73951099 Kripke 1982 Wittgenstein on Rules and Private Language an Elementary Exposition

7/30/2019 73951099 Kripke 1982 Wittgenstein on Rules and Private Language an Elementary Exposition

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SAUL A. KRIPKE

Wittgenstein on Rulesand Private LanguageAn Elementary Exposition

Harvard University PressCambridge, Massachusetts


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Copyright 1982 by Saul A. KripkeAll rights reservedEIGHTH PRINTING, 1995

Printed in the United States of America

Library ofCongress Cataloging in Publication DataKripke, Saul A., 1940-

Wittgenstein on rules and private language.Includes bibliographical references and index.Wittgenstein, Ludwig, 1889-1951. I. TitleB3376.W564K74 192 81-20070AACR2ISBN 0-674-95401-7 (paper)

Contents

Preface1 Introductory2 TheWittgensteinian Paradox3 The Solution and the 'Private Language'ArgumentPostscript Wittgenstein andOtherMindsIndex

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To my parents

Preface

The main partof this work has been delivered at various placesas lectures, series of lectures, or seminars. It constitutes, as Isay, 'an elementary exposition' ofwhat I tak e to b e the c en tr althread of Wittgenstein's later work on the philosophy oflanguage a nd the p hilo so ph y of mathematics, including myinterpretation of the 'private language argument', which onmy view is principally to be explicated in terms of the problemof 'following a r ule' . A p os ts cr ip t p re se nts a no th er problemWittgenstein saw in the conception ofprivate language, whichleads to a discussion of some aspects of his v ie ws on theproblem ofotherminds. Since I stress the strong connection inWittgenstein's later philosophy between the philosophy ofpsychology and the philosophy ofmathematics, I had hopedto add a second postscript on the philosophyofmathematics.Time has no t permitted this, so for the moment the basicremarks on philosophy ofmathematics in the main text mustsuffice.The presentwork is hardly a commentary onWittgenstein's

later philosophy, nor even on Philosophical Investigations.Many well known a nd s ign if ica nt t op ic s - f or e xa mp le , t heidea of 'family resemblances', the concept of 'ce rtainty' - a rehardly mentioned. More important, in the philosophy ofmind itself, a wealth ofmaterial, such as Wittgenstein's viewson intention, memory, dreaming, and th e like, are barel y


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VllI Preface Priface IXglanced at. It is my hope that much of this material becomesfairly clear from an understanding of Wittgenstein's view ofthe central topic.

Many of Wittgenstein's views on the nature of sensationsand sensation language are either only glanced at or areomitted altogether; and, as is stressed in t he text , I adopted thedeliberate policy of avoiding discussion of those sectionsfollowing 243 of the Investigations that are ordinarily calledthe 'private language argument'. I think that many of thesesections - for example, 258ff. - become much clearer whenthey are read in the light of the main argument of the presentwork; but probably some of the exegetical puzzles in some ofthese sec tions (e.g. 265) are no t devoid of residue. Theinterest of these sections is real, bu t in my v iew theirimportance should no t be overstressed, since they representspecial cases of a more general argument. Usually I presentedthis work to sophisticated philosophers, but i t is my hope thatintroductory classes in Wittgenstein could use it in conjunc-tion with other mater ial. In classes it would be helpfulespecially for the instructor to t ryout the Wittgensteinianparadox on the group, and to see what solutions are proposed.Here primarily Imean responses to the paradox that we followthe rule as we do without reason or justification, rather thanthe phi losophical theor ies (disposit ions, quali tative states,etc.), discussed later in the same chapter. It is important for thestudent to feel the problem intuitively. I recommend the sameinitial emphasis to readers who propose to study the presentwork on their own. I also recommend that the student (re)readthe Investigations in the light of the structuring of the argumentproposed in this work. Such a procedure is of specialimportance here, since largely my method is to present theargument as i t struck me, as it presented a problem for me,rather than to concentrate on the exegesis of specific passages.Since I first encountered the 'private language argument'

and the later Wittgenstein generally, and since I came to thinkabou t it in the way expounded here (1962-3), his work onrules has occupied a more central position in discussions of

Wittgenstein's later work. (It had been discussed to someextent. all along.) Some of this discussion, especially thatappeanng after I gave my London, Ontario lecture, can bepresumed to ?av.e been influenced by the present exposition,?ut some of It, m and out of print, can be presumed to bemdependent. I have no t tried to cite similar mater ial in the

l i t e r a ~ u r e , p ~ r t l y because if I made the attempt, I would becertam .to slIght some published work and even more, someunpublIshed work. I have become satisfied, for reasonsmentioned below in the text and footnotes, that publicationstill is no t superfluous.

It deserves emphasis that I do not in this piece of writingattempt to speak for myself , or , except in occasional andminor asides, to say anything about my own views on thesubstantive issues. The primary purpose of this work is thepresentation of a problem and an argumen t, not its critical

e ~ a l u a t i o n . Primarily I can be read, except in a few obviousa S I d e ~ , as almost like an attorney presenting a major philc-sophIcal argument as it struckme. If the work has a main thesisof its own, it is that Wittgenstein's sceptical problem andargument are important, deserving ofserious consideration.

Various people, including at least Rogers Albritton,G. E. M. Anscombe, Irving Block, Michael Dummett,Mar?aretGi.lbert, BarbaraHumphries, ThomasNagel, RobertNozIck, MIchael Slote, and Barry Stroud, influenced thisessay. In addition to theWittgenstein Conference in LondonOntario, 1976, I gave various vers ions of this materialHowison Lec:ures, the University of California, Berkeley,1977; as a senes of lectures in a special colloquium held inBanff, Alberta, 1977; and at aWittgenstein Conference held atTrinity College, Cambridge, England, 1978. Versions were

~ l s o given i? seminars at Princeton University, the first beingm the Spnng Term of 1964-5. Only in these P rince tonseminars did I have time to include the material in thepostscript, so that it has had less benefit of discussion andreaction from others than the rest. No doubt I was influenced

the of my argument and


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x Prefaceseminars. I should especially like to thank Steven Patten andRon Yoshida for their beautifully prepared transcripts of theBanff version, and Irving Block both for his help as editor ofthe volume in which an earlier version of this work appeared,and for inviting me to make this exposition more public at theLondon Conference. Samizdat transcripts of t he ver siongiven at the London Conference have been circulated widelyin Oxford and elsewhere.

An earlier version of the work appeared in I. Block (ed.),Perspectives on the Philosophy of Wittgenstein (Basil Blackwell,Oxford, 1981, xii + 322 pp.). Work on t hat version waspartially supported by a Guggenheim Fellowship, by aVisiting Fellowship at All Souls Col lege, Oxford, by asabbatical f rom Princeton Univers ity, and by the NationalScience Foundation (USA). Work on the present expandedversion was partially supported by a grant from the AmericanCouncil of Learned Societies, by a sabbatical from PrincetonUniversity, and by an Oscar Ewing Research Grant at IndianaUniversity.

,I.I.I

I

Introductory

Wittgenstein's celebrated argument against 'private language'has been discussed so often th at the utility of yet anotherexposition is certainly open to question. Most of the exposit ion which follows occur red to the present wri te r some timeago, in the academic year 1962-3. At that time this approach toWittgenstein's views struck the present writer with the forceof a revelation: what had previously seemed to me to be asomewhat loose argument for a fundamentally implausibleconclusion based on dubious and controversial premises nowappeared to me to be a powerful argument, even if theconclusions seemed even more radical and, in a sense, moreimplausible, than before. I thought at that t ime that I had seenWittgenstein's argument from an angle and emphasis verydifferent from the approach which dominated standardexpositions. Over the years I came to have doubts. First of all,at times I became unsure that I could formulate Wittgenstein'selusive position as a clear argument. Second, the elusive natureof the subject made it p o ~ s i b l e t o in terp ret s ome of thestandard literature as perhaps seeing the argument in the sameway after all. More important, conversations over the yearsshowed that, increasingly, others were seeing the argumentwith the emphases I preferred. Nevertheless, recent expositions by very able interpreters differ enough from the


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2 Introductory Introductory 3following to make me think that a new exposition may still beofuse. IA common view of the 'private language argument' inPhilosophical Investigations assumes that it begins with section243, and that it continues in the sections immediatelyfollowing. 2 This v iew takes the argument to deal primarilywith a problem about 'sensation language'. Further discussionof the argument in this t radi tion, both in support and incriticism, emphasizes such questions as whether the argumentinvokes a form of the verification principle, whether the formin question is justified, whether it is applied correctly tosensation language, whether the argument rests on anexaggerated scepticism about memory, and so on. Some

I Looking through some of t he mos t distinguished commentaries onWittgenstein of the last ten or fifteen years, I find some thatstill treat thediscussion of rules cursorily, virtually no t at all, as if it were a minortopic. Others, who discuss bothWittgenstein's views on the philosophyofmathematics and his views on sensations in detail, treat the discussionof rules as ifit were important for Wittgenstein's views on mathematicsand logical necessity but separateit from 'the privatelanguage argument'.Since Wittgenstein has more than one way of arg uing for a gi venconclusion, and even of presenting a single argument, to defend thepresent exegesis I need no t necessarily arguethat these othercommentaries are in error. Indeed, the y may give important and illuminatingexpositions of facets of the Investigations and its argument deemphasizedor omitted in this essay. Nevertheless, in emphasis they certainly differconsiderably from the present exposition.

