equations and godel

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The ritish Society for the Philosophy of Science

Gdel's Theorem and the MindAuthor(s): Peter SlezakSource: The British Journal for the Philosophy of Science, Vol. 33, No. 1 (Mar., 1982), pp. 41-52

Published by: Oxford University Presson behalf of The British Society for the Philosophy ofScienceStable URL: http://www.jstor.org/stable/687239.

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Brit.J. Phil.Sci. 33 (1982),41-52 Printedn GreatBritain 41

G6del's Theorem and the Mindiby PETER SLEZAKi Truth and Provability2 The Dialectical Argument3 Self Reference4 Speculative ConclusionsJ. R. Lucas has articulated and defended the view that Godel's theoremsimply the falsity of mechanism as a theory of the mind. In this paper I offer anovel analysis of Lucas's argument which shows why it is, in fact, acompelling one, but also that it derives its persuasiveness from a certain kindof subtle confusion which has not been noticed in subsequent discussions.Clarifying this confusion provides further illumination by suggesting somequite different implications of G6del's theorem for the mind.I TRUTH AND PROVABILITYFor the purposes of the following discussion it is helpful to have Lucas'sown statement of his argument:G6del's theorem must apply to cybernetical machines, because it is of the essence ofbeing a machine, that it should be a concrete instantiation of a formal system. Itfollows that given any machine which is consistent and capable of doing simplearithmetic, there is a formula which it is incapable of producing as being true-i.e.,the formula is unprovable-in-the-system-but which we can see to be true. Itfollows that no machine can be a complete or adequatemodel of the mind, that mindsare essentially different from machines.

(Lucas [1961], p. 44)It is important to notice that throughout his argument Lucas repeatedly usesthe expression we see here 'produce as true' to state his central pointregarding the alleged import of Godel's theorem for mechanism. Chihara[1972] has noted the vagueness of this expression and the consequentdifficulty of evaluating Lucas's claims. However, I believe that there is nosuch difficulty since it is possible to go beyond merely noting the vaguenessof Lucas's expression: by paying close attention to G6del's theorem, it ispossible to see precisely what Lucas must have in mind and, moreover, thatthis involves a certain kind of mistake, albeit a compelling one. If weconsider carefully Godel's result independently of Lucas's argument and1The author is grateful to Dan Hausman, Mark Steiner, Frank Vlach and an anonymousreferee of this Journal for valuable comments on earlier versions of this paper.

Received zz January 1980

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42 Peter Slezakarticulate precisely the kind of limitation which it reveals for machines, thenwe may reconstruct what Lucas must intend with his locution and therebyevaluate his thesis. Lucas's notion of 'produce as true' is intended to captureboth what a machine is precluded from doing and also what a mind can do.Accordingly, whatever may be the case with minds, at least we need leave nounclarity with regard to the meaning of this notion in connection withTuring machines in view of the formal rigour of Godel's result.In speaking of what a machine cannot 'produce as true' it is clear thatLucas is referring to G6del's first incompleteness theorem according towhich it is possible to express a statement within the formal systemconcerned (i.e., in the machine's language), but neither the statement nor itsnegation can be formally derived in the system. In other words, the preciselimitation on formal systems revealed by G6del's theorem concerns whatcan be obtained from its axioms by following its rules of derivation. It isimportant to emphasise that the limitation on formal systems or machines isprecisely expressed in terms of formal derivability or provability-in-the-system and it is this strict notion which must underlie Lucas's vague notion'produce as true'. It is worth noting how the exact concept of the limitation isexpressed for Turing machines: the limitation is on the instantaneousdescriptions which can be terminal in a computation of the Turing machineconsisting of a finite sequence of such instantaneous descriptions. Thus, forTuring machines the limitation can be expressed entirely in terms of thenotion of computability. Now it is clear that the limitation shown to exist canbe expressed without any mention of the notion of truth which appears inLucas's expression. The equivalent notions of Turing computability andLambda-definability of Church's theorem define the same class of functionsas G6del's notion of a 'general recursive function', and the limitations onformal systems which can be expressed by these notions do not involve theidea of truth at all. Accordingly, the specifically relevant facet of G6del'stheorem is the undecidability of the statement which is expressible in thesystem. Now it is only in coming to the further conclusion about thenecessary incompleteness of any formal system that we use the idea of truth,and it is the very essence of G6del's result that the notion of truth belongs tothe metalinguistic proof and not to the object language. The profoundimport of G6del's theorem was, indeed, in showing exactly that Hilbert'sformalist program was unrealizable since provability in a given systemcannot be identified with mathematical truth.

