hand sideof Hume’s principle—as some have suggested9—as effecting areconceptualisation of the type of state of affairs depicted by itsright hand side. But—the crucial point—it will be alternativeconceptualisations of the same state of affairs that are in question.An explanation of what it is for two sentences to have the samecontent in these terms will thus, reference of worldly states ofaffairs notwithstanding, locate content firmly in the realm ofsense.

VIII

The option of construing identity in content as a relation at thelevel of sense is both more natural and more attractive; but it isnot without difficulties of its own. The principal difficulty maybe appreciated by reflecting on Frege’s purpose in proposing theDirection Equivalence and Hume’s principle. They are to func-tion as explanations of the (sortal) concepts of direction and num-ber—routes by which a thinker lacking these concepts mightcome to possess them. It is therefore essential that it should bepossible for such a thinker to understand instances of the righthand side of the Direction Equivalence, say, without already pos-sessing the concept of direction, and so without—in advance ofreceiving the explanation—understanding any instances of its lefthand side. But if the content of a sentence is simply identifiedwith the Fregean sense—or Thought—it expresses, it is not clearhow this requirement can be met; indeed, on one reasonable con-strual of sentence sense or Thought, there is a strong appearancethat it cannot be met.

It is indisputable that Frege’s notion of sense, like his notion ofreference, is at least weakly compositional: that is, the sense of anycomplex expression is a function of the senses of its component

9. cf. Crispin Wright’s paper in Heck op. cit., especially section I, my ‘OnDummett’s critique of Wright’s attempt to resuscitate Frege’ in Philosophia Mathem-atica (3) Vol. 2 (1994) pp. 122–47, especially section 2 and the paper by me cited inthe preceding note.

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expressions, just as the reference of a complex expression is a func-tion of the references of its components. As far as sense—but not,of course, reference—goes, however, Frege seems clearly toendorse strong compositionality: the sense of a complexexpression is not merely a function of, but is actually composedof, the senses of its components—the senses of the parts are partsof the sense of the whole. Thus at Grundgesetze Vol. 1, §32, hewrites:

The names, whether simple or themselves composite, of which thename of a truth-value consists, contribute to the expression of thethought, and this contribution of the individual [component] is itssense. If a name is part of the name of a truth-value, then the senseof the former name is part of the thought expressed by the lattername.

It follows that if two sentences have the same sense, anythingwhich is part of the sense of one of them must be part of thesense of the other. Thus if the identity in content between ‘thedirections of lines a and b are the same’ and ‘lines a and b areparallel’ is to consist in the two sentences expressing the samesense, then, since the concept of direction is part of the sense ofthe former, it must be part of the sense of the latter. But then,unless it is possible to grasp the sense of a complete sentencewithout having grasped the senses of its parts, no one who doesnot already possess the concept of direction can understand thestatement that the two lines are parallel.

How, assuming it can be, is this difficulty best surmounted? Ifthe identification of content with strongly compositional sense isto be retained, it will have to be denied that it is necessary, inorder to understand a sentence, to grasp the senses of each ofthe parts of its sense. This in turn will require one to hold thatone and the same sentence-sense may be composed in differentways out of different parts and that in order to understand thesentence, all that is needed is that one knows one way of compos-ing its sense out of part-senses which one knows. On thisaccount, there may be senses which I do not grasp, but whichare parts of the sense of a sentence I understand—my not grasp-ing them does not matter, because they do not figure in my com-positional route to the sense of the whole. Something like thisview—according to which one and the same compound sensemay be decomposable into different parts in different ways—has,of course, been attributed to Frege.


