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&p)). Each of these theories is a truth theory for L in the sense explicated by Tarski: they satisfy Convention 7: That is, Tr(x) is a truth predicate for L in MT, D T and SDT Consider the minimalist theory MI? Every L+-sentence Ramsey(cp) is an instance of RS and thus MT is "the totality of instances" of RS. By construction, MT is just the set of all these sentences. Similarly, the disquotation theory DT, in Tarski's terminology, satisfies Convention T, for (by construction) it proves every L+-sentenceTarski(cp), = Tr( & p), from which Tr( & p)) follows. Within this substitutional logic, one may show that cp is derivable from Cp( & p ) (in particular, using an axiom scheme governing quotations, namely, &p)), just = T iff I+,,;,[Cp(x = & p)] in case, for all e E dom(I+,,,), I+sub~r[Tr(x)] = T. This holds just in case, for all e E dom(I+,,,), e E I',,,[Tr] iff, for some cp E &, I+,ubl,[~ = ) ~ p )Now, finitely many L-sentences, and we can suppose that all of these are instan. each such instance Tr( ) ~ p )But an axiom of DT. Thus, again, there must be ajizite set l- of L-sentences, such that T u DT, i- cp. The above argument then goes through to show that T k cp. Again, we conclude that T u SDT is a conservative extension ofi? These conservativeness results provide consistencyproofs for the deflationary truth theories 07:MT and SDT. They show that by restricting the occurrence of the new predicate Tr in L+-formulasone obtains a consistent theory of truth (and thus avoids the semantic antinomies). I will not examine the proposals for further extending the truth theories so as to define truth for formulas themselves containing Tr. Tarski's 1936 proposal is to introduce a hierarchy of metalanguages L',L", . .. , and truth predicates Tr,, f i , , . .. . Instead, Kripke's (1975) proposal is to maintain a "univocal" truth predicate Tr, but relax the Principle of Bivalence for certain sensitive (ungrounded) formulas (such as "This sentence is false").1° The following are corollaries of Theorem 1. Corollary 1.1 (The contentlessness principle). No non-semantical statement (in L) follows from a deflationary theory of truth unless it is a logical truth." Corollary 1.2 (The irrefutability principle). No non-semantical contingent statement (in L) could refute a deflationary theory of truth. Corollary 1.3 (The consistency principle). A deflationary theory of truth is consistent with any consistent non-semantical theory (in L). A var(i,) A . .. A var(i,), s) t,P(tm(s, i,), . . . , tm(s, i,))] is an axiom. B. Recursion Axioms of TS Vfv's[Sat(neg(f), s) t,1SatV; s)] YF'gVs[sat(conjK g), s) ++(SatK s) A W g , >))I ViV's[Sat(exquV; i), s) t, 3s*(Seq(s*) A Id@, s*, i ) s*>>l. C. E,uplicit Definition of Tr in terms of Sat:
Dejationisrn and Tarski 's Paradise (iii) I+,,,[<E>] = E , for any L-expression
77
E
(iv) I+,,,[Tr] = {cp E &: I k cp). Clearly, I+I= Tr(
=,,
The idea now is to find a single formula Y, (in fact, a definition of Tr(x)) such that Y, t DTr. First, consider T,(x) =: (x =
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Case 2. Suppose that Tr(x) occurs in instances of schemes of T, say, @,, cp. Let @,* be the result of replacin A. These a , * are new schemes in ing Tr(x) by T,(x) in each scheme 0, T. Let the resulting set be A*. We have, Y, t a,t+ a , * , for each i. It follows that A u {Y,) k cp if and only if A* u {Y,) t cp. Thus, A* u {Y,} k cp. But no formulas in A* contain Tr(x). Again, A* k cp. So, T k cp. Thus, T u D T is a conservative extension of T. A similar proof-theoretic argument works to establish the conservativeness of SDT even when axiom schemes are present in T. Although SDT is an explicit definition of Tr, we cannot immediately conclude its conservativeness, because it is framed using apparatus (i.e. substitutional quantification with respect to L-sentences) which exceeds that of the object language L. Still, let us suppose that T v SDT cp, where cp is an L-sentence. So, we have a proof from T u SDTof cp. This is equivalent to a proof . again, any proof will contain only of cp from T u Ilp(Tr(
.. . , @, in A. We know that A u {Y,) k
.
.
'O
See Kripke (1975). See Kirkham (1992) for discussion.
" Quick proof: from the conservativeness theorem, if Tu DT k
Now let T = 0. Hence, if DT 1 cp, then logical truth (in L, of course). QED.