It seems evident that Lucas fails to pay sufficient attention to these crucialdistinctions, for he conflates them all in his expression 'produce as true'.This locution permits Lucas to establish his thesis only by equivocatingbetween provability and truth, since his repeated use of it to express hiscentral point serves only to obscure the precise import of G6del's theorem,namely, that although the undecidable sentence cannot be formally derivedwithin the system, its truth can be known from outside the system. In otherwords, our basis for asserting the truth of the undecidable sentence is a proof



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G6del's Theoremand the Mind 43in the metalanguage which is a vantage point outside the system or outsidethe machine defined by the formal system. When the machine is conceivedin the physical sense, rather than in the mathematical sense, the term'outside' here must be understood literally as meaning a numericallydifferent object from the machine in question. I believe that it is on thispoint that Lucas's argument is particularly confused: I will try to show moreclearly later that Lucas fails to pay attention to the further importantdistinction between the difference of tokens and the difference of types. It isthe latter which he would obviously require to establish his thesis againstmechanism, but it is only the former which follows from G6del's theoremand is utterly trivial.

Thus, when expressed precisely, G6del's theorem shows that, in so far aswe are concerned with the limitations on a machine, we do not need thenotion of truth at all. The limitation can be expressed in terms of what can begenerated by mechanical procedures, and this involves only the com-binatorial notion of the manipulation of uninterpreted symbols. Thesignificance of this point is perhaps easily missed (if it were not for G6del'sresult ) in view of the close association of provability with truth. However,the point can be appreciated even without the actual conclusion of G6del'stheorem, for the association of a truth value with the output of a machinereflects nothing intrinsic to the behaviour of the machine or any of its states.Rather, the attribution of truth values to the states of the machine (outputs)reflects our particular interests in, and uses of, these states. Even to speak of'statements' in connection with the behaviour of a machine tends to disguisethe fact that we choose to interpret the behaviour of the machine inparticular ways, though we need not do so. Indeed, to speak of a Turingmachine as performing 'computations' as we generally do, is already to havetaken a step away from what is, strictly speaking, the actual behaviour of themachine: a 'tape expression' for a Turing machine consisting of symbols inits alphabet is intrinsically no different from any other component of themachine; its special status derives from the fact that we associate ournumerals with the tape expressions, and we are then able to regard theoperations of the machine as calculations on numbers.These facts are clear even on the formalists' view prior to G6del's result,since they simply express the methodology of their enterprise and not itsambitions. The notion of truth is a semantic one involving the relation ofstatements to the world. In the present context, the statements of interest arein the metalanguage and represent what we (minds) know, while the 'world'here corresponds to the formal system or machine. Now, in spite of thecrucial difference between truth and provability, Lucas incorporates themboth in his notion of 'produce as true', and he is thereby able to rely on theresulting equivocation to establish his conclusions. In defending thesuperiority of minds, Lucas employs this notion of his to express both what itis that a machine is unable to do, and also what it is that he or any mind is ableto do. Thus, he makes the contrast 'between what was provable-in-the-



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44 Peter Slezaksystem (the machine's system) and what could be produced by me as true'([1968], p. 147). Lucas's notion is, therefore, central to his argument in thisway because it provides the link between the limitations of any machine andthe abilities of any mind: it is in the ability to 'produce as true' the Godelsentence that minds are supposed to prevail over machines. It is in this waythat Lucas's case is crucially dependent on conflating truth with provability:he is unable to state his thesis without it.