2 Unless otherwise specified (explicitly or contextually), references are toPhilosophical Investigations. The small numbered units of the Investigationsare termed 'sections' (or 'paragraphs'). Page references are used only if asection reference is not possible, as in the secondpart of the Investigations.Throughout I quotethe standardprinted English translation (byG. E. M.Anscombe) and make no attempt to que st ion it except in a very fewinstances. Philosophical Investigations (x+232 pp., parallel German andEnglish text) has undergone several editions since its first publication in1953 but t he par ag raph ing and pagina ti on r emain the same. Thepublishers are Basil Blackwell, Oxford and Macmillan, New York.

This essay does no t proceed by giving detailed exegesis ofWittgenstein 's text but r at he r develops t he a rgumen ts in its own way. Irecommend tha t the reader reread the Investigations in the light of thepresent exegesis and see whether it illuminates thetext.

crucial passages in the discussion following 243 _ forexample, such celebrated sections as 258 and 265 -have beenn o t o r i o ~ s l y o b s c ~ r e to commentators, andit has been thoughtthat theIr proper mterpretation would provide the key to the'private language argument'.

In my view, the real 'private language argument' is to befound in the sections preceding 243. Indeed, in 202 theconclusion is already stated explicitly: "Hence it is not possible toobey a rule 'privately': otherwise thinking one was obeying arule would be the same thing as obeying i t . " I do not think thatWittgenstein here thought of himself as anticipating an argu

m e n ~ he was to give in greater detail later. On thecontrary, thecruCIal considerations are all contained in the discussionleading up to the conclusion s tated in 202. The sectionsfollowing 243 are meant to be read in the light of thepreceding ~ i s c u s s i o n ; difficult as they are in any case, they arel?uch less lIkely to be understood if they are read in isolation.The ' p ~ i v a t e language argument' as applied to sensations is onlya speCIal case of much more general considerations aboutlanguage previously argued; sensations have a crucial role asan ( ~ p p a r ~ n t l y ) convincing counterexample to the generalconSIderatIons previously stated. Wittgenstein therefore goesover. the gro.und ~ g a i n in this special case, marshalling newspeCIfic conSIderatIons appropriate to it. It should be borne inmi?d t h a ~ Philosophical Investigations is not a systematic

~ h I l o s o p h l C a l work where conclusions, once definitely estab-l I s ~ e d , need not be reargued. Rather the Investigations iswntten as a perpetual dialectic, where persisting worries,expressed by the voiceofthe imaginary interlocutor, areneverdefinitively silenced. Since the work is no t presented in theform of a deductive argument with definitive theses as

c o ~ c l u s i o n s , the same ground is covered repeatedly, from thepomt of view of various special cases and from differentangles, with the hope that the en ti re process will help thereader see the problems rightly.The basic structure of Wittgenstein' s approach can bepresented briefly as follows: A certain problem, or in Humean


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4 Introductory Introductory 5terminology, a 'sceptical paradox', is presented concerningthe notion of a rule. Following this, what Hume would havecalled a 'sceptical solution' to the problem is presented. Thereare two areas in which the force, both of the paradox and of itssolution, are most likely to be ignored, and with respect towhich Wittgenstein's basic approach is most likely to seemincredible. One such area is the notion ofa mathematical rule,such as the rule for addition. The other is our talk of our owninner experience, of sensations and other inner states. Intreating both these cases, we should bear in mind the basicconsiderations about rules and language. Although Wittgenstein has already discussed these basic considerations inconsiderable generality, the structure ofWittgenstein's workis such that the special cases of mathematics and psychologyare not simply discussed by citing a general ' result ' alreadyestablished, but by going over these special cases in detail, inthe light of the previous treatment of the general case. By sucha discussion, it is hoped that both mathematics and the mindcan be seen rightly: since the temptations to see them wronglyarise from the neglect of the same basic considerations aboutrules and language, the problems which arise can be expectedto be analogous in the two cases. In my opinion, Wittgensteindid not view his dual interests in the philosophy ofmind andthe philosophy ofmathematics as interests in two separate, atbest loosely related, subjects, as someone might be interestedboth in music and in economics. Wittgenstein thinks of thetwo subjects as involving the same basic considerations. Forthis reason, he calls his investigation of the foundations ofmathematics "analogous t o our investigation of psychology"(p. 23 2). It is no accident that essentially the same basicmaterial on rules is included in both Philosophical Investigationsand in Remarks on the Foundations ofMathematics, 3 both times as] Basil Blackwell , Oxford, 1956, xix+ 204 pp. In the first edition ofRemarks on the Foundations of Mathematics the editors assert (p. vi) thatWittgenstein appears originally to have intended to include some of thematerial on mathematics in Philosophical Investigations.The third edition (1978) includes more material than earlier editions

the basis of the discussions of the philosophies ofmind and ofmathematics, .respectively, which follow.In the following, I am largely trying to present Wittgenstein's argument, or, more accurately, that set of problems

and arguments which I personally have gotten out of readingWittgenstein. With few exceptions, I am not trying to presentviews ofmy own; neither am I trying to endorse or to criticizeWittgenstein's approach. In some cases, I have found a precisestatement of the problems and conc lusions to be elusive.Although one has a strong sense that there is a problem, arigorous statement of it is difficult. I am inclined to think thatWittgenstein's later philosophical style, and the difficulty hefound (see his Preface) in welding his thought into a conventional work presented with organized arguments and conclusions, is not simply a stylistic and literary preference, coupledwith a penchant for a certain degree ofobscurity,4 but stems inpart from the nature ofhis subject. 5

I suspect - for reasons that will become clearer later- that toattempt to present Wittgenstein's argument precisely is tosome extent to falsify it. Probably many ofmy formulationsand recastings of the argument are done in away Wittgensteinwould not himself approve. 6 So the present paper should bethought of as expounding neither 'Wittgenstein's' argumentno r 'Kripke's': rather Wittgenstein's argument as it struckKripke, as it presented a problem for him.

As I have said, I think the basic 'private language argument'precedes section 243, though the sections following 243 are no

and rearranges some of the sections and divisions of earlier editions.When I wrote the present work, I used the f irst edition. Where thereferences differ, the equivalent third edition reference is given in squarebrackets.

4 Personally I feel, however, that the role of stylistic considerations herecannot be denied. It is clear that purely stylisticand literary considerationsmeant a great deal to Wittgenstein. His own stylistic preferenceobviously contributes to the difficulty of his work as well as to its beauty.5 See the discussion of this point in pages 69-70 below.

6 See again the same discussion in pages 69-70.


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6 Introductorydoubt of fundamental importance as well. I propose to discussthe problem of'private language' initiallywithoutmentioningthese latter sections at all. Since these sect ions are oftenthought to be the 'private language argument', to some such aprocedure may seem to be a presentation ofHamlet withoutthe prince. Even if this is so, there are many other interestingcharacters in the play. 77 Looking over what I have written below, I find myselfworried that thereader may lose the main thread of Wittgenstein's argument in theextensive treatment of finer points. In particular, the treatment of thedispositional theory below became so extensive because I heard it urgedmore than once as an answer to the sceptical paradox. That discussionmay contain somewhat more of Kripke's argumentation in support ofWittgenstein rather than exposition of Wittgenstein's own argumentthan does most of the rest of thisessay. (Seenotes 19 and 24 for some of theconnections. The argument is, however, inspired by Wittgenstein'soriginal text. Probably the par t wi th the least direct inspiration fromWittgenstein's text is the argument that our dispositions, like our actualperformance, are ho t potentially infinite. Even this, however, obviouslyhas i ts origin in Wittgenstein's parallel emphasis on t he fact t ha t weexplicitly think of only finitely many cases of any rule.) The treatmentbelow (pp. 38-39) of simplicity is an example of an objection that, as faras I know, Wittgensteinnever considers himself. I think thatmy reply isclearly appropriate, assumi ng th at I have unde rs to od the rest ofWittgenstein's position appropriately. I urge the reader to concentrate,on a first reading, on understanding the intuitive force ofWittgenstein'ssceptical problem and to regard byways such as these as secondary.

2

The WittgensteinianParadox

In 20I Wittgenstein says, "this was ou r paradox: no courseof action could be determined by a rule, because every courseof action can be made out to accord with the rule." In thissection of the present essay, in my own way I wi ll attemptto develop the 'paradox' in question. The 'paradox' is perhapsthe central problem ofPhilosophical Investigations. Even someone who disputes the conclusions regarding 'private language', and the philosophies ofmind, mathematics, and logic,thatWittgenstein draws from his problem, might well regardthe problem itselfas an important contribution to philosophy.It may be regarded as a new form of philosophical scepticism.Following Wittgenstein, I will develop the problem initially

with respect to a mathematical example, though the relevantsceptical problem applies to all meaningful uses oflanguage. I,l ike almost all English speakers, use the word ' plus ' and thesymbol '+ ' to denote a well-known mathematical function,addition. The function is defined for all pairs of positiveintegers. By means of my external symbolic representationand my internal mental representation, I 'grasp' the rule foraddition. One point is crucial to my 'grasp' of this rule.Although I myselfhave computed only finitely many sums inthe past, the rule determines my answer for indefinitely manynew sums that I have never previously considered. This is the

.....I


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8 The Wittgensteinian Paradox The Witt,\?ensteinian Paradox 9whole point of the notion that in learning to add I grasp a rule:my past intentions regarding addit ion determine a uniqueanswer for indefinitely many new cases in the future.Let me suppose, for example, that '68 + 57' is a computation

that I have never performed before. Since I have performed even silently to myself, let a lone in my publicly observablebehavior - only finitely many computations in the past, suchan example surely exists. In fact, the same finitude guaranteesthat there is an example exceeding, in both its arguments, allprevious computations. I shall assume in what follows that'68 + 57' serves for this purpose as well.I perform the computation, obtaining, of course, theanswer '125'. I am confident, perhaps after checking mywork, that' 125' is the correct answer. It is correct both in thearithmetical sense that 125 is the sum of68 and 57, and in themetalinguistic sense that 'plus', as I intended to use that wordin the past, denoted a function which, when applied to thenumbers I called '68' and '57', yields the value 125.