0k
cp, then T C 9. cp. That is, if DT k cp, then cp is a
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One might suggest that these corollaries illustrate a kind of "analyticity" or "contentlessness" that deflationary theories of truth exhibit. Adding them "adds nothing". Indeed, it is these metalogical properties that are closely connected to the idea that the deflationary truth theories illustrate the "redundancy" or "non-substantiality" of truth. Indeed, one might go further: iftruth is non-substantial-as deflationists c l a i p t h e n the theory of truth should be conservative. Roughly: non-substantiality consewativeness.
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3. Non-conservativeness of the full Tarskian theovy of truth Consider the full Tarskian theory of truth (satisfaction) for L, given by Tarski (1936). We shall call this theory TS, for "theory of satisfaction". Our second technical result is that TS not generally conservative over theories in the object language L. The theory TS contains axioms (or a recursive definition) for a primitive satisfaction predicate Sat(x, s). TS is formulated (by Tarski) in a much more powerful metalanguage L', still an extension of L, containing the following extra apparatus: (i) Syntactical predicates and functors: Form(,u), Sen(,u), var(i), etc., with obvious intuitive meanings (",u a is formula", "x is a sentence", "the ith variable", etc.) (ii) Syntactical functors: neg(x), conj(x, y) and exqu(x, i), with obvious intuitive meanings ("the negation ofx", "the conjunction ofx and y", "the existential quantification of x with respect to the ith variable") (iii) The concatenation functor, x A y; (iv) All the terms generated by a suitable quotation-operator <. ..> (v) A two-place satisfaction predicate, Sat(,u, s) (vi) The mathematical predicate, E . We assume that within this (set-theoretical) metalanguage we can define three further notions: (vii) A one-place predicate, Seq(s), meaning "s is a sequence" (viii) A three-place predicate, Id(s, s*, i), meaning "sequences s and s* are identical except possibly at the ith place" (ix) A two-place functor tm(s, i), meaning "the ith term of the sequence s". The axioms governing Sat(x, s) in TS are as follows: A. Basis Axioms of TS
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For each n-place L-primitive predicate symbol P, the formula Vi, ...Vi,Vs[Sat(
A
SatV;
Vx[Tr(x) t,(Sen(x) A Vs(Seq(s) + Sat(x, s))]. This corresponds to Tarski's definition of truth (Tarski 1936, Definition 23). The metatheory TS must be taken also to include (i) Some (suppressed) axioms governing formation rules (e.g. if P is an n-place primitive predicate, then the concatenation of P with the variable sequence x,, . . . ,x,, is an atomic formula; and so on), (ii) Some (suppressed) set-theoretical axioms governing the properties of sequences (e.g. an axiom saying that s is a sequence if and only if s is a complete function on the natural numbers). The full theory of satisfaction TS is then the union of these axioms (syntactical, semantical and set-theoretical). The importance of TS is that, as Tarski showed, it can be shown to derive DT and thus satisfies Convention 7: Briefly, TS i- 07: In short, Tarski showed how to define (in L',not in L) a truth predicate ?i for L within the metatheory TS. The derivation within TS of a materially adequate disquotational "T-sentence" Tr(
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Theorem 2 (The non-conservativeness of the theory of satisfaction.) Let L be the language of arithmetic and let TS be the theory of satisfaction in L for L. Then PA v TS is not a conservative extension of PA. Proof SketchL2We know from Godel's Second Incompleteness Theorem that there is a sentence Con(PA) in L expressing the (syntactical) consistency of PA and which is not derivable from the axioms of PA. However, in PA u TS one may prove that PA is true. That is, it proves the existence of a standard model of PA: the domain of this structure is o (the finite von Neumann ordinals), 0 is 0 and s(x) = x u {x). Thus PA u TS proves the consistency of PA. Hence, within PA u TS one may derive Con(PA). We can construct a proof-theoretic argument for this result using some standard facts about PA and some facts, established by Tarski (1936), about TS. We know from Godel's Second Incompleteness Theorem that: (i) A predicate Prov(x) is expressible in L which satisfies the constraints on a provability predicate for PA." Then the formula Con(PA) =,f ~ P r o v ( ' 0# 0') "asserts" that PA is (syntactically) consistent. (ii) not: (PA i- Con(PA)). Let PA(S) = PA v TS. Let True(PA) be the L+-formula Vx(Prov(x) -+ Tr(x)), which asserts (intuitively speaking) that PA is true (that is, anything provable in PA is true). Then we have two further important facts: (iii) TS k Tr('cpl) tt cp, for any sentence cp of L (iv) PA(S) t True(PA). Fact (iii) just states that TS satisfies Tarski's Convention T, that is, it derives the materially adequate T-theorems. Equivalently, TS t DT. Fact (iv) just states that PA (S) proves the formula which says that PA is true (that is, that every theorem of PA is true). In fact, this result is equivalent to Tarski's (1936) Theorem 5 , which Tarski expresses as P r Tr. Informally, Tarski proves this by showing first that each axiom of the object theory T is true.14Next, one shows that the rules of inference yres e n e truth, so that anything derived from true assumptions is also true." Thus, all the theorems of the object theory Tare true. "This fact is well known to mathematical logicians. I am indebted to John Burgess for pointing this out (private cornrnunication). l 3 See Boolos and Jeffrey (1989, Ch. 16). More exactly, the relation "m is the godel number of a proof-sequence in PA whose final formula has godel number n" (i.e. "m is a proof of n") is a recursive (decidable) relation of numbers. It is thus represented in P.4 by a formula PrAm, n). Then, Prov(x) is defined by 3mPrAm, x). This predicate Prov(x) has important logical properties. E.g. if PA t cp then PA r Prov( 9'). l 4 The axioms of the object theory Tare also axioms of the metatheory T u TS. That the metatheory contains an explicit definition of "cp is an axiom of T'.