Avoiding Lucas's expression, then, we might ask: What is the precisesignificance of our being able to know something is true but which themachine cannot prove? From the above discussion it is already clear thatwhat we can do is know the truth of the Godel sentence by being able tocompare what it says (about itself) with the facts (i.e., that it is not derivablewithin the system). However, since this is done in the metalanguage, to saythis much is, in fact, only to re-state Godel's result and not to establishanything about the relative abilities of minds and machines. As we haveseen, the limitation for the machine concerns what it can grind out astheorems in a mechanical way from its axioms. For these reasons Lucas'scomparison of the machine's abilities with our knowledge of the truth is notwarranted because we have not yet made sense of the machine knowing thetruth of any statement at all-that is, semantics as opposed to syntax. Wesee, therefore, that Lucas's locution obscures both the precise character ofthe limitations on machines, and also what is involved in our knowing thetruth of the unprovable sentence.Criticisms by Benacerraf [1967] have prompted Lucas [1968] to make anattempt at clarifying his notion 'produce as true' and its relation to truth andprovability. In doing so, however, I believe that Lucas succeeds only inbringing clearly into relief his confusions on the matter. Pressed intoexplaining his notion, Lucas tries to distinguish it from the formal notion ofprovability-in-the-system, for he says:I took care not to use the word in such a sense, but to frame the contrast between whatwas provable-in-the-system (the machine's system) and what could be produced byme as true. .... Provability has been construed by mathematical logicians for ageneration as a syntactical term with a very precise definition, and it could beconfusing to import loose notions of what I can see to be true into this well-disciplined and useful concept. (Lucas [1968], p. 147)Indeed But while these motives are laudable, the remarks are plainlyinconsistent with what Lucas had written in his original [1961] paper. Forthere, as we have already seen, Lucas explicitly identifies the two notions:. . given any machine . .. there is a formula which it is incapable of producing asbeing true-i.e., the formula is unprovable-in-the-system ...

([196I], p. 44)Lucas's later disclaimer simply belies the clear, indeed only intelligible,intent of his argument. This argument, such as it is, can only go through if



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G6del's Theorem and the Mind 45Lucas's expression serves to characterise both the limitations on a machineand also that respect in which a mind is allegedly not limited. Hence theargument requires the conflation of truth and provability to reach itsconclusion.

Clear support for this analysis of Lucas's argument comes from the factthat he is forced into making our knowing the truth of a statement into amysterious affair. He asserts ([1968], p. 148) 'I think I know what truth is,but I know I cannot tell anybody else exactly what it is.' Then, after someobscure discussion Lucas simply concludes: 'And therefore truth cannot beprecisely defined.' However, for the purpose at hand we need no furtherunderstanding of the concept beyond the observation that the truth of theundecidable sentence is learned by comparing what it says with the facts.That a machine is unable to do this does not follow in any way from G6del'sproof. The problem of determining the truth of a sentence (i.e., understand-ing its meaning) is, indeed, one which we have good reason to believe istractable and not any limitation in principle on the capacities of machines.This is the problem of 'semantic information processing' and an area ofimportant research in 'artificial intelligence'. However, be that as it may, mypoint here is only that there is no reason to believe that machines havelimitations in this regard and, in particular, none which follow from G6del'stheorem.