It is clear that, if the multiple decomposition account is accept-able, it gives a way of solving the particular difficulty with whichwe are concerned, since it plainly allows that a thinker may graspthe compositional sense of ‘lines a and b are parallel’ withoutalready possessing the concept of direction, even though that is,on one decomposition, part of the sense expressed by the sen-tence. I do not know whether it is acceptable. It does seem to meto have at least two awkward consequences. One of them is animmediate corollary of the very feature in virtue of which theaccount solves the present difficulty—since, on the proposal weare considering, the sense of ‘the directions of lines a and b arethe same’ is the same as that of ‘lines a and b are parallel’, itfollows that a thinker who lacks the concept of direction maynevertheless grasp the sense of the direction-identity. Hence ifunderstanding a sentence is equated with grasping the thought itexpresses, it follows that a thinker may understand ‘the directionsof lines a and b are the same’ without having any idea of what adirection is. As against this, it may be felt that only a thinkerwho possesses the concept of direction can understand this sen-tence. The other somewhat awkward consequence, which isclearly closely related to the first, concerns reports of belief. Onany view, such as Frege’s, according to which (i) the reference ofa complex expression is a function of the references of its compo-nents and (ii) the reference of words in the subordinate clause ofa belief-report is to their customary sense, sentences expressingthe same sense must be interchangeable salva veritate withinbelief-reports. Thus on the present proposal, if a thinker believesthat lines a and b are parallel, it will follow that—irrespective ofwhether she possesses the concept of direction—she believes thattheir directions are the same. But once again, it may be thoughtthat only a thinker who has the concept of direction can properlybe credited with beliefs about directions. I do not claim thateither of these points amounts to a decisive objection to thepresent proposal, but I do think they give a motive, if one isneeded, to consider an alternative solution to the difficulty.

IX

What is needed is to explain how a thinker who understandsstatements of line-parallelism, but does not yet possess theconcept of direction, may be enabled to acquire it by being given

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the Direction equivalence. On the explanation I favour, it isunnecessary to maintain that the two halves of the Directionequivalence have the same strongly compositional sense, orexpress the same thought. It should rather be understood as astipulation that instances of the left hand side are to have thesame truth-condition as corresponding instances of the righthand side, in a (relatively weak) sense of ‘same truth-condition’which allows that two sentences may have the same truth-con-dition even though they involve deployment of different con-cepts. If we assume, for the moment, that a satisfactory criterionof identity in truth-condition can be provided, the explanationthen proceeds as follows. From the stipulation of the Directionequivalence, the trainee knows that ‘the direction of line aGthedirection of line b’ is to have the same truth-condition as ‘lines aand b are parallel’, and knows what that condition is. But shealso possesses, as Frege emphasises, the general concept of ident-ity, and knows that the target sentence is being explained as oneapt for making a new kind of identity-statement. She furtherknows that if this is to be the case, the expressions flanking thesign of identity must be singular terms and that these new singu-lar terms are complex, involving as they do familiar singularterms for lines, so that the explanatory intention must be tointroduce ‘the direction of...’ as standing for a function from linesto objects of some kind. If she now asks herself—as she is inposition to do—what kind of objects these new direction-termsstand for, she can answer: objects so related to lines that theyare identical just in case the lines are parallel. She can gather, inother words, that directions simply are (on this explanation)objects for whose identity (and therefore for whose existence) itis necessary and sufficient that the corresponding statement ofline-parallelism be true. She thus acquires the concept of direc-tion, and with it, a way of reconceptualising familiar facts aboutparallelism among lines.

How should the notion of sameness in truth-conditionrequired to underpin this explanation be characterised? As I havesaid, it is to be relatively weak in the sense that it should allowtwo sentences to have the same truth-condition in cases wherethey involve deployment of different concepts. On what is per-haps the weakest account that is at all plausible, two sentenceshave the same truth-condition if and only if they are necessarily


