## Ketland.fm

Deflationism and the Gödel
Jeffrey Ketland
1. Introduction
Neil Tennant (Tennant 2002) has recently replied to an argument givenby Stewart Shapiro (Shapiro 1998) and myself (Ketland 1999) concern-ing the incompatibility of the deﬂationary conception of truth with theGödelian incompleteness phenomena in mathematics. Call this argu-ment the Reflection Argument, since it turns on the truth-theoreticjustiﬁcation of reﬂection principles.1 The Reﬂection Argument con-cerns the substantiality of the notion of truth. To introduce the idea,consider the soundness statement, All theorems of Peano arithmetic (PA) are true. By Gödel’s Second Incompleteness Theorem, this single statement isdeductively stronger than the whole theory PA, at least if PA is consist-ent. For PA does not imply the consistency statement Con(PA), whilethe soundness statement ‘All theorems of PA are true’ does implyCon(PA) (modulo disquotation axioms for truth). But surely if weaccept PA and we also grasp the notion of truth, we see that we shouldaccept ‘All theorems of PA are true’. This shows that accepting the axi-oms and rules (and thus theorems) of PA is logically weaker thanaccepting ‘All theorems of PA are true’. There is a logical differencebetween accepting each theorem of PA and accepting the single reﬂec-tive proposition that each theorem of PA is true.
2. The reflection argument
In order to answer Tennant’s reply, it is necessary to rehearse in detailthe arguments given by Shapiro and myself. The reason is that Tennant 1 Hartry Field in his interesting reply (Field 1999) to Shapiro calls it the ‘Conservativeness Argu- Mind, Vol. 114 . 453 . January 2005
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doi:10.1093/mind/fzi075
misconstrues the central points of the argument that Shapiro and Igave. Tennant introduces a so-called ‘semantical argument’, and in thisconnection he cites Kleene, Dummett and others. It should be stressedthat this ‘semantical argument’ is quite diﬀerent from the argumentsgiven by Shapiro and myself. Our argument concerned the justifica-tion of the reﬂective principle ‘All theorems of S are true’, given thatone already accepts a mathematical theory S. It also concerns theconsequent justification of weaker reﬂection principles of the formProvS(⎡␸⎤) t␸, given acceptance of S. Such principles are logical conse-quences of ‘All theorems of S are true’, given the disquotation scheme,and are a weaker way of expressing in the language of S that anythingprovable in S is true. Shapiro and I argue that the justification for reﬂection principles involves the notion of truth. Tennant, however, provides no response tothis justiﬁcation given by Shapiro and myself. Tennant seems, in part, tobe worried that the global reﬂection principle ‘All theorems of S aretrue’ contains a truth predicate and a reﬂection principle like ProvS (⎡␸⎤) t␸ does not. It is hard to believe that this could be considered of signi- ﬁcance, since we are concerned with the justification of these principles,and not their conceptual content. Of course, ProvS(⎡␸⎤) t␸ is equiva-lent to ProvS(⎡␸⎤) t T((⎡␸⎤)), if we have a disquotational truth predicate. Rather, the Reﬂection Argument is about the proof-theoretic strength of truth-theoretic extensions of formal systems. Tennant’sreply involves the idea that one might adopt reﬂection principles suchas ProvS((⎡␸⎤)) t ␸ without any justiﬁcation at all. Tennant doesn’tappreciate that the Reﬂection Argument aims to explain why someonewho accepts S should also accept the instances of ProvS(⎡␸⎤) t␸.
Let us rehearse this argument. The deﬂationist claims that truth is insubstantial. Shapiro and I both analysed the notion of ‘non-substan-tiality’ of truth via the notion of conservation or conservativeness. Sha-piro wrote, … adding a truth predicate to the original theory [should] not allow us toprove anything in the original language that we could not prove before weadded the truth predicate … [Conservativeness is] essential to deﬂationism.