It is interesting to notice further that Lucas even imputes precisely hisown mistake to the mechanist, claiming that it is the mechanist who mustidentify truth with provability:The mechanist, regarding man as something less than men, namely machines,regards their concept of truth as again something less, namely provability-in-a-given-system ... ([1968], p. 148)Lucas has his straw-mechanist ask: 'If truth is not provability-in-a-given-system, what is it?', as if the mechanist is somehow unable to understand orto cope with the implication of G6del's theorem concerning the divergenceof truth and provability. Needless to say, 'the mechanist' need not useLucas's special locution as he further suggests ([1968], p. 148). All of this isobviously unwarranted and only reflects Lucas's conviction that G6del'stheorem must, somehow, be an embarrassment to the mechanist. However,in so far as the notion of truth is concerned, making it vague and mysteriousonly serves Lucas's purpose by suggesting that it is somehow essentiallyinvolved in our little-understood mental processes and not in the behaviourof machines.

2 THE DIALECTICAL ARGUMENTThe separation of truth from provability and the fact that the truth of theundecidable sentence is known by argument in the metalanguage aresignificant facts which must be kept in mind if Lucas's argument is to be



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46 Peter Slezakfully understood. These facts are intimately related to the peculiar strategyof argumentation that Lucas insists must be employed in establishing hiscase. He attempts to turn the vice of his confusions into a virtue by claimingthat his argument is of a particular, special kind:The argument is dialectical. It is an argument between two persons, not a proofsequence constructed by one. I believe that the dialectical form reveals theunderlying logic better than any monologue can.... (Lucas [1968], p. 154)However, contrary to Lucas, I want to show that this form of argumentationreveals only a confusion over the object language-metalanguage distinc-tion. So far I wanted to show that by his argument about what a machine'cannot produce as true', if Lucas has said anything at all, it is nothingbeyond what ca'n be stated more precisely and more perspicuously in statingG6del's result. In other words, at best, Lucas has only re-stated G6del'stheorem. I want now to show that in attempting to exploit G6del's result toprove the superiority of minds over machines, it is the metalinguisticcharacter of the proof which forces Lucas to present his argument in thedialectical form. He explains:We can use [the argument] . . .against those who, finding a formula their firstmachine cannot produce as being true, concede that that machine is indeedinadequate, but thereupon seek to construct a second, more adequate, machine, inwhich the formula can be produced as true. This they can indeed do: but then thesecond machine will have a G6delian formula all of its own, constructed by applyingG6del's procedure to the formal system which represents its (the second machine's)own enlarged scheme of operations. And this formula the second machine will not beable to produce asbeing true, while a mind will be able to see that it is true. ... And soit will go on. However complicated a machine we construct, it will, if it is a machine,correspond to a formalsystem, which in turn will be liable to the G6del procedure forfinding a formula unprovable-in-that-system. This formula the machine will beunable to produce as being true, although a mind can see that it is true. And so themachine will still not be an adequate model of the mind. ... the mind always has thelast word. (Lucas [196i], p. 48)This argument is remarkable for its intuitive compellingness and, indeed,allowing for Lucas's misleading locution, it is correct as far as it goes instating the G6delian limitation on any machine. Notice in particular that thedialectical argumentation reflects the way in which the truth of themachine's G6del sentence is known, and so it is intimately related to theproblems we have already seen. Thus, in spite of its superficial plausibility, Iwant to show that the argument is misleading in subtle ways. In beginning tobring this out clearly, it is helpful to look more closely at what is involved inthe dialectical form of argument. Lucas ([1968], p. 146) explains further:The argument is a dialectical one. It is not a direct proof that the mind is somethingmore than a machine, but a schema of disproof for any particular version ofmechanism thatmay be put forward. If the mechanist maintains any specific thesis, Ishow that a contradiction ensues. But only if. It depends on the mechanist makingthe first move and putting forward his claim for inspection.