alike in truth-value. This may appear too weak for present pur-poses, for two reasons. First, on this account, arbitrary pairsof necessary truths share the same truth-condition, but are notplausibly regarded as alternative conceptualisations of the samestate of affairs. Second, if Hume’s principle and the DirectionEquivalence are to be viewed as analytic, respectively, of the con-cepts of number and direction, it seems that their left and rightsides should stand in a more intimate relation than merely strictequivalence. On reflection, it appears that neither point consti-tutes as compelling reason against identifying the notion of con-tent we are seeking to elucidate with that of truth-condition inthis weak sense. The first would be decisive, if the claim werethat two sentences having the same content is not only necessarybut also sufficient for one to be properly viewed as recarving thecontent of the other. But a defender of the Fregean account hasno need to make so strong a claim: he can claim that coincidencein truth-condition in this weak sense suffices as far as the require-ment of identity in content goes, but point out that this does notpreclude the imposition of further conditions on the sentencesinvolved, if Frege’s procedure is to be properly applied. As tothe second, he may agree that, pending clarification of the notionof necessity deployed in this characterisation, we cannot be surethat every biconditional linking two sentences with the sametruth-condition will be analytic in some reasonable sense. Butthis too need raise no problem for the Fregean account, so longas Hume’s principle may be so regarded—as it presumably maybe if it is put forward as a stipulation, playing a role akin to thatof a Carnapian explication of the concept of number.

It is thus unclear, in the absence of further reasons to doubtthe adequacy of identifying content with truth-condition in thisweak sense, that a defender of the Fregean account need haverecourse to any stronger or more refined notion. It is, neverthe-less, worth enquiring whether we can circumscribe appropriatelyfiner-grained notion of content—one for which two sentences’identity in content would suffice for one to be regarded asexpressing a reconceptualisation of the state of affairs depictedby the other.

A considerably stronger requirement which might be proposedis that two sentences share their truth-condition if and only ifanyone who understands both of them is able to tell immediately

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(i.e. without inference) that they must have the same truth-value.This is in fact the requirement which Michael Dummett takesFrege to have accepted as both necessary and sufficient for twosentences to express the same thought.10 Dummett argues, cor-rectly in my view, that this gives only a necessary, not a sufficient,condition for identity of thought expressed. But might it be takenas both necessary and sufficient for identity in truth-condition?The answer, I believe, is that whilst it is certainly sufficient, it isprobably too exacting to be acceptable as a necessary condition.A pair of sentences such as ‘There are fathers’ are ‘There aremothers’ are plausibly regarded as sharing their truth-conditionin a suitable sense, but would not count as doing so by this cri-terion, since some reasoning is needed to see that they cannotdiffer in truth-value. Weakening the requirement so as to allowthat reasoning may be involved in determining that the sentencesmust coincide in truth-value threatens, however, to render thecriterion too weak. Even if only—but any—deductive reasoningis permitted, arbitrary pairs of necessary truths which are separ-ately knowable a priori—such as ‘There are infinitely many pri-mes’ and ‘CH is independent of ZFC’, or distinct and otherwiseunrelated instances of a logically valid schema – will qualify ashaving the same truth-condition, as will any contingent statementP and a logical complication of that statement such as ‘(P &Q)v(P & ™Q)’. But it will, in general, be wildly implausible toregard one of a pair of sentences so related as reconceptualisingthe state of affairs represented by the other.

One way to amend our criterion is to restrict the kind ofreasoning allowed. Classical entailments typically depend onlyupon the semantic contributions of key items of logical vocabu-lary, and are in general insensitive to those made by non-logicalwords. A more selective notion of entailment—introduced byCrispin Wright for other purposes, but well-suited to ours—isthat of compact entailment. Wright’s aim is not to provide an all-purpose replacement for classical entailment, but to circumscribea narrower species of it which, unlike classical entailment itself,essentially exploits the non-logical content of the premisses andconclusion. As Wright explains it, ‘an entailment is defined as

10. cf. Dummett 1991, p. 171.


compact just in case it is liable to disruption by uniform replace-ment of any non-logical constituent in its premisses (but not inits conclusion)’.11 In other words:

A1,..., An compactly entail Biff (i) A1,..., An entail B, and

(ii) for any non-logical constituent E occurring in A1,..., An,there is some substitution E ′yE which applied uniformlythrough A1,..., An, yields A′1 ,..., A′n which do not entail B