(Shapiro 1998, p. 497) … if truth is non-substantial—as deﬂationists claim—then the theory oftruth should be conservative. Roughly, non-substantiality ≡ conservative-ness (Ketland 1999, p. 79) Mind, Vol. 114 . 453 . January 2005
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Deflationism and The Gödel Phenomena: Reply to Tennant We both pointed out that, at least in some circumstances, the additionof axioms for the notion of truth to a mathematical base theory Sallows one to prove more about the non-truth-theoretic domain thanwhat was provable before in S. In particular, if the base theory is some-thing like PA and the truth axioms are Tarski’s truth axioms, then theglobal reﬂection statement becomes provable in the truth-theoreticextension. So, the truth-theoretic extension of PA is a non-conservativeextension.2 Hence, the notion of truth must be substantial. Hence, deﬂ-ationism about truth is mistaken.3 In even more detail, the Reﬂection Argument ﬁrst proposes that, (A) A deflationary conception of truth should be committed to some sort of conservation constraint for its favoured truth axi-oms. On a deﬂationary conception of truth, given one’s antecedent non-semantical base theory S, one should not be able to prove new resultsfor this domain merely by invoking axioms or rules for the notion oftruth. We both noted that the disquotational truth axioms (suitablyrestricted, as Tarski noted, to avoid paradox) do indeed satisfy this con-servation constraint. To illustrate, extending PA by the restricted dis-quotational T-scheme T (⎡␸⎤) t␸ yields a conservative extension, atleast so long as the instances are ‘Tarskian’ (i.e., ␸ belongs to the objectlanguage, and thus does not contain the truth predicate). The second part of the Reﬂection Argument concerns reflective rea- soning. As Shapiro and I both noted, we should like to be able to prove,given a base theory S, the global reﬂection principle ‘All theorems of Sare true’. Indeed, it seems that we should adopt this as an adequacy con-straint on theories of truth: 2 Unbeknownst to me, Solomon Feferman (Feferman 1991) had given a detailed analysis of pre- cisely this scenario (i.e., the properties of truth-theoretic extensions of mathematical base theo-ries), and had given a philosophical justiﬁcation similar to the one given by Shapiro and myself.
Namely, the justiﬁcation of reﬂection principles.
3 For an analogy, note that the addition of second-order comprehension axioms for the exist- ence of sets of numbers to PA similarly yields a non-conservative extension. For example, the pas-sage from PA to full second-order arithmetic Z2 involves adding the comprehension principle for sets (of numbers) deﬁnable by any arbitrary second-order formula. This yields a highly non-con-servative extension of PA. Note that the truth-theoretic extension of PA actually ‘lives inside’ aweaker second-order subsystem of Z2. More exactly, the truth extension Tr(PA) is inter-translata- ble with the second-order system ACA, which contains comprehension for arithmetically deﬁna-ble sets (with second-order parameters) plus full induction. Note ﬁnally that the proof thatCon(PA) is a theorem of ACA goes ﬁrst via formalizing the Tarskian truth deﬁnition for LPAwithin ACA, and then shows that ‘All theorems of PA are true’ is a theorem of ACA using the in-duction principle.
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(B) The base theory S ought to combine with truth axioms in such a way that reﬂection principles become provable. Any adequate theory of truth should be able to prove the equivalence of a(possibly inﬁnitely axiomatized) theory T and its truth True(T) (that is, themetalanguage formula ᭙x(ProvT(x) t Tr(x))). (Ketland 1999, p. 90).