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From (iv), we infer, PA(S) k ~ r o v ( ' 0# 0') -+ Tr('0 # 0'). And, using (iii), PA(S) t Prov('0 # 0') + 0 # 0. But PA(S) k i(O # 0). Thus PA(S) t Con(PA). To summarize, adding the theory TS in L' of truthlsatisfaction for the language of arithmetic to PA permits the deduction of the consistency of PA, which is an assertion not deducible within PA itself (assuming PA is consistent (which it is, of course)). It is worth mentioning the second-order case. If T is a categorical second-order theory (say axiomatic second-order Peano Arithmetic, PA2), then any new theorems in T u TS are L-assertions which already hold in the (effectively unique) model of T and are thus semantically implied by T. Godel's Incompleteness Theorem still applies, but now as an incompleteness of the second-order deducibility relation, t . In the first-order case, Godel's (1930) Completeness Theorem ensures that the proof- theoretic concept t is coextensive with the model-theoretic concept !=. But in the second-order case, Godel's (1931) Incompleteness Theorem ensures that t is a proper subset of k . For example, Con(PA2)is not a theorem (k ) of PA2but is a logical consequence ( k ) of PA2.Indeed, the second-order formula PA2 & iCon(PA2) is consistent but unsatisfiable (logically false). In any case, adding the satisfaction theory TS allows the deduction of some of these previously underivable logical implications (of course, PA u TS is not complete: it is an axiomatic extension of PA, so it may be godelized too!).
4. w-incompleteness, laws of truth and non-standard models The deflationary truth theory D T has four further interesting logical properties: (i) D T has (in a sense) "non-standard models"; (ii) D T cannot prove certain general laws of truth (like the law for negation); (iii) D T can prove "all the instances" of such general laws, so it has a property analogous to a-incompleteness; (iv) D T does not implicitly define Tr (or "fix the extension" of Tr). First we prove (i) by exhibiting a non-standard model ofDT For example, we can exhibit an interpretation of the metalanguage L' which is a model l S More exactly, the metatheory T u TS can prove: if A is a set of true L-sentences and cp is a deductive consequence of A, then (O is true.