Now suppose I encounter a bizarre sceptic. This scepticquestions my certainty about my answer, in what Ijust calledthe 'metalinguistic' sense. Perhaps, he suggests, as I used theterm 'plus' in the past, the answer I intended for '68+ 57'should have been '5'! Of course the sceptic's suggestion isobviously insane. My initial response to such a suggestionmight be that the challenger should go back to school and learnto add. Let the challenger, however, continue. After all, hesays, if ! am now so confident that, as I used the symbol'+',my intention was that '68+ 57' should turn out to denote 125,this cannot be because I explicitly gavemyselfinstructions that125 is the result of performing the addition in this particularinstance. By hypothesis, I did no such thing. But ofcourse theidea is that, in this new instance, I should apply the very samefunction or rule that I applied so many t imes in the past. Butwho is to say what function this was? In the past I gavemyselfonly a finite number of examples instantiating this function.All, we have supposed, involved numbers smaller than 57. Soperhaps in the pas t I used 'plus' and '+ ' to denote a function

which I will call 'quus' and symbolize by 'EB'. It is defined by:xEBy=x+y, ifx, y < 57= 5 otherwise.

Who is to say that this is not the function 1previouslymeant by'+'?The sceptic claims (or feigns to claim) that I am nowmisinterpreting my own previous usage. By 'plus ', he says, 1always meant quus;8 now, under the influence of some insanefrenzy, or a bout ofLSD, I have come to misinterpretmy ownprevIous usage.Ridiculous and fantastic though i t is, the scept ic 's hypo-thesis is no t logically impossible. To see this, assume thecommon sense hypothesis that by '+ ' I did mean addition.Then it would be possible, though surprising, that under theinfluence of a momentary 'high', I should misinterpret all mypast uses ofthe plus sign as symbolizing the quus function, andproceed, in conflict with my previous linguistic intentions, tocompute 68 plus 57 as 5. (I would have made a mistake, not inmathematics, but in the supposition that I had accorded withmy previous linguistic intentions.) The sceptic is proposingthat 1have made a mistake precisely of this kind, but with aplus and quus reversed.Now if the sceptic proposes his hypothesis sincerely, he is

crazy; such a bizarre hypothesis as the proposal that I alwaysmeant quus is absolutely wild. Wild it indubitably is, no doubtit is false; bu t if it is false, there must be some fact about mypast usage that can be cited to refute it. For although thehypothesis is wild, it does not seem to be apriori impossible.8 Perhaps I should make a remark about such expressions as "By 'plus' Imeant quus (or plus)," "By 'green' I meant green," etc . I am no t familiarwithan accepted felicitous conventionto indicate the objectof the verb 't omean' . There are two problems. First, if one says, "By 'the woman whodiscovered radium' I meant the woman who discovered radium," theobject can be interpreted in two ways. It may stand fo r a woman (MarieCurie), in which case the assertion is true only if 'meant' is used to meanreferred to (as it can be used); or it may be used to denote the meaning ofthe quoted expression, not a woman, in which case the assertion is true


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10 The Wittgensteinian Paradox The Wittgensteinian Paradox I I

,

Of course this bizarre hypothesis, and the references toLSD, or to an insane frenzy, are in a sense merely a dramaticdevice. The basic point is this. Ordinarily, I suppose that, incomputing '68+ 57' as I do, I do not simply make anunjustified leap in the dark. I fol low direct ions I previouslygave myself that uniquely determine that in this new instance Ishould say '125'. What are these directions? By hypothesis, Inever explicitly told myself that I should say '125' in this veryinstance. No r can I say that I should simply 'do the same thing

with 'meant' used in the ordinary sense. Second, as is illustrated by'referred to', 'green', 'quus', etc. above, as objects of 'meant' , onemustuse various expressions as objec ts in an awkward manner contrary tonormal grammar. (Frege's difficulties concerning unsaturatedness arerelated.) Both problems tempt one to put the object in quotation marks,l ike the subject; but such a usage conf licts with the convention ofphilosophicallogic that a quotation denotes the expression quoted. Somespecial 'meaning marks', as proposed for example by Dav id Kaplan,could be useful here. If one is content to ignore the first difficulty andalways use 'mean' to mean denote (for most purposes of the presentpaper, such a reading would suit at least as well as an intensional one;often I speak as ifit is a numericalfunction that is meant by plus), thesecondproblemmight lead one to nominalizethe objects- 'plus' denotes the plu'sfunction, 'green' denotes greenness, etc. I contemplated using italics(" 'plus ' means plus"; "'mean' may mean denote"), but I dec ided tha tnormally (except when italics areotherwise appropriate, especially whena neologism like 'quus' is introduced for the first t ime), I will write theobject of 't o mean' as an ordinary roman object. The convention I haveadopted reads awkwardly in the written language but sounds ratherreasonable in the spoken language.

Since use-mention distinctions are significant for the argument as Ig ive i t, I try to remember to use quotation marks when an expression ismentioned. However, quotation marks are also used for other purposeswhere they might be invoked in normal non-philosophical Englishwriting (for example, in the case of ' ' 'meaning marks'" in the previousparagraph, or " 'quasi-quotation'" in the next sentence). Readers familiarwith Quine's 'quasi-quotation' will be aware that in some cases I useordinary quota tion where logical purity would require that I usequasi-quotation or some similar device. I have not t ri ed to be carefulabout this matter, since I am confident that in practice readers will not beconfused.

I always did,' if this means 'compute according to the ruleexhibited by my previous examples.' That rule could just aswell have been the rule for quaddition (the quus function) asfor addition. The idea that in fact quaddition is what I meant,that in a sudden frenzy I have changed my previous usage,dramatizes the problem.

In the discussion below the challenge posed by the sceptictakes two forms. First, he questions whether there is any factthat I meant plus, not quus, tha t will answer his scepticalchallenge. Second, he questions whether I have any reason tobe so confident that now I should answer '125' rather than '5'.The two forms of the challenge are related. I am confident thatI should answer '125' because I am confident that this answeralso accords with what I meant. Neither the accuracy of mycomputation nor ofmy memory is under dispute. Soit oughtto be ag,reed that if l meant plus, then unless I wish to changemy usage, I am justified in answering (indeed compelled toanswer) '125', no t '5'. An answer to t he sceptic must satisfytwo conditions. First, it must give an account ofwhat fact it is(aboutmy mental state) that constitutesmy meaning plus, no tquus. Bu t further, there is a condition tha t any putat ivecandidate for such a fact must satisfy. It must, in some sense,show how I amjustified in giving the answer '125' to '68+57'.The 'directions' mentioned in the previous paragraph, thatdetermine what I should do in each instance, must somehowbe 'contained' in any candidate for the fact as to what I meant.Otherwise, the sceptic has not been answered when he holdsthat my present response is arbitrary. Exactly how thiscondition operates will become much clearer below, after wediscuss Wittgenstein's paradox on an intuitive level, when weconsider various philosophical theories as to what the fact thatI meant plus might consist in. There will be many specificobjections to these theories. But all fail to give a candidatefor afact as to what I meant that would show that only '125', no t'5', is the answer I 'ought' to give.The ground rules ofou r formulation of the problem should

be made clear. For the sceptic to converse with me at all, we


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must have a common language. So I am supposing that thesceptic, provisionally, is not q u e ~ t i o n i n g my present use of theword 'plus'; he agrees that, according to my present usage, '68plus 57' denotes 125. Not only does he agree with me on this,he conducts the entire debate with me in my language as Ipresently use it. He merely questions whether my present usageagrees with my past usage, whether I am presently conformingto my previous linguistic intentions. The problem is no t "Howdo I know that 68 plus 57 is 12 5?", which should be answeredby giving an arithmetical computation, but rather "How do Iknow t ha t '68 plus 57', as I meant 'plus' in the past, shoulddenote 125?" If the word 'plus' as I used i t in the past, denotedthe quus function, not the plus function (,quaddition' ratherthan addition), then my past intention was suchthat, asked forthe value of'68 plus 57', I should have replied '5'I put the problem in this way so as to avoid confusingquestions about whether the discussion is taking place 'bothinside and outside language' in some illegitimate sense.9 If weare querying the meaning of the word 'plus', how can we use it(and variants, like 'quus') at the same time? So I suppose thatthe sceptic assumes that he and I agree in our present uses of theword 'plus': we both use i t t o denote addition. He does not - atleast initial ly - deny or doubt that addit ion is a genuinefunction, defined on all pairs of integers, nor does he deny thatwe can speak of it. Rather he asks why I now believe that by'plus' in the past, I meant addition rather than quaddition. If Imeant the former, then to accord with my previous usage Ishould say' 125' when asked to give the result ofcalculating '68plus 57'. If ! meant the latter, I should say '5'

The present exposition tends to differ from Wittgenstein'soriginal formulations in taking somewhat greater care to makeexplicit a distinction between use and mention, and betweenquestions about present and past usage. About the presentexample Wittgenstein might simply ask, "How do I knowthat I should respond '125' to the query '68+ 57'?" or "How do9 I be lieve I got the phrase "both inside and outside language" from a

conversation with Rogers Albritton.