By Gödel’s incompleteness theorems, this adequacy constraint contra-dicts the conservation constraint on the deﬂationary conception oftruth. For if S is a consistent and suﬃciently rich mathematical basetheory, then there is a consistency statement Con(S) expressible in thelanguage of S, but which is unprovable in S, by Gödel’s Second Incom-pleteness Theorem. And from ‘All theorems of S are true’ we candeduce ‘S is consistent’. So, if we could prove ‘All theorems of S are true’,we must have generated a non-conservative extension of S. The third part of the Reﬂection Argument concerns the speciﬁcation of just such a (consistent) theory of truth with the requisite non-con-servation property: Tarski’s inductive definition of truth (for arithmetic):
(TAt) An atomic sentence t = u is true iﬀ the value of t
(T¬) ¬␸ is true iﬀ ␸ is not true;
(T!) ␸ ! ␺ is true iﬀ ␸ is true and ␺ is true;
(T᭙) ᭙x␸ is true iﬀ, for each n, ␸(n) is true;
Shapiro and I both pointed out that, starting with PA as non-semanti-cal base theory, the extension obtained by adding Tarski’s inductive def-inition is non-conservative.4 In Ketland 1999, this truth-theoreticextension is called PA(S), following Richard Kaye’s terminology. Fefer-man refers to it as Tr(PA). Crucially, Tr(PA) proves, Global Reflection Principle: ᭙x(ProvPA(x) t T(x)),
which expresses ‘All theorems of PA are true’. Consequently, Tr(PA)proves all the resulting instances of both the local and uniform reﬂec-tion schemes, 4 It is essential here that the induction scheme in PA be expanded to formulas in the richer lan- guage L(T) containing the truth predicate. If the induction scheme is not expanded, then the ex-tension by Tarski’s axioms is indeed conservative. This is the basis of Hartry Field’s reply (Field Mind, Vol. 114 . 453 . January 2005
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Deflationism and The Gödel Phenomena: Reply to Tennant Local Reflection Scheme:5 ProvPA(⎡␸⎤)t ␸.
Uniform Reflection Scheme: ᭙x(ProvPA (⎡␸((x))⎤)t ␸(x)).
Indeed, Tr(PA) proves that all instances of local and uniform reﬂectionare true, and thus it proves the consistency of local and uniform reﬂec-tion (and thus the global reﬂection principle is logically stronger thanthe local and uniform principles).
To summarize, adding Tarski’s axioms to PA yields proofs of various reﬂection principles for PA. Indeed, this logical fact provides an expla-nation of the ‘reﬂective reasoning’ which begins with acceptance of PAand then yields acceptance of the global reﬂection principle. Onemight, and I think should, insist that something should explain why thisis so. Shapiro and I argued (and implicitly, also Feferman) that it is ourunderstanding of the notion of truth for the base language whichexplains why it is so.6 Finally, Shapiro and I both noted that when one has proved the Glo- bal Reﬂection Principle in the truth-theoretic extension, then one has aquick proof of the Gödel sentence G.7 That is, Tr(PA) ٛ G. Of course, onthe standard interpretation of Gödel’s results, one is committed to G ifone is committed to PA. More generally, Conditional epistemic obligation:
If one accepts a mathematical base theory S, then one is committedto accepting a number of further statements in the language of thebase theory (and one of these is the Gödel sentence G). Gödel’s theorems show the inadequacy of single formal systems … howeverat the same time they point to the possibility of systematically generatinglarger and larger systems whose acceptability is implicit in acceptance of thestarting theory. The engines for that purpose are what have come to be calledreflection principles. (Feferman 1991, p. 1)… which statements in the base language L of S … ought to be accepted ifone has accepted the basic axioms and rules of S? The answer is given as an 5 By Löb’s Theorem, PA does not prove the reﬂection principle ProvPA (⎡␸⎤)t ␸ unless it already proves ␸. In particular, PA does not prove ProvPA(⎡0 = 1⎤) t 0 = 1. Thus, PA does not prove ¬ProvPA(⎡0=1⎤), the consistency statement Con(PA), which arithmetically expresses ‘PA is consist-ent’.
6 In Feferman’s article, more powerful ‘self-applicative truth axioms’ are used to deﬁne what he calls the ‘reﬂective closure’ of mathematical base theories. Such extensions are even stronger thanthe simple Tarskian extensions under consideration here.
7 I also cited Tarski, who made a similar point nearly 70 years ago. See below.
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ordinary theory Ref(S) formulated in a language L(T, F) … where T and Fare partial truth and falsity predicates which are self-applicable in the sensethat they apply to (codes of) statements of L(T, F) … Thus, for example, wemay reason in Ref(PA) by induction about the truth of statements whichcontain the notion of truth, and so arrive at statements of the form: ᭙x[ProvPA(x) t T(x)], and by iterating this kind of argument derive iterat-ed reﬂection principles for arithmetic. (Feferman 1991, p. 2).