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of DT but in which the general "law of negation" fails. This proves (ii). Then we show that, despite D7"s failure to prove this law, it still has the power to prove "all" of its instances, so it is analogous to certain incomplete axiomatizations of arithmetic (like Robinson Arithmetic Q). Finally, we prove (iv), which is a refutation of the oft-made claim that Tarski's Convention T (on its own) "fixes the extension" of "true". 4.1 iVon-standard model
Let I be any L-interpretation. Construct I+ by setting where a is not an element of dom(l) (i) dom(I+)= dom(l) u {a, 01, (ii) P[Sen] = {cp: cp E & j u {a) (iiia) if cp E ,,L, for some n, then i-[neg](cp) = 7 c p (iiib) I+[neg](a) = a (iiic) if x is neither an L-formula nor a, then P[neg](x) = 0 (iv) I+[Tr] = {cp E &: I k cpj u {a) (v) P[<E>]= E, for any L-expression E . I+is surely a model of DT, because, for any L-sentence cp, I' k P(
4.2 D T does not imply the general law of negation Consider the classical law of negation, NEG, which is just the Lt-formula Vx(Sen(x) + [Tr(neg(x)) ++ ~Tr(x)]) It is easy to check that this formula is false in I+,because, as we noted above, the non-standard element a and its NEGATION are TRUE SENTENCES. 4.3 D T is "w-incomplete
"
Despite the fact that DT does not imply NEG, it still implies all the "instances" ofNEG. That is, for any cp, D T 1 Tr(<~cp>)t,~Tr(
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Compare this with the notion of a-incompleteness from mathematical logic. A theory T i n the language of arithmetic L is a-incomplete if and only if there is a formula P(x) such that (i) For all natural numbers n, T k P(E) (ii) T does not imply VxP(x). Obviously, we may adopt an analogous notion for theories that talk about syntactical items, like closed formulas. That is, T implies P(
4.4 D T does not implicitly define Tr(x) A number of authors writing about Tarski's theory have claimed that Convention T (in effect, the infinite list of T-sentences: i.e. the theories D T and MT) "fixes the extension of", or implicitly defines, "true". Examples of such authors are Quine (1953, p. 136) and Haack (1978, p. 100). Certainly, if we set up two disquotation theories DT, and DT, governing truth predicates Tr,(x) and Tr,(x), then since DT, i- Tr,(
Theorem 4 (Convention T does not implicitly define "true"). Let T be a consistent axiomatic extension of PA. Then T u D T does not implicitly define Tr . Proof: Because Textends PA, (i) the syntax of L may be formalized within Tin L and (ii) Tarski's Indefinability Theorem applies. Now, assume that T u D T implicitly defines Tr. Then, by the Beth Definability Theorem, For a detailed presentation of Q, see Boolos and Jeffrey (1989, Ch. 14). "The Beth Definability Theorem (Beth 1953) is an important metatheorem for Jirst-order logic. Implicit definability was first clearly explained by Padoa (1903). A theory Timplicitly defines a (concept expressed by) aprimitive symbol S(x) in the language of Tif any pair of models I,and I, of Twith identical domains and which agree on the extensions of all symbols except S also agree on the extension of S. Beth's theorem says that if a symbol S is implicitly definable in a first-order theory T, then it is explicitly dejnable in T. So, there is a formula Y(x) not containing S such that T i- Vx(S(x) t,Y(x)). See Boolos and Jeffrey (1989, pp. 245-9). l6
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there exists an explicit definition of Tr in T u DT. Thus, T u D T k Vx(Tr(x) t,Y(x)), where Y(x) is an L-formula. Thus, for each closed Lformula cp, T u DT 1- Y(
5. Deflationary theories are incomplete theories of truth The deflationary theories of truth are, I claim, incomplete accounts of the concept of truth. The Tarskian theory of satisfaction is a much more complete account of our conception of truth. For example Tarski's theory TS (unlike the deflationary theories DT and MT) yields all the usual general theorems expected from a (classical) theory of truth: For any closed formula cp, l c p is true if and only if cp is not true. If C is a set of true closed formulas, then any deductive consequence of C is true. For any set of closed formulas C, if C is true then C is consistent. .. . and so on. Consider the attempt to express "T is true" within the language of T. The statement "T is true" means "for any closed formula cp, if cp is an element of T, then cp is true". There are two problems. First, by Tarski's Indefinability Theorem, "true" may not be definable in T (e.g. arithmetical truth is not (first-order) definable in (first-order, complete) arithmetic, Th(N)). Second, some theories are not axiomatizable (e.g. Th(N)), so cp's being an element of Tneed not mean that there is a recursive axiom system thatproves cp. But we can make certain progress. Let Tbe a consistent axiomatizable extension of first-order PA. Now, let cp be any L-sentence, we want to express (1) If cp is provable in T then cp is true. A provability predicate (for 7)Prov(x) is expressible in T and we can express (1) first in the metalanguage L+by (2) Prov('cpl) + Tr('cpl). Then we can "disquote" the truth-predication and obtain the L-formula, (3) prov('cpl) + cp. Any such formula is called a "reflection principle" (see Boolos & Jeffrey 1989, p. 283). Let Rej(7) be the set of all these reflection principles in L. One might think of this infinite collection of sentences ReJ(7) as expressing (at least, partially) within the language of an axiomatic theory T the "truth of T". Now, it is possible to show that, if T is a consistent axioma-
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tizable extension of PA, then not-(T t ReJ(T)).lSActually, ReJ(7) implies Con(T)! So T u ReJ(7) is certainly not a conservative extension of T. Furthermore, one may show that adding our deflationary truth theory D T to T is insufficient to derive the metalanguage formula which fully expresses the "truth of T". As before, we define the "truth of T" as the L+formula: True(T) Yx(Prov(x) + Tr(x)). It is then possible to prove the following theorem, Theorem 5: Let T be a consistent axiomatic extension ofPA. Then T u D T does not imply True(T).