The Wittgensteinian Paradox12 The Witt,\?ensteinian Paradox

j

13I know that '68+ 57' comes out 125?" I have found that whenthe problem is formulated this way, some listeners hear it as asceptical problem about arithmetic: "How do I know that68+ 57 is 125?" (Why not answer this question with amathematical proof?) At least at this stage, scepticism about

a r i t h m e t ~ c s h o u ~ d not be taken to be in question: we mayassume, If we wIsh, that 68+ 57 is 125. Even if the question isreformulated 'metalinguistically' as "How do I know that'plus', as I use it, deno tes a funct ion tha t, when applied to 68and 57, yields 125?", one may answer, "Sure ly I know that'plus' denotes the plus function and accordingly that '68 plus57' denotes 68 plus 57. But i f! know arithmetic, I know that 68plus 57 is 125. So I know that '68 plus 57' denotes 125!" Andsurely, if ! use language at all, I cannot doubt coherently that

' p ~ u s ' , as I now use it, denotes plus! Perhaps I cannot (at least atthIS stage) doubt this about my present usage. But I can doubtthat my past usage of 'plus' denoted plus. The previousremarks - about a frenzy and LSD - should make this quiteclear.. Let ~ e repeat the problem. The sceptic doubts whether anyInstructIons I gave myself in the past compel (or justify) theanswer '125' rather than '5'. He puts the challenge in terms ofasceptical hypothesis about a change in my usage. Perhapswhen I u s ~ d the term 'plus' in the past, I always meant quus: byhypothesIs I never gave myself any explicit directions thatwere incompatible with such a supposition.Of course, ultimately, if the sceptic is right, the concepts ofmeaning and of intending one function rather than anotherwill make no sense. For the sceptic holds that no fact about mypast history - nothing that was ever in my mind, or in myexternal behavior - establishes that I meant plus rather thanquus. (Nor, of course, does any fact estab lish tha t I meantquus!) But if this is correct; there canofcourse be no fact aboutwhich function I meant, and if there canbe no fact about whichparticular function I meant in the past, there can be none in thepresent either. But before we pull the rug out from under ou row n feet, we begin by speaking as if the notion that at present


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14 The Wittgensteinian Paradox The Wittgensteinian Paradox ISwe mean a certain function by 'plus' is unquestioned andunquestionable. Only past usages are to be questioned.Otherwise, we will be unable to formulate our problem.Another important rule of the game is that there are no

limitations, in particular, no behaviorist limitations, on thefacts that may be cited to answer the sceptic. The evidence isnot tobe confined to thatavailable to anexternalobserver, whocan observe my overt behavior but not my internal mentalstate. It would be interesting if nothing in my external be-havior could show whether I meant plus or quus, butsomething aboutmy inner state could. But the problem here ismore radical. Wittgenstein's philosophy of mind has oftenbeen viewed as behavioristic, bu t to the extent thatWittgen-stein may (or may not) be hosti le to the 'inner', no suchhostility is to be assumed as a premise; it is to be argued as aconclusion. So whatever 'looking into my mind' may be, thesceptic asserts that even ifGod were to do it, he still could no tdetermine that I meant addition by 'plus'.

This feature ofWittgenstein contrasts, for example, withQuine's discussion of the 'indeterminacy of translation'. 10There are many points of contact between Quine's discussionand Wittgenstein's. Quine, however, is more than content toassume that only behavioral evidence is to be admitted into hisdiscussion. Wittgenstein, by contrast, undertakes an extensiveintrospectiveI I investigation, and the results of the investiga-10 See W. V. Quine, Word and Object (MIT, The Technology Press,Cambridge, Massachusetts, 1960, xi + 294 pp.), especially chapter 2,'Translationand Meaning' (pp. 26-79). Seealso Ontological Relativity andOther Essays (Columbia University Press, New York and London, 1969,viii+165 pp.), especially the first three chapters (pp. 1-90); and see also..On the Reasons for the Indeterminacy of Translation," The Journal ofPhilosophy, vol. 67 (1970), pp. 178-83.Quine's views are discussed further below, see pp. 55-7.

I I I do not mean the term ' introspective' to be laden with philosophicaldoctrine. Ofcourse much of the baggage that has accompanied this termwould be objectionable to Wittgensteinin particular. I simply mean thathe makes use, in his discussion, of ourown memories and knowledge ofour 'inner' experiences.

tion, as we shall see, form a key feature of his argument.Further, the way the sceptical doubt is presented is notbehavioristic. It is presented from the 'inside'. Whereas Quinepresents the problem about meaning in terms of a linguist,trying to guess what someone else means by his words on thebasis of his behavior, Wittgenstein's challenge can be pre-sented to me as a question about myself: was there some pastfact about me - what I 'meant' by plus- that mandates what Ishould do now?To return to the sceptic. The sceptic argues that when I

answered '125' to the problem '68+57', my answer was anunjustified leap in the dark; my past mental history is equallycompatible with the hypothesis that I meant quus, andtherefore should have said'5'. We can put the problem thisway: When asked for the answer to '68+57', I unhesitatinglyand automatically produced '125', but it would seem that ifpreviously I never performed this computation explicitly Imight just as well have answered '5'. Nothingjustifies a bruteinclination to answer one way rather than another.Many readers, I should suppose, have long been impatient

to protest that ou r problem arises only because ofa ridiculousmodel of the instruction I gave myself regarding 'addition'.Surely I did not merely give myself some finite number ofexamples, from which I am supposed to extrapolate the wholetable ("Let'+' be the function instantiated by the followingexamples: ... "). No doubt infinitely many functions arecompatible with that. Rather I learned - and internalizedinstructions for - a rule which determines how addition is tobecontinued. What was the rule? Well, say, to takei t in its mostprimitive form: suppose we wish to add x and y. Take a hugebunch ofmarbles. First count ou t x marbles in one heap. Thencount ou t y marbles in another. Pu t the two heaps togetherandcount ou t the number ofmarbles in the union thus formed.The result is x+y. This set of directions, I may suppose, Iexplicitly gave myself at some earlier time. It is engraved onmy mind as on a slate. It is incompatible with the hypothesisthat Imeant quus. It is this set ofdirections, not the finite list of


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16 The Wittgensteinian Paradox The Wit(l?etlsteinian Paradox 17particular additions I performed in the past, thatjustif ies anddetermines my present response. This consideration is, afterall, reinforced when we think what I really do when I add 68and 57. I do no t reply automatically with the answer '125' no rdo I consult some non-existent past instructions that I shouldanswer '125' in this case. Rather I proceed according to analgorithm for addition that I previously learned. The algorithmis more sophisticated and practically applicable than theprimitive one just described, but there is no diffe rence inprinciple.

Despite the initial plausibility of this objection, the sceptic'sresponse is all too obvious. True, if 'count', as I used the wordin the past, referred to the act of counting (and my other pastwords are correctly interpreted in the s tanda rd way), then'plus' must have stood for addition. But I applied 'count', like'plus', to only finitely many past cases. Thus the sceptic canquestionmy present interpretation ofmy past usage of'count'as he did with 'plus'. In particular, he can claim that by 'count 'I formerly meant quount, where to 'quount' aheap is to countitin the ordinary sense, unless the heap was formed as the unionof two heaps, one ofwhich has 57 or more items, in which caseone must automatically give the answer' 5'. I t is clear that if inthe past 'counting' meant quounting, and if! follow the rulefor 'plus'that was quoted so triumphantly to thesceptic, Imustadmit that ' 68+57' must yield the answer '5'. Here I havesupposed that previously 'count' was never applied to heapsformed as the union of sub-heaps either of which has 57 ormore elements, bu t if this particular upper bound does no twork, another will do. For the point is perfectly general: if'plus' is explained i n t erms of 'counting', a non-standardinterpretation of thelatter will yield a non-standardinterpretation of the former. 1212 The same objection scotches a related suggestion. It might be urged that

thequus function is ruled ou t as an interpretation of' +' because it fails tosatisfy some of thelawsI accept for' + ' (for example, it is no t associative;we could have defined it so as no t even to be commutative). One mighteven observe that, on the natural numbers, addition is the only functionthat satisfies certain laws that I accept- the 'recursion equations' for +: (x)

It is pointless ofcourse to protest tha t I intended the result ofcounting a heap to be independent of its composition in terms ofsub-heaps. Let me have said this to myself as explicitly aspossible: the sceptic will smilingly reply that once again I ammisinterpreting my past usage, that actually 'independent'formerly meant qUindependent, where 'qu independent'means ...

Here of course I am expounding Wittgenstein's wellknown remarks about "a rule for interpreting a rule" . It istempting to answer the sceptic by appealing from one rule toanother more 'basic' rule. But the sceptical move can berepeated at the more 'basic' level also. Eventually the processmust stop - "justifications come to an end somewhere" - and Iam left with a rule which is completely unreduced to anyother. How can I justify my present application of such a rule,when a sceptic could easily interpret i t so as to yield any of anindefinite number of other results? It set;ms that my application of it is an unjus tified stab in the dark. I apply the ruleblindly.