The effect of (ii) is to disallow entailments in which the premissescontain parts which are ‘passengers’, in the intuitive sense thattheir specific content makes no essential contribution to the hold-ing of the entailment. Thus classical entailments of arbitrary logi-cal truths by other logical truths—such as that of ‘P v ™P’ by ‘Qv ™Q’—and of arbitrary contingent statements by their logicalcomplications—such as that of ‘P’ by ‘(P & Q) v (P & ™Q)’—fail of compactness (in these cases, because no uniform substi-tution on ‘Q’ can disrupt either).12 Employing this notion, wemay formulate a stronger notion of truth-condition (and with it,a more refined criterion of identity in content) as follows:

Two sentences have the same truth-condition (content) iff anyonewho understands both of them can tell, without determining theirtruth-values individually, and by reasoning involving only com-pact entailments,13 that they have the same truth-value

As I have said, I have not myself been able to uncover any clearreason why a defender of the Fregean account should not workwith the weaker and cruder notion of truth-condition. But shouldsuch a reason be forthcoming, he may fall back on the strongerand more refined one just sketched. In terms of the notion oftruth-condition (in either its more or its less exacting form), we

11. cf. Wright ‘The Verification Principle: Another Puncture—Another Patch’ MindVol. 98, (1989), p. 611–22. Compact entailment is introduced on pp. 611–12.

12. A small snag with compact entailment as defined is that it is not reflexive, sinceno logical truth compactly entails itself. But this defect is easily handled by allowingany entailment as compact if it is a substitution-instance of an entailment that iscompact by the definition above. Thanks to Peter Milne for drawing attention to thepoint, and the remedy.

13. This condition is required, even in the presence of the restriction to reasoning bycompact entailment. Without it, arbitrary pairs of necessary truths, each of whichcan be established by purely compact reasoning, would qualify as having the sametruth-condition.

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may introduce a derivative notion of states of affairs: two sen-tences will represent the same state of affairs if and only if theyhave the same truth-condition. We may then speak—conveniently, but not ineliminably—of one sentence involving areconceptualisation of the same state of affairs as another.

Corresponding instances of the left and right hand sides ofHume’s principle and the Direction Equivalence will coincide intruth-condition in both of the senses described. Indeed, they willsatisfy the stronger requirement imposed by the Fregean criterionfor thought-identity, since anyone who understands a statementof direction-identity via the stipulation of the Direction Equival-ence can tell, without inference, that it must have the same truth-value as the corresponding statement of parallelism. But thepoint, of course, was to elucidate the general notion of truth-condition that is in play—only if that is done can the intendedeffect of the stipulation be clear.

X

My aim in this paper has been a severly limited one. I have triedto show that none of the objections Potter brings against theview that arithmetic may be based upon Hume’s principle in away which justifies taking it to be analytic in a reasonable senseis effective. I have argued that this view does not require us toclaim—as Potter appears to suppose—that Hume’s principle isa purely logical one, but agreed with him that, if that it is to beviable, an account is needed of the way in which instances of itsleft hand side may be taken to have the same content as corre-sponding instances of its right hand side. If the account I havetried to supply is acceptable, a neo-Fregean philosophy of arith-metic remains, so far, a competitive option. There are, of course,other reasons, besides those specifically advanced by Potter,which have persuaded others to think that Hume’s principle can-not be analytic. I do not believe any of them is compelling, butit has obviously been beyond the scope of this paper to discussthem.14,15

14. See the papers by Boolos and by Wright, both entitled ‘Is Hume’s principleanalytic?’, cited in note 6.

15. Thanks to Gary Kemp, Adam Rieger and Crispin Wright for helpful commentsor discussion. This paper was written during my tenure of a British AcademyResearch Readership; I am grateful to the Academy for its generous support.

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Intuition and Reflection in Arithmetic: Bob Hale