=,,
Proof. Assume that T u DT t True(T). Then, T u D T I- Vx(Prov(x) + Tr(x)). Thus, T u D T t ~ r o v ( ' 0# 0') + Tr('0 # 0'). So, T u D T t Con(T). Thus, by conservativeness, T t Con(T). This is impossible, by Godel's Second Incompleteness Theorem. The same holds for the other conservative deflationary truth theories MT and SDT, Let us quickly look at how this works for Robinson Arithmetic Q, which is finitely axiomatized. Of course, we may suppose that Q is a single (finite) axiom in L, and then we can "express its truth" in L+as a single formula Tr(' Q'). Q also has provability predicates, say Prov(x) again, and we can also "express it truth" as True(Q): again, the L+-formula, Yx(Prov(x) + Tr(x)) . What we can then show is that Theorem 6: (i) Q u D T does not imply True(Q). (ii) Q u DT does not imply Tr('Q1) t,True(Q). Proof: The proof of (i) is analogous to the above Theorem 5. The proof of (ii) resides in the fact that D T t Tr('Q1) H Q (by construction!). So, Q u D T t Tr('Q1). If Q u D T implied Tr('Q1) H True(Q), then it would also imply True(Q), which it does not by (i). So (ii) is proved. This further strengthens the case for thinking that the deflationary truth theories really are weak and incomplete theories of truth.
6. Tarski's theory of truth and Godel sentences The non-conservativeness of Tarski's theory of satisfaction is very interesting. As before, let T i n L be a consistent axiomatic extension of PA. Then let G be a Godel sentence for 7: Now, G "says that" G is not provable in T So, G is true if and only if G is not provable in 7: This can be formal"Use Lob's Theorem (see Boolos and Jeffrey 1989, p. 187), which says that, if T t Prov('cp7) + cp, then T t cp. So if T implied Rej(T), it would have to imply every sentence cp, and T would be inconsistent.
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ized within T. Indeed, T t G t,1Prov('G1). There is such a sentence G, by the Diagonal Lemma (or Fixed Point Theorem).
Theorem 7 (Provability of Godel sentences). Let T be a consistent axiomatic extension of PA. Let G be a Godel sentence for T. Then T u TS t G. Proof: By the basic property of TS (Tarski 1936, Theorem 5), T u TS t True(T). Thus, T v TS t Prov('cpl) -+ Tr('cpl), for all cp E &. And, T t G t,iProv('G1). Thus, T u TS t Prov('G1) + Tr('G1)). By "disquotation", T v TS t prov('G1) -+ G. Thus, T u TS t 1 G + G. Thus, by simple logic, T v TS t G. We can certainly "recognize" that a Godel sentence G for T is true (on the assumption that Titself is true), but our knowledge of its truth does not obtain from correct formal derivations within the theory T to which it applies. For example, one way of recognizing the commutativity of addition for natural numbers (i.e. the truth of the formula VxVy(x + y = y + x)) is to assume that each axiom of PA is true and to derive 'dx'dy(x + y = y + x)), using the induction scheme, from these axioms of PA. But this does not work for G(PA). For G(PA), although true, is not a consequence ofPA (if PA is consistent, etc.). How then do we "recognize the truth" of G? According to an argument associated with Lucas (1961) and, more recently, Roger Penrose (1989), this recognition involves some kind of non-computational "insight" (see Penrose 1989, Ch. 4). Although I (like them) am inclined to disagree with the computational theory of mind, I think they are wrong on this matter, for: G is deducible from the strengthened theory: namely, Tplus the standard Tarskian theory oftruth for the language of T. We can give a more informal and perhaps more instructive proof of Theorem 7 as follows. We have the Fixed Point Theorem: (FPT) T implies that G is true if and only if G is not provable in T, plus the generalized "Equivalence Principle": (EP) T + TS implies that T is true. Then we proceed as follows:
(1) T + TS implies that, for any cp, if cp is provable in T then cp is true [EPI (2) T + TS implies that if G is provable in T then G is true [I] (3) T + TS implies that if G is not true then G is true (4) T + TS implies that G is true (5) T + TS implies G
[2, FpT]
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In any case, a deflationary theory of truth cannot achieve such "insight" (i.e. deduction). It is conservative, so T u "deflationary theory" does not imply G. Indeed, the generalized equivalence principle EP fails for 07: If I am right, our ability to recognize the truth of Godel sentences involves a theory of truth (Tarski's) which signijicantly transcends the dejationary theories. To summarize, an adequate theory of truth looks as if it must be nonconservative. Indeed, it is bound to be non-conservative if it satisfies the generalized "equivalence principle" above. Tarski's theory does the job nicely. But the deflationary theories are conservative. So they are inadequate.