Normally, when we consider a mathematical rule such asaddition, we think ofourselves as guided in ou r application ofitto each new instance. Just this is the difference betweensomeone who computes new values of a function andsomeone who calls out numbers at random. Given my pastintentions regarding the symbol'+', one and only one answer

(x+o=x) and (x) (y) (x+y'=(x+y)') where the stroke or dash indicatessuccessor; these equations are sometimes called a 'definition' of addition.The problem is that the other s igns used in these laws ( the universalquantifiers, the equality sign) havebeen applied in only a finite number ofinstances, and they can be given non-standard interpretations that will fitnon-standard interpretations of'+' . Thus for example '(x)' might meanfor every x


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18 The Wittgensteinian Paradox The W i t ~ ~ e n s t e i n i a n Paradox 19is dictated as the one appropriate to '68+57'. On the otherhand, although an intelligence tester may suppose that there isonly one possible continuation to the sequence 2,4,6, 8, ... ,mathematical and philosophical sophisticates know t hat anindefinite number of rules (even rules stated in terms ofmathematical functions as conventional as ordinary polynomials) arecompatible with anysuchfiniteinitialsegment. Soif the tester urges me to respond, after 2, 4, 6, 8, ... , with theunique appropriate next number, the proper response is thatno such unique number exists, nor is there any unique (ruledetermined) infinite sequence that continues the given one.The problem can t hen be put this way: Did I myself , in thedirections for the future that I gave myself regarding '+',really differ from the intelligence tester? True, I may n otmerely stipulate that '+' is to be a function instantiated by afinite number ofcomputations. In addition, I may give myselfdirections for the further computation of'+', stated in termsof other functions and rules. In turn, I may give myselfdirections for the further computation of these functions andrules, and so on. Eventually, however, the process must stop,with 'ultimate' functions and rules that I have s tipulated formyself only by a finite number of examples, just as in theintelligence test. If so, is no t my procedure as arbitrary as thatof the man who guesses the continuation of the intelligencetest? In what sense is my actual computation procedure,following an algorithm that yields '125', more justified bymypast instructions than an alternative .procedure that wouldhave resulted in OS'? Am I no t simply following an unjustifiable impulse?!313 Few readers, I s uppose, wi ll by this time be tempted to appeal a

determination to "go on the same way" as before. Indeed, I mention it atthis point p rima ri ly t o remove a possible misunderstanding of thesceptical argument, no t to counter a possible reply to it. Some followersof Wittgenstein - perhaps occasionally Wittgenstein himself - havethought tha t his point involves a rejection of 'absolute identity' (asopposed to some kind of ' relative' identity). I do not see tha t thi s is so ,whether or not doctrines of ' relative ' identity are correct on o thergrounds. Let identity beas 'absolute' as one pleases: itholds only between

O f course, these problems apply throughout language andare no t confined to mathematical examples, though it is withmathematical examples that t hey can be most smoothlybrought out. I think that I havelearned the term ' table' in sucha way that it will apply to indefinitely many future items. So Ican apply the term to a new situation, say when I enter theEiffel Tower for the first time and see a table at the base. Can Ianswer a sceptic who supposes that by 'table' in the past Imeant tabair, where a ' taba ir ' is anything that is a table notfound at the base of the Eiffel Tower, or a chair found there?Did I think explicitly of the EiffelTower when I first 'graspedthe concept of ' a table, gave myselfdirections forwhat Imeantby 'table'? And even if I did think of the Tower, cannot anydirections I gave myself mentioning it be reinterpretedcompatibly with the sceptic's hypothesis? Most important

each thing and itself. Then the plus function is identical with itself, andthe guus function is identical with itself. None of thiswill tell me whetherI referred to the plus function or t o t he guus funct ion in the past, northerefore will it tell me which to use in order to apply the same functionnow.

Wittgenstein does insist (2 I5- 16) that the law of identity ('everything is identicalwith itself) gives noway out of this problem. It shouldbe clear enough that this is so (whether or not the maxim should berejected as 'useless'). Wittgenstein sometimes writes (225-27) as if theway we givea responsein a new case determines what we call the 'same',as if the meaning of ,same' varies from case to case. Whatever impressionthis gives, it need not relate to doctrines of relative and absoluteidentity.The point (which can be ful ly understood only after the third sectionof the present work) can be put this way: If someone who computed,+ ' as we do for small arguments gave bizarre responses, in the styleof 'guus', for larger arguments , and ins is ted tha t he was 'going on thesame way as before' , we would not acknowledge his c la im tha t he was'going on in t he s ame way' as for the small arguments. What we callthe 'right' response determines what we call 'going on in the same way'.None of thi s in i tsel f implies tha t identity is ' relative' in senses that'relative identity' has been used elsewhere in the literature.In fairness to Peter Geach, the leading advocate of the 'relativity' ofidentity, I should mention (lest thereader assume I had him in mind) thathe is not one of those I have heard expound Wittgenstein's doctrine asdependent on a denial of ,absolute' identity.


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14 See Nelson Goodman, Fact, Fiction, and Forecast (]rd ed., Bobbs-Merrill;Indianapolis, 1973, xiv+I]1 pp .) , especiall y ch. III, 4, pp. 72-8r.

15 The exact definition of 'grue' is unimportant. It is best to suppose thatpastobjects were grue if and only if they were (then) green while presentobjects aregrue if andonly if they are (now) blue. Strictlyspeaking, this isnot Goodman's original idea, but it is probably most convenient forpresent purposes. Sometimes Goodman writes this way as well.

16 'Schmolor', with a slightly different spelling, appears in Joseph Ullian,"More on 'Grue ' and Grue," The Philosophical Review, vol. 70 (1961),pp. 3 8 ~ .

for the 'private language' argument, the point of courseapplies to predicates of sensations, visual impressions, thelike, as well: "How do I know that in working ou t the senes+ 2Imust write "20,004,20,006" and not "20,004, 20,008"?- (Theguestion: "How do I know that this color is 'red'?" issimilar.)" (Remarks on the Foundations ofMathematics, I, 3.) Thepassage strikingly illustrates a central thesis of this essay: t.hatWittgenstein regards the fundamental problems of the phIlosophy ofmathematics and of the 'private language argument'- the problem of sensation language - as at root identical,stemming from his paradox. The whole of3 is a succinct andbeautiful statement of theWittgensteinian paradox; indeed thewhole initial section of part I of Remarks on the Foundationsof Mathematics is a development of the problem with specialreference to mathematics and logical inference. It has beensupposed that all I need to do to determine my use of the word'green' is to have a!1 image, a sample, of green that I bring tomind whenever I apply the word in the future. When I use thisto justifymy application of' green' to a new object, should notthe sceptical problem be obvious to any reader ofGoodman?14Perhaps by 'green', in the past I meant grue, 15 and the colorimage, which indeed was grue, was meant to direct me toapply theword 'green' togrue objects always. If the blue objectbefore me now is grue, then it falls in the extension of'green',as I meant it in the past. It is no help to suppose that in the pastIstipulated that 'green' was to apply to all and only those things'o f the same color as' the sample. The sceptic can reinterpret'same color' as same schmolor, 16 where things have the sameschmolor if . . .

20 The Wittgensteinian Paradox The Wittgensteinian Paradox 2 ILet us return to the example of 'plus' and 'gUllS'. We have

just summarized the problem in terms of the basis of mypresent particular response: what tells me tha t I shou ld say'125' and no t '5'? Of course the problem can .be puteguivalently in terms of the sceptical gue.ry regardIng. mypresent intent: nothing in my mental hIstory estabhsheswhether I meant plus or guus. So formula ted, the problemmay appear to be epistemological - how can anyone ~ n o ~which of these I meant? Given, however, that everythIng Inmy mental history is compatible both with the conclusion ~ h ~ tI meant plus and with the conclusion that I meant ~ u u s , It ISclear tha t the sceptical challenge is no t really an epIstemological one. It purports to show t ha t nothing in ~ ~ mentalhistory of past behavior - not even what an ommSCIent Godwould know - could establish whether I meant plus or guus.But then i t appears to follow that there was no fact about methat constitutedmy having meant plus rather than guus. Howcould there be, if nothing in my internal mental history orexternal behavior will answer the scepticwho supposes that infact I meant gUllS? If there was no such thing as my meaningplus rather than guus in the past, neither can there be any suchthing in the present. When we initially presented the ~ a r a d o x ,we per fo rce used language, t ak ing p re sent meamngs forgranted. Now we see, as we expected, that this provisionalconcession was indeed fictive. There can be no fact as to what Imean by 'plus', or any other word at any time. The laddermust finally be kicked away. .