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G6del's Theoremand the Mind 47It is necessary to ask the mechanist, not the machine. The machine cannot answer thequestion whether it can prove-or cannot prove-the G6delian formula in-its-system. But the question is an askable one, and one which we can press on themechanist.... ([I968], p. 152)From these quotations I think it can be clearly seen that Lucas is introducinga further confusion into the situation in claiming a special status for hisargument. It should be obvious that the relevant facts in assessing the thesisof mechanism are what the machine can or cannot do, and not whether wecan ask it of the mechanist to tell us or whether he makes the first move. Weare concerned with objective facts about the machine's capabilities vis-a-visthose of minds, and these facts are not inherently or exclusively obtainableby engaging the mechanist in debate. To the extent that it makes sense totalk of any competition here, it must be between minds and machines, notbetween minds and any mechanist thesis. With this further confusion Lucasis just obscuring what could be the only force of his own argument: There isnothing 'dialectical' or otherwise peculiar about the argument. It mustsimply amount to the claim that in any competition between a mind and amachine, there is a certain respect in which the mind can always beat themachine. In fact, of course, Lucas is forced to equivocate in presenting hisdiscussion, for on some occasions he speaks of his argument as applyingagainst 'any form of mechanism' which might be put forward as a thesis, andon other occasions he correctly speaks of the specific machine which must besuggested as a model of the mind. It seems clear that the truth or falsity ofmechanism could hardly depend, as Lucas insists, on whether anybodyactually put forward any particular thesis. On this account mechanism couldlose by default.To see Lucas's mistake more precisely, we may usefully look at hisresponse to a possible objection to his argument. Since the procedure forconstructing the G6del formula for any machine is a standard procedure, amachine could be constructed which would carry it out and it could repeatthe procedure as often as required. This would amount to adding thesuccessive G6del sentences as axioms to the formal systems. Lucas ([1961],p. 48) responds to this objection in a most revealing way:Yet even so, the matter is not settled: for the machine with a G6delizing operator, aswe might call it, is a differentmachine from the machines without such an operator;and, although the machine with the operator would be able to do those things inwhich the machines without the operator were outclassed by a mind, yet we mightexpect a mind, faced with a machine that possessed a G6delizing operator, to takethis into account and out-G6del the new machine, G6delizing operatorand all. Thishas, in fact, proved to be the case. Even if we adjoin to a formal system the infinite setof axioms consisting of the successive G6delian formulae, the resulting system is stillincomplete, and contains a formula which cannot be proved-in-the-system,although a rational being can, standing outside the system, see that it is true .... In asense, just because the mind has the last word, it can always pick a hole in any formalsystem presented to it as a model of its own workings.



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48 Peter SlezakAlthough Lucas does not notice it, it is evident that he has given here a'reductio ad absurdum' of his own case. It need not be a 'rational being' or amind which could play this game and always have the last word. Just such amachine as Lucas describes here could repeatedly 'out-G6del' any othermachine whose machine table (i.e., program or G6del number) it was givensimply by virtue of being outside the formal system in the relevant sense.Lucas himself acknowledges that it is through being outside the system thatthe truth of the G6del formula can be known, but he gives no reason forthinking that this must be a rational being or a mind as opposed to anothermachine. Thus, following Lucas's argument, a machine with a G6delizingoperator could play this game of 'one-upmanship' against another machineand conclude that it (the first machine) is not a machine at all Being able toplay the game is a consequence only of being a numerically different entityor token and not something other than a machine. Lucas might just as wellhave claimed not to be a machine on the grounds that however high anumber any machine could produce, he could always have the last word byproducing a number which is higher.Lucas's ability to play his dialectical game rests entirely on the fact that heis able to determine the truth of its unprovable sentence by virtue of being'outside the system', that is outside the machine. This does, in fact, meanthat he is 'not the same' as the machine, but this is only in the sense that he isa numerically different object, perhaps a different machine. Of course, thisconclusion is entirely trivial in its consequences for mechanism and amountsonly to another way of stating the metalinguistic character of G6del's proof.To refute mechanism, however, Lucas needs to show that he is 'not thesame' in the sense of being a different kind or type from any machine, but thislatter conclusion does not follow from his argument. By being able to knowthe truth of a machine's G6del sentence, Lucas proves that he cannot be 'it',i.e., not the same machine in the sense of necessarily having a differentprogram or 'software' and in this sense a different token, but he claims tohave shown that he is a different kind in not being a machine at all.