7. Tarski 1936 revisited Some of the technical material presented above appeared, in slightly different clothing, in Tarski's classic 1936 essay, "The Concept of Truth in Formalized Languages". 7.1 The consetvativeness of DT Tarski (1936) proves the following theorem: THEOREM 111: if the class of all provable sentences of the metatheory is consistent and if we add to the metatheory the symbol "Tr" as a new primitive sign, and all the theorems that are described in conditions (a)and (P) of the Convention T as new axioms [i.e. all the "T-sentences"], then the class of all provable sentences in the metatheory is consistent. (Tarski 1936, p. 256) This theorem is certainly implied by the conservativeness of DT (our Theorem 1, above). If T u DT is a conservative extension of a consistent theory T, then T u DT must also be consistent. Actually, Theorem I11 implies conservativeness also. Theorem I11 says that, for any consistent theory T in L, T u DT is consistent. So, if T u DT is inconsistent, then so is 7: Suppose that T u DT t cp, where cp is an L-sentence. Then, T u { i c p } u D T is inconsistent. By Theorem 111, T u { i c p } must be inconsistent. Thus, T 1 cp.
7.2 The "w-incompleteness" of DT
Shortly after Tarski's introduction and proof sketch of Theorem 111, we read: The value of the result is considerably diminished by the fact that the axioms mentioned in Theorem I11 [i.e. the axioms of DT] have a very restricted deductive power. A theory of truth founded on
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them would be a highly incomplete system, which would lack the most important and most fruitful theorems. (Tarski 1936, p. 25'7, emphasis added.) To illustrate this, Tarski discusses the formula iTr(x) v iTr(neg(x)). He first points out (in effect) that DT proves iTr(
7.3 D T does not implicitly dejne Tr Again, this idea appeared originally in Tarski's paper: Thus it seems natural to require that the axioms of the theory of truth, together with the original axioms of the metatheory, should constitute a categorical system. It can be shown that this postulate coincides in the present case with another postulate, according to which the axiom system of the theory of truth should unambiguously determine the extension of the symbol "Tr " which occurs in it, and in the following sense: if we introduce into the metatheory, alongside this symbol, another primitive sign, e.g., the symbol "Tr"', and set up analogous axioms for it, then the statement "Tv = Tr"' must be provable. But this postulate cannot be satisfied. For it is not difficult to prove that in the contrary case the concept of truth could be defined exclusively by means of tenns belonging to the morphology of the language, which would be in palpable contradiction with Theorem I [Tarski's Indefinability Theorem]. (Tarski 1936, p. 258, emphasis added.) It is interesting that Tarski's proof, which he does not give explicitly, involves a similar argument to the Beth Definability Theorem, which was not in fact proved until later (Beth 1953). 7.4 The non-conservativeness of the full Tarskian theory of truth
Finally, Tarski discusses what amounts to the non-conservativeness result, in particular, the provability of (undecidable in T ) Godel sentences in the overall truth-theoretic metatheory T u TS. He writes: The definition of truth allows the consistency of a deductive science to be proved on the basis of a metatheory which is of higher order than the theory itself. On the other hand, it follows from Godel's investigations that it is in general impossible to prove the consistency of a theory if the proof is sought on the basis of a metatheory of equal or lower order. Moreover, Godel has given a method for constructing sentences whick-assuming the theory concerned to be consistent--cannot be decided in any direction in this theory. All sentences constructed according to Godel's methodpossess the property that it can be established whether they are true or false on the basis of the metatheory of higher order
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having a correct definition of truth. Consequently, it is possible to reach a decision regarding these sentences, i.e. they can be either proved or disproved. (Tarski 1936, p. 274, emphasis added.) Tarski discusses Godel's method of obtaining a sentence G "which satisfies the following condition: G is not provable if and only ifp, where 'p' represents the whole sentence G". Tarski then goes on to show that this sentence G is "actually undecidable and at the same time true". He concludes: By establishing the truth of the sentence G we have eo ips0 . . . also proved G itself in the metatheory. .. . [Tlhe sentence G which is undecidable in the original theory becomes a decidable sentence in the enriched theory. (Tarski 1936, p. 276) To be brief, most of the technical details in $ $ 2 - 6 of this paper amount to little more than a restatement, in the modern context of deflationism, of Tarski's own discoveries in his 1936 essay.