In short, we can explain the conditional epistemic obligation using thenotion of truth.8 But is the notion of truth indispensable in this expla-nation of conditional epistemic obligation? It looks as if the deﬂationistis cornered. For if the notion of truth is indispensable to this explana-tion of conditional epistemic obligation, then the axioms for notion oftruth (which are essential to the proof of the reﬂection principles andother statements) must be non-conservative.9 And this violates the deﬂa-tionary conservation constraint. If the deﬂationist insists on the conser-vation constraint, then she cannot explain why, given that we acceptsome base theory S, we ought to accept the stronger reﬂective statement‘All theorems of S are true’. The deﬂationist cannot have it both ways. Itseems that there are two options: (i) Either abandon the conservation constraint, thereby becoming some sort of substantialist about truth; (ii) Or abandon the adequacy condition. And furthermore, oﬀer some non-truth-theoretic analysis of the conditional epistemicobligation. This is the Reﬂection Argument against deﬂationism. 3. Tennant’s ‘Semantical Argument’
Tennant replies to both Shapiro 1998 and Ketland 1999. For reasonsobscure to me, Tennant thinks that Shapiro and I presented diﬀerent 8 The statements that one is committed to in accepting a base theory S might be called the re- flective consequences of S. These reﬂective consequences include the theory’s Gödel sentence G, theconsistency statement Con(S), plus the local and uniform reﬂection schemes. Extending a basetheory S with axioms for truth is therefore one way of deductively extracting these reﬂective conse-quences.
9 Note that if one adds something equivalent to adding the notion of truth, then the situation remains the same. Given that adding set theory (or, say, substitutional quantiﬁers) may be suﬃ-cient to permit an explicit deﬁnition of a Tarskian compositional truth predicate, this fact strength-ens the Reﬂection Argument. Indeed, as noted above, the proof of the reﬂection principlesproceeds precisely by formalizing the notion of truth, shows that the truth deﬁnition satisﬁes Tar-ski’s inductive clauses, and proving that all axioms of the base theory are true, that deducibilitypreserves truth, and so on.
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Deflationism and The Gödel Phenomena: Reply to Tennant arguments. Setting this aside, Tennant is primarily worried that weexplained the conditional epistemic obligation using the notion oftruth. Given the dilemma above, Tennant’s proposal is that we do nothave to invoke the Tarskian truth axioms (as an extension of the basetheory) to justify the reﬂection principles. In particular, Tennant is con-cerned with the consequence that the Gödel sentence is a theorem ofthe non-conservative truth-theoretic extension. Citing Kleene, Dum-mett and others, Tennant writes, … a certain interpretive dogma had managed to establish itself in responseto the Gödel phenomena: The substantialist dogma: The way in which the semantical argument estab-
lishes the truth of the Gödel sentence requires that the notion of truth be
substantial. (Tennant 2002, p. 557)
In particular, Tennant refers here to what he calls the ‘semantical argu-ment’, Semantical argument for the truth of the Gödel sentence:
G is a universally quantiﬁed sentence … Every numerical instance of thatpredicate is provable in the system S … Proof in S guarantees truth. Henceevery numerical instance of G is true. So, since G is simply the universalquantiﬁcation over those numerical instances, it too must be true. (Tennant Note that a crucial assumption in Tennant’s ‘semantical argument’ isthat proof in S guarantees truth. This is assumed without argument. ButShapiro, Feferman and I do not assume this: we prove it. We give anargument whose conclusion is that proof in S guarantees truth. So howis Tennant’s ‘semantical argument’ related to anything discussed byShapiro and myself? Tennant writes, A case will be made below for the following conclusion: the deﬂationist hasproperly deﬂationary means for attaining the insight that the undecidableGödel sentence for any sound theory of arithmetic is one that ought to be as-serted, rather than denied. (Tennant 2002, p. 553) Thus, Tennant’s ‘semantic argument’ can be phrased in just one line: Semantical Argument: If S is sound, then the Gödel sentence G is
true.
Tennant appears to be arguing that the deﬂationist ‘has properly deﬂ-ationary means for attaining the insight that’ G follows from the sound-ness of S. I ﬁnd this misinterpretation of the debate bizarre. It isacceptable to ﬁnitists, intuitionists, deﬂationists, nihilists, deconstruc-tionists, etc., that G follows from the soundness of S. After all, G is true if Mind, Vol. 114 . 453 . January 2005
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and only if G is not a theorem of S. On the assumption that S is sound,G is true if G is a theorem of S. So, G is not a theorem of S. So, G is true.