3 SELF REFERENCEAt a first approximation, then, I believe we have exposed in its essentials thefatal flaw in Lucas's refutation of mechanism. However, the interest of theissues raised has by no means been exhausted, and I want to show that it isthrough pursuing the matters in more detail that we can see what might bethe genuine import of G6del's theorem for mechanism.

In discussing the role of truth in G6del's theorem, I noted that it is not theincompleteness result as such which is specifically relevant to mechanism.Rather it is the notion of undecidability which is pertinent, for it is in termsof this notion that the limitations on formal systems can be preciselyexpressed. Now a fundamentally important feature of the undecidablesentence (which is clear from its form) is that it is essentially 'self-



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Godel's Theoremand the Mind 49referential'. It is self-referential in the precise sense that it is constructed bythe process of 'diagonalisation' as used in Cantor's proof of the non-denumerability of the real numbers. Corresponding to G6del's arithmetis-ation, any Turing machine can be represented in the symbols used by themachine itself. This representation corresponds to the Godel number of themachine. Now, in the case of Turing machines, the diagonal argument restsprecisely on this fact that any Turing machine can be described by a finitesequence of symbols, and such finite sequences of symbols are just whatTuring machines themselves act upon. The undecidable sentence is self-referential in the sense that it results from the operation of diagonalisation,and it is the attempt to compute this sentence by the machine whose G6delnumber it refers to which is impossible. Thus, the limitation on any Turingmachine is one which is essentially concerned with the machine's relation toitself as this is embodied in the uncomputable sentence. The limitation onany formal system is with regard to a sentence expressible in the systemwhich says of itself that it cannot be proved in that very system. In this sense,the G6del limitation involves a sentence which is capable of an 'indexical'interpretation in view of its specificity to the formal system in which it isexpressed.

Thus, if we are to speak at all of the limitations on machines with a view toderiving any possible consequences for mechanism, we must keep in mindthe precise nature of these limitations in the respects I have just indicated.That is, in comparing the abilities of minds and machines, the kind of task inquestion must be accurately characterised before any verdict can be reached.I am concerned to suggest that, in fact, Lucas's dialectical argument relieson improperly formulating the task on which minds are alleged to prevailover machines. In understanding the nature of the task that a Turingmachine is unable to perform, it must be noticed that there are two quitedifferent kinds of self-reference at play. The first is that of the G6del sentencein relation to itself as just indicated. However, the second kind of self-reference is that of the formal system or machine in relation to itself. This issimply the fact that the G6del sentence can be expressed in the symbolism ofthe formal system. Thus, the relation 'being a proof in the system' can beexpressed in the symbolism of the system itself. This second kind of self-reference is just the idea of mapping employed by G6del through whichmetamathematical statements about a formalised system can be translatedinto actual arithmetical statements within the system itself. So, in the formerkind of self-reference we have a statement which, as Godel himself pointsout, resembles the statement of the Liar paradox, for it says of itself that it isnot provable. However, in the latter kind of self-reference it is the formalsystem or machine which is able to refer to its self. Here we have statementswhich are by the machine, but which are also about the machine.

Now, the specific limitations on any formal system or machine involve astatement which is self-referential in both of these ways. It is in terms of thischaracterisation of the task that we must assess the relative abilities of minds