8. Conclusion: de$ating de$atiorzism It seems to me that if the result of adding "higher-level" axioms to some "base theory" T yields new theorems expressible in the language of the base theory T but not derivable in T, and we have reasons for thinking that these extra theorems are themselves true, then these axioms could not be considered redundant (equivalently, the augmented theory T + new axioms could not be considered dispensable in favour of 7). The non-conservativeness results in $3 and $6 show that adding the axioms of the full Tarskian theory of satisfaction (for L) to a theory Tin L, need not yield a conservative extension. And we have reasons for thinking that the extra theorems are themselves true. Part of the basic (not necessarily deflationist) idea about truth is that a particular statement cp and its "truth" Tr(
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T u "deflationary truth theory" does not imply True(T). Moreover, the "recognizability" of Godel sentences further emphasizes these points. Indeed, the ability to "see" that G(T) and Con(T) are in fact true is a fundamental element of understanding the significance Godel's Incompleteness Theorems: we can "see" that G(7) and Con(7) are true, even though the consistent axiomatic theory T itself cannot prove them. There are truths that cannot be proved. The notions of truth and proof come apart:
. . . perhaps the most significant consequence of [Godel's First Incompleteness Theorem] is what it says about the notions of truth (in the standard interpretation of the language of arithmetic) and theoremhood, or provability (in any particular formal theory): that they are in no sense the same. (Boolos and Jeffrey 1989, p. 180) We have seen that our results about the logical properties of Tarski's theory of truth help explain this phenomenon. Suppose we accept a standard axiomatization of arithmetic (PA, say). It seems correct to say that we also (implicitly) accept its truth, and thus we surely then think that it is consistent. Tarski's Indefinability Theorem tells us that we cannot define a truth predicate for L in the language L of PA, but we can (using Godelian techniques) express the consistency of PA in L. So, we have accepted PA, we think it's true, and we seem to be committed to thinking it consistent. But the consistency of PA is not deducible from the axioms of PA, by Godel's Second Incompleteness Theorem. Nevertheless, the consistency of PA is a true statement, ifPA is consistent. How do we "know it"? What we have shown is that by adding a strong enough theory of truth (the theory of satisfaction for the language of arithmetic), we can deduce the truth of PA (i.e. True(PA)) and hence the consistency of PA (i.e. Con(PA)) from this truth-theoretic strengthening ofPA. However, we have also shown that the deflationary theories of truth are powerless to achieve this deduction, for they are conservative (anything derivable with them is derivable without them). Let us gather together the main results: (1) (2) (3) (4)
The deflationary theories DT, MT and SDT are conservative; D T and MT are "a-incomplete"; Neither DT nor MT implicitly defines "true"; The standard Tarskian theory of truthlsatisfaction TS is non-conservative;
(5) T u TS t True(T); ( 6 ) But T u "deflationary truth theory" does not imply True(7); (7) T u TS implies Godel sentences: T u TS t Con(7); T u TS t G.
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To summarize the argument, if "deflationism about truth" is construed as the following claims,
Deflationism About Truth: (a) Deflationary theories of truth constitute "all there is to truth" (b) Anything explained truth-theoretically could be explained nontruth-theoretically (i.e. adding a theory of truth adds "no new content" to a non-truth theoretic base theory) then deflationism is false. Formalized theories based on the deflationary conception of truth are incomplete with respect to our prior grasp of the "truth about truth". If I am right, there is more to truth than is expressed by the deflationary truth theories. In the introduction, I hinted at a close analogy between the indispensability of mathematics and the indispensability of a substantial (Tarskian) t h e o y of truth. Field's deflationary programme aims to show that mathematical theories (like standard set theory, ZFC) are convenient fictions: ultimately redundant and (in principle) dispensable from any scientific applications. This programme founders on the non-conservativeness of adding mathematics to "mathematics-free" nominalistic theories, like the theory of the gravitational field in Euclidean spacetime presented by Field (1980). I have argued that Tarskian truth theory is in some way analogous. I would like to conclude by suggesting that this analogy between the indispensability of (Tarskian) theories of truth and the indispensability of mathematical theories deserves more intensive investigation. In the meantime, no-one will drive us out of Tarski's truth-theoreticparadi~e!'~ Dept. of Philosophy, Logic & Scient$c Method JEFFREY KETLAND London School of Economics & Political Science Houghton Street London, WC2A 2AE UK j,j. ketlandalse.ac. uk
REFERENCES Ayer, A J. 1936: Language, Truth and Logic. Second edition. London: Pelican Books, 1946. Beth, E. W. 1953: "On Padoa's Method in the Theory of Definition". Indagationes Mathematicae 15, pp. 330-9. Boolos, G. and Jeffrey, R. C. 1989: Computability and Logic. Third edition. Cambridge: Cambridge University Press. l9 I would like to thank John Burgess, David Miller, Colin Howson, Stewart Shapiro and Richard Kaye for several helpful discussions. This work was supported by the British Academy.