Discharging, we conclude that the soundness of S implies G. This con-clusion is acceptable to anyone. But what does this have to do with what Shapiro and I discuss? Sha- piro and I do of course use this fact, but we provide a crucial separateargument for S’s soundness. In contrast, Tennant provides no justiﬁca-tion for the crucial premiss, that the theory S is sound. So, the actualargument given by Shapiro and myself is not the ‘semantical argument’given by Tennant, or the argument given by Kleene, Dummett, et al.
For example, Dummett appears to be unaware of the central point dis-cussed by Feferman, Shapiro and myself. This is the point that addingaxioms for the notion of truth to S results in the provability of the sound-ness of S, and, consequently, reflection principles for S. This idea is notnew. As Tarski put it, All sentences constructed according to Gödel’s method possess the propertythat it can be established whether they are true or false on the basis of the me-tatheory of higher order having a correct deﬁnition of truth. (Tarski 1936,p. 274) Indeed, this point is possibly what Gödel was intimating when he wrotein his original 1931 paper, … [T]he true reason for the incompleteness inherent in all formal systemsof mathematics is that the formation of ever higher types can be continuedinto the transﬁnite … while in any formal system at most denumerablymany of them are available. For it can be shown that the undecidable prop-ositions constructed here become decidable whenever appropriate highertypes are added (e.g., of type ␻ for the system P). An analogous situation pre-vails for the axiom system of set theory. (Gödel 1931, footnote 48a) Intriguingly, in a letter to A. W. Burks, Gödel wrote, … the theorem of mine that von Neumann refers to is … the fact that a com-plete epistemological description of a language A cannot be given in the samelanguage A, because the concept of truth of sentences in A cannot be deﬁnedin A. It is this theorem which is the true reason for the existence of undecid-able propositions containing arithmetic. (von Neumann 1966, pp. 55–6) That is, the substantiality of truth highlighted by Shapiro and myself isnot only connected to the Gödelian incompleteness phenomena, butalso intimately related to the indefinability of truth.
The Reﬂection Argument against deﬂationism and for the substanti- ality of truth concerns the non-conservation properties of truth-theo-retic extensions, and how one justifies reﬂection statements such as ‘Alltheorems of S are true’. It is not solely about what follows from such Mind, Vol. 114 . 453 . January 2005
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Deflationism and The Gödel Phenomena: Reply to Tennant statements. So, it seems to me that Tennant is mistaken in his analysisof the actual arguments given by Shapiro and myself.
4. Tennant on the provability of G in truth-theoretic extensions
For a range of base theories S, the Tarski truth-theoretic extension Tr(S)proves the global reﬂection principle for S. A consequence is that thereis a proof in Tr(S) of the Gödel sentence G for S. And, of course, G isunprovable in the base theory S, if S is consistent. In the section in myarticle on the demonstrability of the Gödel sentence, I remarked, How do we ‘recognize the truth’ of G? … G is deducible from the strengthened theory: namely, T plus the standardTarskian theory of truth for the language of T.
… a deﬂationary theory cannot achieve such ‘insight’ (i.e., deduction). It isconservative, so T ʜ ‘deﬂationary theory’ does not imply G. Indeed, the gen-eralized equivalence principle fails for DT. If I am right, our ability to recog-nize the truth of Gödel sentences involves a theory of truth (Tarski’s) whichsigniﬁcantly transcends the deﬂationary theories. (Ketland 1999, pp. 87–8.) The ‘generalized equivalence principle’ referred to is the adequacyrequirement mentioned earlier that the truth-theoretic extension of abase theory S should prove ‘All theorems of S are true’. This is in fact thecase for any suﬃciently rich base theory S, axiomatized by ﬁnitely manyaxioms and schemes.10 Omitting my reference to the adequacy constraint on truth-theoretic extensions, Tennant quotes my claim ‘… our ability to recognize thetruth of Gödel sentences involves a theory of truth (Tarski’s) whichsigniﬁcantly transcends the deﬂationary theories’. (This is the only sen-tence of my article quoted by Tennant.) Tennant continues, Ketland, as we shall see below, errs by assuming without argument that Tar-ski’s theory of truth is the only way that we can come to recognize the truthof Gödel sentences. (Tennant 2002, p. 567) I did not make this assumption. First, if Tarski’s theory of truth pro-vides at least one way of ‘recognizing the truth of Gödel sentences’, thenthis fact alone contradicts deﬂationism (for a deﬂationary theory oftruth should be conservative). The entirely diﬀerent assumption, whichI did not make, that this ‘is the only way’ is irrelevant. Second, asPutnam and others have stressed in their replies to Lucas’s argumentthat Gödel’s theorem implies that ‘minds are not machines’, any con-sistency proof for a formal system S yields G as a bonus. So, any theory 10 See Feferman 1991 for details on the constraints that the base theory S must satisfy.