This, then, is the sceptical paradox. When I respond In oneway rather than another to such a problem as '68+57', I canhave no justif ication for one response rather than another.Since the sceptic who supposes that I meant guus cannot beanswered, there is no fact about me that distinguishes b e t ~ e e nmy meaning plus and my meaning guus. Indeed, there. IS nofact about me that distinguishes between my meanIng adefinite function by 'plus' (which determines my responses innew cases) and my meaning nothing at all.Sometimes when I have contemplated the situation, I havehad something of an eerie feeling. Even now as I write, I feel


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22 The WittRensteinian Paradox The Witt,!?ensteinian Paradox 23confident that there is something in my mind - the meaning Iattach to the 'plus' s ign - tha t instructs me what I ought t o do inall future cases. I do not predict what I will do - see thediscussion immediately below - but instruct myself what Iought to do to conform to the meaning. (Were I now to make aprediction of my future behavior, i t would have substantivecontent only because it already makes sense, in terms of theinstructions I give myself, to ask whether my intentions willbe conformed to or not.) But when I concentrate on wha t isnow in my mind, what instructions can be found there? Hqwcan Ibe said to be acting on the basis of these instructions whenI act in the future? The infinitely many cases of thetable are notin my mind for my future self to consult. To say that there is ageneral rule in my mind that tells me how to add in the.futureis only to throw the problem back on to other rules that alsoseem to be given only in terms of finitely many cases. Whatcan the re be in my mind t ha t I make use of when I act in thefuture? It seems that the entire idea of meaning vanishes intothin air.Can we escape these incredible conclusions? Let me firstdiscuss a response that I have heard more than once inconversation on this topic. According to this response, thefallacy in the argument that no fact about me constitutes mymeaning plus lies in the assumpt ion that such a fact mustconsist in an occurrent men ta l state. Indeed the scepticalargument shows that my entire occurrent past mental historymight have been the same whether I meant plus or quus, bu tall this shows is that the fact tha t I mean t plus (rather thanquus) is to be analyzed dispositionally, rather than in terms ofoccurrent mental states. Since Ryle's The Concept oj Mind,dispositional analyses have been influential; Wittgenstein'sown later work is of course one of the inspirations for suchanalyses, and some may think that he himself wishes tosuggest a dispositional solution to his paradox.

The dispositional analysis I have heard proposed is simple.To mean addition by '+' is to be disposed, when asked for anysum 'x+y' to give the sum of x and y as the answer (in

particular, to say '125' when queried about '68+ 57'); to meanquus is to be disposed when queried about any arguments, torespond with their quum (in particular to answer '5' whenqueried about '68+ 57'). True, my actual thoughts andresponses in the past do not differentiate between the plus andthe quus hypotheses; but, even i n the past, there weredispositional facts about me that did make such a differentiation. To say that in fact I meant plus in the past is to say - assurely was the case! - that had I been queried about '68 + 57', Iwould have answered '125'. By hypothesis I was not in factasked, but the disposition was present none the less.

To a good extent this reply immediately ought to appear tobe misdirected, of f target. For the sceptic created an air ofpuzzlement as to my justification for responding '125' ratherthan '5 ' to the addition problem as queried. He thinks myresponse is no better than a stab in the dark. Does thesuggested reply advance matters? How does it justify mychoice of'12S'? What it says is: "'125' is the response you aredisposed to give , and (perhaps the rep ly adds) i t would alsohave been your response in the past." Well and good, I knowthat' 12 5' is the response I am disposed to give (I am actuallygiving it!), and maybe it is helpful to be t old - as a matter ofbrute fact - that I would have given the same response in thepast. Ho w does any of this indicate that - now or in the past'125' was an answer justified in terms of instructions I gavemyself, rather than a mere jack-in-the-box unjustified andarbitrary response? Am 1supposed tojustify my present beliefthat I meant addition, not quaddit ion, and hence shouldanswer '125', in terms of a hypothesis about my past dispositions? (Do I record and investigate the past physiology of mybrain?) Why am 1so sure that one particular hypothesis of thiskind is correct, when all my past thoughts can be construedeither so that I meant plus or so that I meant quus?Alternatively, is the hypothesis to refer to my present dispositions alone, which would hence give the r ight answer bydefinition?Nothing is more contrary to ou r ordinary view - or


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The Wittgensteinian Paradoxcandidate for what the fact as to what I mean might be, It ISworth examining some problems with the view in moredetail.As I said, probably some have read Wittgenstein himself as

favoring a dispositional analysis. I think that on the contrary,although Wittgenstein's views have dispositional elements,any such analysis is inconsistent with Wittgenstein's view. ly19 Russell's The Allalysis ofMilld (George Allen and Unwin, London, in the

Muirhead Library of Philosophy, 310 pp.) already gives dispositionalanalyses of certain mental concepts: see especially, Lecture III, "Desireand Feeling," pp. 58-76. (The objectof a desire, for example, is roughlydefined as that thing which, when obtained, will cause the activity of thesubject due to the desire to cease.) The book is explicitly influenced byWatson ian behavior ism; see t he preface and the first chapter. I aminclined to conjecture thatWittgenstein's philosophicaldevelopment wasinfluenced considerably by this work, both in the respects in which hesympathizes with behavioristic and dispositional views, and to the extentthat he opposes them. I take Philosophical Remarks (Basil Blackwell,Oxford, 1975, 357 pp. , translated by R. Hargreaves and R. White),zlff., to express a rejection of Russell's theory of desire, as stated inLecture III of The Analysis of Mind. The discussion of Russell's theorypl ayed, I t hi nk , an important role in Wittgenstein 's development: theproblem of the relation of a desire, expec tat ion, etc. , to its obj ec t('intentionality') is one of the important forms Wittgenstein's problemabout meaning and rules takes in the Investigations. Clearly the sceptic, byproposing his bizarre interpretations ofwhat I previously meant, can getbizarre results as to what (in the present) does, or does not, satisfy my pastdesires or expectations, or what constitutes obedience to an order I gave.Russell's theory parallels the dispositional theory of meaning in the textby giving a causal dispositional account of desire. Just as the dispositionaltheory holds that the value I meant '+' to have for two particulararguments m and n is, by definition, the answer I would give if queriedabout 'm+n', so Russell characterizes the thing I desired as the thingwhich, were I to get it, would quiet my 'searching' activity. I think thatevenin the Investi.l?ations, as in Philosophical Remarks (whichstems from anearlier period), Wittgenstein still rejects Russell's dispositional theorybecause it makes the relation between a desire and its objectan 'external'relation (PR, ZI), although in the Investigations, unlike PhilosophicalRemarks, he no longer bases this v iew on the 'pic ture theory' of theTractatus. Wittgenstein's view t ha t th e r ela ti on between the de sir e(expectat ion, e tc .) and its objec t must be 'internal', not 'external',

24Wittgenstein's - than is the supposition that "whatever isgoing to seem ri ght t o me is right." (258). t h e " c ~ n . t r a r y ,"that only means that here we can't talk about nght (I.bld.). Acandidate for what constitutes the state of my meanmg onefunction, rather than another, by a given function sign, o u g ~ tt o b e such that, whatever in fact I (am disposed to) do, there ISa unique thing that I should do. Is no t the dispositional viewsimply an equation of performance and correctness? A s s u ~ -ing determinism, even if ! mean to denote no number theoretICfunction in particularby the sign ' *' , thento the same extent asit is true for '+', it is true here that for any two arguments mand n, there is a uniquely determined answer p that I wouldgive.I7 (I choose one at random, as we would n o r m ~ l l y say,bu t causally the answer is determined.) The d I f f e ~ e n c ebetween this case and the case of the'+' function is that m theformer case, but not in the latter, my uniquely determined1 b 11 d ' . h" , 18answer can proper y e ca e ng t or wrong . . .So it does seem that a dispositional account mIsconceIvesthe sceptic's problem - to find a past fact that justifies .mypresent response. As a candidate for a. 'fact' t ~ ~ t determmeswhat I mean, it fails to satisfy the baSIC condItion on s uch acandidate, stressed above on p. I I , that it should tell me what Iought to do in each new instance. U l t ~ m a t e l y , alm?st allobjections to the dispositional account bOll down to thIS one.However, since the disposi tional is t does offer a popular17 We will see immediately below that for arbitrarily large m and n, thisassertionis no t really true even for '+'. That is why I say that the assertionis true for'+' and the meaningless ,*, 't o the same extent' .

18 Imight have introduced '* ' to meannothing in particular e.ven. thoughanswer I arbitrarily choose for 'm*n' is, through some qmrk m my bramstructure, uniquely determined independently of the t ime an d othercircumstanceswhen I am asked the question. Itmight, in addition, evenbe the case that I consciously resolve, once I have chosen a particularanswer to 'm*n', to stick to it if the query is repeated for any particularcase, yet nevertheless I think of'*' as meaning no. f ~ n . c t i o ~ in ' p a r t i c u ~ a . r .What I will no t say is that my particular answer IS nght o r w ron g Interms of the meaning I assigned to '*', as I will for'+', since there is nosuch meaning.