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Burgess, J. P. and Rosen, G. 1997: A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Clarendon Press. David, Marian 1994: Correspondence and Disquotation. Oxford: Oxford University Press. Field, H. H. 1980: Science Without Numbers. Princeton: Princeton University Press. -1982: "Realism and Anti-Realism About Mathematics", in Field 1989, pp. 53-78 . Originally published in 1982 in Philosophical Topics 13. -1986: "The Deflationary Conception of Truth", in G. Macdonald and C. Wright (eds.), Fact, Science and Value: Essays on A.J. Ayer 's Language, Truth and Logic. Oxford: Blackwell, 1986, pp. 55-1 17. -1988: "Realism, Mathematics and Modality", in Field 1989, pp. 227-8 1. Originally published in 1988 in Philosophical Topics 19. -1989: Realism, Mathematics and Modality. Oxford: Blackwell. Frege, Gottlob 1892: "On Sense and Meaning", in Geach and Black 1980. Originally published in 1892 as " ~ b e rSinn und Bedeutung", in Zeitsckrift fur Pkilosophie undpkilosophische Kritik 100. Geach, P. and Black, M. (eds.) 1980: Philosophical Writings of Gottlob Frege. New Jersey: Barnes and Noble Books. Haack, Susan 1978: Philosophy of Logics. Cambridge: Cambridge University Press. Hellman, Geoffrey 1989: Mathematics Without Numbers. Oxford: Clarendon Press. Honvich, Paul 1990: Truth. Oxford: Blackwell. Kirkham, Richard 1992: Theories of Truth. Cambridge, MA.: Bradford Books, M.I.T. Press. Kripke, Saul 1975. "Outline of a Theory of Truth". Journal ofPhilosophy 72, pp. 69C716. Reprinted in Robert L. Martin (ed.) 1984, Recent Essays on Truth and the Liar Paradox. Oxford: Oxford University Press. Leeds, S. 1978: "Theories of Reference and Truth". Erkenntnis 13, pp. 111-29. Lucas, J. R. 1961: "Minds, Machines and Godel". Philosophy 36, pp. 120-4. Padoa, A. 1903: "Le Probleme No. 2 de M. David Hilbert", L 'Enseignement Math. V , pp. 85-91. Penrose, Roger 1989: The Emperor's New Mind. London: Vintage Books, 1990. Putnam, Hilary 1971: Philosophy oflogic. New York: Harper. Reprinted in H. Putnam 1979, Mathematics, Matter and Method. Philosophical
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Papers. Volume 1. Second edition. Cambridge: Cambridge University Press. -198 1: Reason, Truth and History. Cambridge: Cambridge University Press. Quine, W. V. 1948: "On What There Is", in Quine 1980, pp. 1-19 . Originally published in Review ofMetaphysics 2. -1953: "Notes on the Theory of Reference", in Quine 1980, pp. 13&8 Originally published in 1953 in the first edition of Quine 1980. Word and Object. Cambridge, M A . : M.I.T. Press. -1960: Philosophy of Logic. Cambridge, MA.: Harvard University -1970: Press. Second edition, 1986. 1980: From a Logical Point of View: Nine Logico-Philosophical Essays. Second edition, revised. Cambridge, MA.: Harvard University Press. From Stimulus to Science. Cambridge, MA.: Harvard Univer-1995. sity Press. Ramsey, F. P. 1927: "Facts and Propositions". Proceedings of the Aristotelian Society, suppl. vol., 7, pp. 153-71. Reprinted in G. Pitcher (ed.) 1962: Truth. Englewood Cliffs, NJ: Prentice Hall. Shapiro, Stewart 1983: "Conservativeness and Incompleteness". Journal of Philosophy, 80, pp. 521-31. Reprinted in W. D. Hart (ed.) 1996: Philosophy of Mathematics. Oxford: Oxford University Press. Spinoza, B. 1677: On the Improvement of the Understanding, in B . Spinoza 1955, On the Improvement of the Understanding. The Ethics. Correspondence (translation by R. H. M. Elwes), New York: Dover Publications Inc. Originally published posthumously in 1677 as Tractutus de Intellectus Emendatione in his Opera Postuma. Tarski, Alfred 1936: "The Concept of Truth in Formalized Languages", in J. H. Woodger (trans., ed.), Logic, Semantics, Metamathematics: Papers by Alfred Tarski from 1922-1938. Oxford: Clarendon Press, 1956, pp. 152-278. Originally published in 1936 as "Der Wahrheitsbegriff in den formalisierten Sprachen" in Studia Philosophica I.