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which proves the consistency of S yields a proof of G. Indeed, insofar aswe have any good reasons for thinking S consistent, we necessarily havegood reasons for thinking G is true. And these good reasons mightthemselves be empirical-inductive. Lucas appears to assume that thehuman mind has a special capacity: the capacity to reliably recognizeany consistent system to be consistent. As Putnam and others stressed,that human minds possess this capacity is not implied by Gödel’sincompleteness theorems. It is an entirely open empirical questionabout the structure of the human mind. In any case, these are irrelevant matters for the present discussion, 5. Tennant’s strategy for ‘proving’ G
Tennant expresses the essence of his proposal as follows, My counter to both Shapiro and Ketland will amount to this: the deﬂationisthas ‘philosophically modest’ means for carrying out the so-called ‘semanti-cal’ argument for the “truth” of the Gödel sentence. (Tennant 2002, p. 553)My main purpose in this paper will have been achieved if I can simply showthat there is a ‘deﬂationary way’ of faithfully carrying out the so-called ‘se-mantical argument’ for the truth of the independent Gödel sentence. (Ten-nant 2002, p. 557) Tennant arrives at a weak reﬂection principle, uniform primitive recur- sive reflection (UPRF), ᭙x(ProvPA( (⎡␸((x))⎤)t ␸(x)) itive recursive predicate. It is not diﬃcult to show that G is a theorem ofPA ʜ UPRF. Of course. But, unlike Feferman, Shapiro and myself, Ten-nant does not say why someone who accepts PA ought to accept UPRF.
As explained above, our main aim is—pace Ketland—to show precisely thatthere is another way to recognize the truth of Gödel sentences, which doesnot invoke anything like a Tarskian theory of truth signiﬁcantly transcend-ing the deﬂationary theories. If I succeed in this constructive aim, then thatwill be dispositive; for a deﬂationist rendering of the semantical argumentfor the truth of the Gödel sentence will refute Ketland’s objection. (Tennant2002, p. 567) As pointed out above, the existence of ‘another way to recognize thetruth of Gödel sentences’ does not touch the arguments given. Theexistence of such ‘another way’ has no bearing on the existence of theproof of G in the Tarskian truth-theoretic extension. The point is thatthe truth-theoretic extension Tr(PA) proves G. This should not happenon a deﬂationary view of truth. But it does happen.
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Deflationism and The Gödel Phenomena: Reply to Tennant Tennant again refers to his ‘semantical argument’, not the argument that Shapiro and I discussed. The argument that we give is based on theprovability of the global reﬂection principle using the compositionalTarskian truth axioms, a point that Feferman, Shapiro and I make clear.