The W i t ( ~ e n s t e i n i a n Paradox 25


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26 The Wittgmsteinian Paradox The W i t ( ~ m s t e i n i a l 1 Paradox 27First, we must state the simple dispositional analysis. I t

gives a criterion that will tell me what number theoreticfunction cp I mean by a binary function symbol], namely:The referent cpo f ] is thatuniquebinary function cp such that Iam disposed, if queried about ' f(m, n)', where 'm' and 'n' ,ar,enumerals denoting particular numbers m and 11, to reply p ,where 'p' is a numeral denoting cp(m, n). The criterion is m ~ a n tto enable us to 'read off' which function I mean by a glVenfunction symbol from my disposition. The cases of additionand quaddition above would simply be special cases of such ascheme ofdefinition. 20The dispositional theory attempts to avoid the problem of

the finiteness ofmy actual past performance by appealing to adisposition. But in doing so, i t ignores an obvious fact: notonly my actual performance, but also the total ity ofdispositions, is finite. It is no t true, for example, that Ifquenedabout the sum of any two numbers, no matter how large, Iwill reply with their actual sum, for some pairs ofnumbers are

parallels corresponding morals drawn about meaning in my text below(the relation of meaning and intention to future actionis 'normative, no tdescr ip tive ', p. 37 below). Sections 42


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28 The Wittgensteinian Paradox The Witt,qwsteinian Paradox 29ceteris paribus clause really means is something like this: If Isomehow were to be given the means to carry out myintentions with respect to numbers that presently are too longfor me to add (or to grasp), and if I were to carry ou t theseintentions, then if queried about 'm+n' for somebig m and n, Iwould respond with their sum (and not with their quum).Such a counterfactual conditional is true enough, bu t it is ofnohelp against the sceptic. I t presupposes a prior notion of myhaving an intention to mean one function rather than anotherby '+'. It is in virtue of a fact of this kind about me that theconditional is true. But of course the sceptic is challenging theexistence of just such a fact; his challenge must be met byspecifying its nature. Granted that I mean addition by '+',then of course if I were to act in accordance with myintentions, I would respond, given any pair of numbers to becombined by '+', with their sum; but equally, granted that Imean quaddition, if I were to act in accordance with myintentions, Iwould respondwith the quum. One cannot favorone conditional rather than anotherwithout circularity.

Recapitulating briefly: if the dispositionalist attempts todefine which function I meant as the function determined bythe answer I am disposed to give for a rbit rari ly l argearguments, he ignores the fact that my dispositions extend toonly finitely many cases. Ifhe tries to appeal to my responsesunder idealized conditions that overcome this finiteness, hewill succeed only if the idealization includes a specification thatI will still respond, under these idealized conditions, accordingto the infinite table of the function I actually meant. But thenthe circularity of the procedure is evident. The idealizeddispositions are determinate only because it is already settledwhich function I meant.The dispositionalist labors under yet another , equa lly

potent, difficulty, which was foreshadowed above when Irecalled Wittgenstein's remark that, if 'right' makes sense, i tcannot be the case that whatever seems right to me is (bydefinition) right. Most of us have disposit ions to make

mistakes. 21 For example, when asked to add certain numberssome people forget to 'carry' . They are thus disposed, forthese numbers , to give an answer differing f rom the usualaddition table. Normally, we say that such people havemade amistake. That means, that for them as for us, '+' meansaddition, bu t for certain numbers they are no t disposed to givethe answer they should give, if they are to accord with the tableof the function they actually meant. Bu t the dispositionalistcannot say this. According to him, the func-tion someonemeans is to be read of f from his d isposi tions; i t cannot be2l However, in the s logan quoted and in 202, Wittgenstein seems to be

more concerned with the question, "Am I right in thinking that I am stillapplying the same rule?" than with the question "Ismy application of therule right?" Relatively few of us have the disposition- as far as I know bizarrely to cease to app ly a g iv en r ul e if once we were applying it.Perhaps there is a corrosive substance presentin my brain already (whoseaction will be 'triggered' if I am given a certa in addition problem) thatwillleadme to forget how to add . Imight, once this substance is secreted,start giving bizarre answers to additionproblems- answers that conformto a quus-like rule, or to n o discernible pattern at all. Even if I do thinkthat I am following the same rule, in fact I am not.Now, when I asser t that I def initely mean addition by 'plus' , am I

making a prediction about my future behavior, asserting that there is nosuch corrosive acid? To put the matter differently: Iassert that the presentmeaning I give to '+' determines values for arbitrarilylargeamounts. I donot predict that I wil l come out with these values, or even that I wil l useanything l ike the 'right' procedures' to get them. A disposition to goberserk, to change the rule, e tc ., may b e in me already, wai ti ng to betriggered by the right s timu lu s. I make no assertion about suchpossibilities when I say that my use ofthe '+' sign determines values forevery pair of arguments. Much less do I assert that thevalues I will comeou t with under these circumstances are, by definition, the values thataccord with what is meant.These possibilities, and the case mentioned above with '* ' , when I am

disposed to respond even though I follow no rule from the beginning,should be borne in mind in addition t o t he garden-variety possibility oferror mentioned in the text. Note that in the case of '*' , it s eemsintuit ively possible that I could be under th e impr es si un t ha t I wasfol lowing a rule even though I was fol lowing none - see the analogouscase of reading on pp. 45-6 below, in reference to r66.


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22 Lest I be misunderstood, I hope it is clear tha t in saying this I do no tmyself reject Chomsky's competence-performance distinction. On thecontrary, I personally find that the familiar arguments for the distinction(and for the attendant notion of grammatical rule) have great persuasiveforce. The present work is intended to expound my understanding of

presupposed in advance which function is meant. In thepresent instance a certain unique function (call it 'skaddition')corresponds in its table exactly to the subject's dispositions,including his dispositions to make mistakes. (Waive thedifficulty that the subject's dispositions are fmite: suppose hehas a disposition to r espond to any pair of arguments.) So,where common sense holds that the subject means the sameaddition function as everyone else but systematically makescomputational mistakes, the disposi'cionalist seems forced tohold that the subject makes no computational mistakes, butmeans a non-standard function ('skaddition') by '+'. Recallthat the dispositionalist held that we would detect someonewho meant quus by '+' via his disposition to respond with' 5'for arguments ~ 5 7 . In the s ame way, he wi ll 'detect ' that aquite ordinary, though fallible, subject means some nonstandard function by '+ '.Once again, the difficulty cannot be surmounted by a ceterisparibus clause, by a clause excluding 'noise', orby a distinctionbetween 'competence' and 'performance'. No doubt a disposit ion to give the true sum in response to each addition problem'is part ofmy 'competence', ifby this we meansimply that suchan answer accords with the rule I intended, or ifwe mean that,if allmy dispositions to make mistakes were removed, Iwouldgive the correct answer. (Again I waive the finiteness of mycapacity.) But a di sposi tion to make a mistake is simply adisposition to give an answer other than the one that accords with thefunction I meant. To presuppose this concept in the presentdiscussion is of course viciously circular. If! meant addition,my 'erroneous' actual disposition is to be ignored; if ! meantskaddition, it should not be. Nothing in the notion of my'competence' as thus def ined can possibly tell me whichalternative to adopt. 22 Alternatively, we might try to specify

The Wittgensteinian Paradox30 The Wittgensteinian Paradox 31the 'noise' to be ignored without presupposing a prior notionof which function is meant . A lit tle exper imentat ion willreveal the futility ofsuch an effort. Recall that the subject has a

Wittgenstein's position, no t my own ; but I certainly do not mean,exegetically, to assert that Wittgenstein himselfwould reject the distinction. But what is important here is that the notion of'competence' is itselfnot a dispositional notion. It is normative, not descriptive, in the senseexplained in the text.The point is that our understanding of the notion of 'competence' is

dependent o n our understanding of the idea of 'following a rule' , as isargued in the discussion above. Wittgenstein would reject the idea that'competence' can be de fined in terms of an idealized dispositional ormechanical model, and used without circularity to explicate the notion offollowing a rule. Only after the sceptical problem about rules has beenresolved can we then define 'competence' in terms of rule-following.Although notions of 'competence' and 'performance' differ (at least)f rom wri ter to wri te r, I see no reason why linguists need assume that'competence' is defined prior to rule-following. Although the remarks inthe text warn against the use of the 'competence' notion as a solution toour problem, in no way are they arguments against the notion itself.

Nevertheless, given the sceptical nature of Wittgenstein's solution tohis problem (as this solution is explained below), i t is clear that ifWittgenstein's standpoint is accepted, the notion of'competence' will beseenin a light radicallydifferent from the way it implicitly is seen inmuchof the literature oflinguistics. For ifstatements attributing rule-followingare neither to be regarded as stating facts, nor to be thought of asexplaining our behavior (seesection3 below), itwould seemthat the use ofthe ideas of rules and of competence in l inguis tics needs ser iousreconsideration, even if these notions are no t rendered 'meaningless'.(Depending on one's standpoint, one might view the tension revealedhere between modern linguistics and Wittgenstein's sceptical critique ascasting doubt on the linguistics, or on Wittgenstein's sceptical critiqueor both.) These questions would arise even if, as throughout the presenttext, we deal with rules, like addition, that are stated explicitly. Theserules we think of ourselves as grasping consciously; in the absence ofWittgenstein's sceptical arguments, we wou ld see no problem in theassumption that each particular answer we produce is justified by o ur'grasp' of the rules. The problems are compounded if, as in linguistics,the rules are thoughtof as tacit, to be reconstructed by the scientist andinferred as an explanation of behavior. The matter deserves an extendeddiscussion elsewhere. (See also pp. 97 to 99 and n. 77 below.)