D



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50 Peter Slezakand machines in any competition between them. However, we do not evenneed to consider the relatively obscure questions of what the mind can doregarding the task in order to perceive Lucas's error (though I will return tothese obscure questions later). Since the inherent limitation encountered bya machine is a specific one concerning itself insofar as it is unable to computea self-referring statement, It is utterly trivial for Lucas to observe that he cansurmount this specific limitation for this specific machine. Clearly, this isjust what Lucas's dialectical argument amounted to, but it is neithersurprising nor interesting that a mind or another machine is able to prove theparticular statement which is undecidable for a specified machine. Since thelimitation for a machine is a peculiar one concerning itself, it is of noconsequence for the thesis of mechanism that Lucas can overcome it. Ananalogy can perhaps illustrate Lucas's logic here: If a person P is unable tobend over and touch his own toes, and Lucas claims that he is better than Pbecause he can do that task, if Lucas means by 'that task' that he can bendover and touch P's toes, it is only a joke. In the same way Lucas claims to bebetter than any machine because he can prove what is undecidable for it.Again, Lucas is clearly correct in claiming that he can perform the taskwhich the machine cannot, and that this shows that he is not the same as themachine, but the subtle and misleading error is that this is only true in thetrivial sense that he is a different token. He could well be the same type-namely, another machine. Thus, in appraising the relevance of Godel'stheorem to mechanism, the appropriate question, if it makes sense at all, iswhether Lucas could prove his own Godel formula from his own axioms andhis own rules. However we might manage to interpret this question, wemust at least be clear that it is the relevant one to ask in comparing minds andmachines in relation to G6del's theorem. The problem, then, is not whatLucas can know about the machine, but rather what Lucas can know abouthimself.

4 SPECULATIVE CONCLUSIONSFrom his own reconstruction of Lucas's argument, Benacerraf ([1967], p.30) concludes that, if we are Turing machines, then it seems that we may beprecluded from obeying the injunction 'know thyself '. Benacerraf construesthis in a sense which suggests that empirical psychology appears to beimpossible. However, this radical and paradoxical conclusion does notfollow directly from Godel's theorem, but only from Benacerraf's re-construction of certain assumptions in Lucas's argument. In fact, althoughspace does not permit it here, I believe that Benacerraf 's conclusion can beshown to follow from a failure to observe the indexical, self-referentialfeatures of the relevant statements and to this extent is an accuratereconstruction of Lucas. However, I believe that properly interpreted, theindexicality implicit in Benacerraf's own reconstruction entails the sameconclusions I want to suggest here-namely, that the limitations which may



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52 Peter SlezakREFERENCESBENACERRAF,P. [1967]: 'God, The Devil, and G6del', The Monist, 51, pp. 9-32.CHIHARA, . [1972]: 'On Alleged Refutations of Mechanism Using G6del's Incompleteness

Results', Journal of Philosophy, 64, pp. 507-26.GUNDERSON, K. [I1970]: 'Asymmetries and Mind-Body Perplexities', in M. Radner and S.Winokur (eds.): Minnesota Studies in the Philosophy of Science, vol. 4. Minneapolis: TheUniversity of Minnesota Press.HERZBERGER,H. [1970]: 'Paradoxes of Grounding in Semantics', Journal of Philosophy, 67,

pp. 145-67.HOFSTADTER, D. R. [1979]: Godel, Escher, Bach: An Eternal Golden Braid. New York: BasicBooks, Inc., Publishers.LUCAS,J. R. [1961]: 'Minds, Machines and G6del', Philosophy, 36, pp. 1 2--I7, reprinted inA. R. Anderson (ed.) [1964]: Minds and Machines. Englewood Cliffs, N.J.: Prentice-Hall.LUCAS,J. R. [1968]: 'Satan Stultified: A Rejoinder to Paul Benacerraf' The Monist, 52, pp.

145-58.MACKAY,D. M. [1960]: 'On the Logical Indeterminacy of a Free Choice', Mind, 64, pp.31-40.MINSKY, M. [1968]: 'Matter, Mind and Models', in M. Minsky (ed.): Semantic InformationProcessing.Cambridge, Mass.: MIT Press.POPPER,K. R. [1950]: 'Indeterminism in Quantum Physics and in Classical Physics', TheBritish Journal for the Philosophy of Science, z, pp. I 17-33.SLEZAK, .: 'Descartes's Diagonal Deduction.' Unpublished.

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