Equipped with just the restricted disquotation axioms, the deﬂationistcannot prove the reﬂection statements (i.e., the global, uniform andlocal reﬂection principles). But the substantialist can prove these in themore powerful truth-theoretic extension. The global reﬂection principle does of course imply the local and uniform reﬂection schemes. Shapiro and I (and Feferman) were con-cerned with the justiﬁcation of these statements. The problem withTennant’s proposal is the absence of any justiﬁcation for accepting anyof these reﬂection principles. One might similarly ‘prove’ G by merelyassuming G, or even ‘prove’ the Riemann Hypothesis by merely assum-ing the Riemann Hypothesis. Tennant claims that his aim is to show‘that there is another way to recognize the truth of Gödel sentences’. Hedoes this by assuming without argument a principle that he has notjustiﬁed, and in fact a principle that Feferman, Shapiro and I showedhow to prove. We have also argued that deﬂationist or prosententialist strategies are avail-able for ‘reﬂecting’ upon essentially incomplete formal systems in the proc-ess of continually extending them. This avoids any epistemic shortfall incomparison with substantialism. (Tennant 2002, p. 580) But Tennant has not succeeded in showing that such strategies are‘available’, unless assuming something without argument makes it ‘avail-able’. Part of the point of the articles by Feferman, Shapiro and myselfwas to show how to prove reﬂection principles, which, ‘ought to beaccepted if one has accepted the basic notions and schematic principlesof that theory’ (Feferman 1991, p. 44). On Tennant’s proposal, instead ofproving the reﬂection principles in the manner proposed by Feferman,Shapiro and myself, the deﬂationist may simply assume them. As far as Ican see, in the absence of the sort of truth-theoretic justiﬁcation givenby Feferman, Shapiro and myself, Tennant’s proposal is that the deﬂa-tionist may assume these principles without argument. No reason, argu-ment or explanation for adopting the reﬂection principles is given byTennant. If we avoid explaining why acceptance of mathematical theoryS rationally obliges further acceptance of reﬂection principles, then thisis apparently ‘philosophically modest’. I ﬁnd this curious. It is ratherlike saying that if we avoid explaining a phenomenon, we achieve ‘phil-osophical modesty’. Presumably, the ideal way to achieve such ‘mod- Mind, Vol. 114 . 453 . January 2005
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esty’ in the scientiﬁc arena would be to abandon scientiﬁc explanationaltogether. 6. An analogy: ad hoc instrumentalist strategies in science
Consider an analogy. The scientist who is a realist about the magnetic ﬁeld B thinks that such a physical ﬁeld exists and that Maxwell’s equa-
tions for B, E, J, ␳, etc., are in fact correct, and thus that the properties
of B are described by these equations. (For example, the spatial diver-
gence of B is zero, and the curl of B is given by the sum of the current J
and the time derivative of E.) On this basis, she proves that if you have a
circular current loop with current i, then the physical magnetic ﬁeld
distribution B nearby looks like this (gives a diagram). She also knows
that if the magnetic ﬁeld distribution looks like this, then the pattern of
orientation of small iron ﬁlings near the current loop will look like that
(gives another diagram.) That is, she knows that the iron ﬁlings line up
along the integral curves of the ﬁeld B. Upon experimental checking,
we actually observe that whenever we set up such a current loop, then
we actually get precisely the observable iron ﬁling pattern predicted by
Maxwell’s equations
. On this basis, we can reasonably claim that Max-
well’s equations explain the observed pattern.
Consider an instrumentalist, like Thomas Kuhn or Bas van Fraassen, who thinks that the magnetic ﬁeld B is a ﬁction. The instrumentalist
refuses to concede the explanatory success of Maxwell’s theory, and
If you have a current loop with current i, then the pattern of orienta-tion of small iron ﬁlings near the current loop will look like that(diagram given).
The instrumentalist might insist that this is suﬃcient to ‘explain’ whywe observe the particular pattern. After all, this would be ‘philosophi-cally modest’ and would not assume the existence of ‘ﬁctions’, such asthe magnetic ﬁeld, and so on. But no serious scientist is satisﬁed with this ad hoc instrumentalist manoeuvre. Originally, we had no idea what the pattern would look
like, but using the abstract theory, we made a prediction which was
subsequently veriﬁed. This increases our conﬁdence in the correctness
of the abstract laws (diﬀerential equations relating the magnetic ﬁeld B
to the current J) used to obtain the prediction. It would be quite ad hoc
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Deflationism and The Gödel Phenomena: Reply to Tennant to repudiate the highly successful explanation given by the abstract law,and propose instead the weakest possible ‘explanation’ of the pattern.