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The Wittgensteinian Paradoxsystematic disposition to forget to car ry in cer ta in circumstances: he tends to give a uniformly erroneous answer whenwell rested, in a pleasant environment free of clutter, etc. Onecannot repair matters by ur ging t ha t the subject ~ o u l deventually respond with the right answer after c o ~ r e c t l o ~others. First, there are uneducable subjects who WIll persIst mtheir error even after persistent correction. Second, what ismeant by 'correction by others'? Ifit means rejection by othersof ,wrong' answers (answers that do no t accord with the rulethe speaker means) and suggestion of the right . a n ~ w e r (rheanswer that does accord), then again the account IS CIrcular. Ifrandom intervention is allowed (that is, the 'corrections' maybe arbitrary, whether they are 'right' or 'wrong' ), t h e ~ ,although educable subjects may be induced to c ~ r r e c t theIrwrong answers, suggestible subjects may also be mduced toreplace the ir cor rect answers with erroneous. ones.amended dispositional statement will, then, provIde no criterion for the function that is really meant.

The dispositional theory, as stated, assumes tha t whichfunction I meant is determined bymy dispositions to computeits values in pa rt icul ar cases. In fact, this is not so. S i ~ c edispositions cover only a finite segment of the total f u n ~ t l o ?and since they may deviate from its true values, two mdIviduals may agree on their computations in particular ~ a s e seven though they are actually computing different functIons.Hence the dispositional view is no t correct.In d iscussions , I have sometimes heard a var iant of thedispositional account. The argument goes as .follows: thesceptic argues, in essence, that I am free to gI.ve any newanswer to an addition problem, since I can alwaysmterpret myprevious intentions appropriately. how can this ?e? AsDummett put the objection: "A machme can follow. thI.S r u l ~ ;whence does a human being gain a freedom of chOIce m thISmatter which a machine does not possess?"23 The objection is23 M. A. E. Dummett, "Wittgenstein's Philosophy of Mathematics," The

P h i l ~ s o p h i c a l Review, vol. 68 (1959), pp. 324-48, see p. 331.; re.printed inGeorge Pitcher (ed.), Wittgenstein: The Philosophical InvestIgatIOns (Mac-

The Wittgensteinian Paradox 33really a form of the dispositional account, for that account canbe viewed as if i t interpreted us as machines, whose outputmechanically yields the correct result.

We can interpret the objector as arguing that the rule can beembodied in a machine that computes the relevant function. If !bui ld such a machine, it will s imply grind ou t the rightanswer, in any particular case, to any particular additionproblem. The answer that the machine would give is, then,the answer that I intended.

Th e term 'machine' is here, as often elsewhere in philosophy, ambiguous. Few ofus are in a position to build a machineor draw up a program to embody our intentions; and if atechnician performs the task for me, the sceptic can asklegitimately whether the technician has performed his taskcorrectly. Suppose, however, that I am fortunate enough to besuch an exper t tha t I have the technical facility required toembody my ow n intentions in a computing machine, and Istate that the machine is definitive ofmy own intentions. Nowthe word 'machine' here may refer to anyone of variousthings. It may refer to a machine program that I draw up,embodying my intentions as to the operation of the machine.Then exactly the same problems arise for the program as forthe original symbol '+' : the sceptic can feign to believe that theprogram, too, ought to be interpreted in a quus-like manner.To say that a program is not something that I wrote down onpaper, bu t an abstract mathematical object, gets us no further.The problem then simply takes the form of the question: whatprogram (in the sense of abstract mathematical object) corresponds to the 'program' I have written on paper (in accordancewith the way I meant it)? ('Machine' often seems to mean aprogram in one of these senses: a Turing 'machine', forexample, would be better called a 'Turing program'.) Finally,however, I may build a concrete machine, made ofmetal and

millan, 1966, pp. 420-47), see p. 428. The quoted objection need no tnecessari ly be taken to express Dummett 's own ultimate view of thematter.


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gears (or transistors and wires), and declare that i t embodiesthe function I intend by '+' : t he values that i t gives are thevalues of the funct ion I intend. However, there are severalproblems with this. First, even if I say that the machineembodies the function in this sense, I must do so in terms ofinstructions (machine 'language', coding devices) that tell mehow to interpret themachine; further, Imust declare explicitlythat the function always takes values as given, in accordancewith the chosen code, by the machine. But then the sceptic isfree to interpret all these instructions in a non-standard,'guus-like' way. Waiving this problem, there are two othershere is where the previous discussion of the dispositional viewcomes in. I cannot really insist that the values of the functionare given by the machine. First, the machine is a finite object,accepting only finitely many numbers as input and yieldingonly finitely many as output - others are simply too big.Indefinitely many programs extend the actual finite behaviorof the machine. Usually this is ignored because the designer ofthe machine intended it to fulfill just one program, but in thepresent context such an approach to the intentions of thedesigner simply gives the sceptic his wedge to interpret in anon-standard way. (Indeed, the appeal to the designer'sprogram makes the physical machine superfluous; only theprogram is really relevant. The machine as physical object is ofvalue only if the intended function can somehow be read of ffrom the physical object alone.) Second, in practice it hardly islikely that I really intend to entrust the values of a function tothe operation of a physical machine, even for that finiteportion of the function for which the machine can operate.Actual machines can malfunction: through melting wires orslipping gears they may give the wrong answer. Ho w is itdetermined when a malfunction occurs? By reference to theprogram of the machine, as intended by its designer, notsimply by reference to t he machine itself. Depending on theintent of the designer, any particular phenomenon mayormay not count as a machine 'malfunct ion' . A programmerwith suitable intentions might even have intended to make use

34 The Wittgensteinian Paradox The Wittgensteinian Paradox 35of the fact that wires melt or gears slip, so that a machine that is'malfunctioning' for me is behaving perfect ly for him.Whether a machine ever malfunctions and, if so, when, is not aproperty of the machine itselfas a physical object but is welldefined only in t erms of its program, as stipu la ted by itsdesigner. Given the program, once again the physical object issuperfluous for the purpose of determining what function ismeant. Then, as before, the sceptic can concent rat e hisobjections on the program. The last two criticisms of the useof the physical machine as a way out ofscepticism- its finitudeand the possibility of malfunction - obviously parallel twocorresponding objections to the dispositional account. 24

24 Wittgenstein discusses machines explicitly in I93-S. See the paralleldiscussion in Remarks on the Foundations ofMathematics, part I, II8-30,especially IIg-26; see also, e.g., II [III], 87, and III [IV], 48--gthere. The cri ticisms in the text of the dispositional analysis and of theuse of machines to solve the problem are inspired by these sections.In particular, Wittgenstein himself draws the distinction between themachine as an abstract program ("der Maschine, als Symbol" I93) andthe actual physical machine, which is subject to breakdown ("do weforget the possibility of their bending, breakingoff, melting, and soon?"(I93)). The dispositional theory views the subject himself as a kind ofmachine, whose potential actions embody the funct ion. So in this sensethe dispositional theory and the idea of the machine-as-embodying-thefunction are really one. Wittgenstein's attitude toward both is the same:they confuse the 'hardness of a rule' with the 'hardness of a material '(RFM, II [III], 87). On my interpretation, then, Wittgenstein agreeswith his interlocutor (I94 and I9S) that the sense in which all the valuesof the funct ion are already present is not simply causal, although hedisagrees with t he idea t ha t t he f ut ur e use is already present in somemysterious non-causal way.

Although, inan attempt to followWittgenstein, I haveemphasized thedistinction between concrete physica l machines and the ir abs tr ac tprograms in what I have written above, it might be instructive to look atthe outcome when the limitation of machines is idealized as in themodern theory of automata. A finite automaton, as usually defined, hasonly finitely many states, receives only finitely many distinct inputs, andhas only finitely many outputs, but it is idealized in two respects: it has noproblem ofmalfunction, and its lifetime (without any decay or wearingou t of i ts par ts) is inf inite. Such a machine can, i n a sense, performcomputations on arbitrarily large whole numbers. If i t has notations for


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The Wittgensteinian Paradox The Witt,gensteinian Paradox 37the s ingle d ig it s from zero through nine, inclusive , i t can rece ivearbitrarily large positive whole numbers as inputs simply by being giventheir digits one by one. (We cannot do this, since our effective lifetimesare finite, and there is a minimum t ime needed for us to understand anysingledigit.) Suchan automaton can addaccording to theusual algorithmin decimal notation (the digitsfor thenumbers beingadded should be fedinto the machine starting from the last digits of both summands andgoing backwards, as in the usual algorithm). However, it canbe provedthat, in t he same ordinary dec imal notat ion, such a machine cannotmultiply. Any function computed by such a machine that purports to bemultiplication will, for large enough arguments, exhibit 'quus-like' (orrather, 'quimes-like') properties at sufficiently large arguments. Even ifwe were idealized as finite automata, a dispositional theory would yieldunacceptable results.

Suppose we idealized even further and considered a Turing machinewhich has atape tousewhich is infinitein both directions. Sucha machinehas infinite extent at every moment, in addit ion to an infinite l ifet imewithout malfunctions. Turing machines can multiply correctly, but it iswell known t ha t even her e t he re are many functions we can defineexplicitly that canbe computed by no suchmachine. A crudedispositionaltheory would attribute to us a non-s tandard interpretat ion (or nointerpretation at all) for any such function. (Seeabove, note 20.)

I have found that both the crude dispositional theory and thefunction-as-embodied-in-a-machine come up frequently when Wittgen-stein's paradox is discussed. For this reason, and because of their closerelation to Wittgenstein's text, I have expounded these theories, thoughsometimes I havewondered whether the discussionof them is excessivelylong. On the other hand, I have resisted t he t empt at ion to discuss'functionalism' explicitly, even though various forms of i t have been soattract

73951099 Kripke 1982 Wittgenstein on Rules and Private Language an Elementary Exposition

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