7. Summary.
Based on remarks of Kleene, Dummett and others, Tennant describes a‘semantical argument’, which is roughly that if a base theory S is sound,then its Gödel sentence G is true. This somewhat obvious fact has littlebearing on the actual arguments given by Shapiro and myself, concern-ing the non-conservation properties of certain compositional theoriesof truth and the fact that one may prove reﬂection principles usingthese truth axioms. Repudiating the truth-theoretic explanation of thereﬂection principles given by Feferman, Shapiro, and myself, Tennantmerely posits a reﬂection principle, without providing justiﬁcation forit, thereby illustrating what Bertrand Russell once called ‘the advan-tages of theft over honest toil’.
Tennant’s proposed extension doesn’t directly invoke the notion of truth, just as, in the above analogy, the instrumentalist’s ad hoc obser-
vation conditional doesn’t refer to the magnetic ﬁeld B. But that is irrel-
evant, because it is not conceptual content that is at stake, but rather the
justification. The weak reﬂection principle that Tennant proposes (uni-
form primitive recursive reﬂection) is introduced without justiﬁcation,
and is thus ad hoc. Neither Shapiro nor I have denied that it might be
possible for Tennant (or the deﬂationist) to provide some other non-
truth-theoretic
form of justiﬁcation of reﬂection principles. But Ten-
nant has not—at least not yet— provided any such alternative justiﬁca-
tion.
In contrast, Feferman, Shapiro and I do indeed provide a justiﬁcation— a sort of explanatory reason— for accepting the reﬂec-tion principles. And this particular justiﬁcation is truth-theoretic. Takethe axioms you initially accept, and then reﬂect upon these axiomsusing your independent grasp of the notion of truth for the language(e.g., as given by Tarski’s inductive deﬁnition). Thus equipped, we canprove that each axiom is true; we can prove that deductive inference pre-serves truth. We thus obtain a proof that each theorem is true. This latteris the Global Reﬂection Principle, which implies the uniform and localprinciples. This justiﬁcation of reﬂection principles is clearly truth-the-oretic, and this is the essence of the Reﬂection Argument given byShapiro and myself. Mind, Vol. 114 . 453 . January 2005
Ketland 2005
School of Philosophy, Psychology and Language SciencesUniversity of EdinburghGeorge SquareEdinburgh EH8 [email protected] References
Feferman, Solomon 1991: ‘Reﬂecting on Incompleteness’, Journal of
Symbolic Logic, 46, pp. 1–49.
Field, Hartry 1999: ‘Deﬂating the Conservativeness Argument’, Journal of Philosophy, 96, pp. 533–40.
Gödel, Kurt 1931: ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’. Monatshefte für Mathema-tik und Physik 38, pp. 173–98. English translation, ‘On FormallyUndecidable Propositions of Principia Mathematica and RelatedSystems I’, in S. Feferman et al 1986 (eds), Kurt Gödel: CollectedWorks. Volume I. Oxford: Oxford University Press.
Ketland, Jeﬀrey 1999: ‘Deﬂationism and Tarski’s Paradise’, Mind, 108, Shapiro, Stewart. 1998: ‘Truth and Proof—Through Thick and Thin’, Journal of Philosophy 95, pp. 493–521 Tarski, Alfred. 1935/6: ‘Der Wahrheitsbegriﬀ in den formalisierten Sprachen’, Studia Philosophica I, pp. 261–405. English translation‘The Concept of Truth in Formalized Languages’ appears in A. Tar-ski 1956, Logic, Semantics, Metamathematics: Papers by Alfred Tarskifrom 19221938. Edited and translated by J. H. Woodger. Oxford:Clarendon Press. Tennant, Neil 2002: ‘Deﬂationism and the Gödel Phenomena’, Mind, von Neumann, John 1966: Theory of Self-Reproducing Automata. A. W.
Burks (ed.). Urbana: University of Illinois Press.
Mind, Vol. 114 . 453 . January 2005
Ketland 2005

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Sept 2007 To my Dear Sweet Ladies, This is a month of changes. Two weeks ago, we were on the farm in Minnesota, then we came here to live in Abu Dhabi in a high rise overlooking the *Arabian* Gulf (where we were last year. Some call it the *Persian* Gulf). Next week we will be moving everything across the country to what they call a villa, but what we would call a cement and clay tile, single fami