Deflationism and Tarski's Paradise Jeffrey Ketland Mind, New Series, Vol. 108, No. 429. (Jan., 1999), pp. 69-94. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28199901%292%3A108%3A429%3C69%3ADATP%3E2.0.CO%3B2-%23 Mind is currently published by Oxford University Press.
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Deflationism and Tarski 3 Paradise JEFFREY KETLAND
Deflationism about truth is a pot-pourri, variously claiming that truth is redundant, or is constituted by the totality of "T-sentences", or is a purely logical device (required solely for disquotational purposes or for re-expressing finitarily infinite conjunctions andlor disjunctions). In 1980, Hartry Field proposed what might be called a "deflationary theory of mathematics", in which it is alleged that all uses of mathematics within science are dispensable. Field's criterion for the dispensability of mathematics turns on a property of theories, called conservativeness. I present some technical results, some of which may be found in Tarski (1936), concerning the logical properties of truth theories; in particular, concerning the conservativeness of adding a truth theory for an object level language to any theory expressed in it. It transpires that various deflationary truth theories behave somewhat differently from the standard Tarskian truth theory. These results suggest that Tarskian theories of truth are not redundant or dispensable. Finally, I hint at an analogy between the behaviour of mathematical theories and of standard (Tarskian) theories of truth with respect to their indispensability to, as Quine would put, our "scientific world-view".
I . Defiationism about truth and mathematics 1.1 Deflationism about truth
According to the doctrine known as deflationism about truth, the concept of truth (along perhaps with satisfaction and other alethic concepts) is redundant and dispensable: the idea perhaps is that we need to "deflate" the correspondence notion that truth expresses a substantial or theoretically significant language-world relation. Associated with the deflationary conception are two schemes, Redundancy Scheme (RS) The proposition thatp is true if and only i f p Disquotation Schemes (DS,)The sentence "p7'is true if and only i f p Mind, Val. 108 . 4 2 9 . January 1999
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(DS,) The predicate "F" is true of x,, . . . ,x, if and only if Fx, .. . 4 1 ,
An early deflationist hint can be found in Frege (1892):
One can indeed say "The thought that 5 is a prime number is true". But close examination shows that nothing more has been said than in the simple sentence "5 is a prime number". (Frege 1892, p. 34, emphasis added.) The redundancy theory of truth was advocated by Ramsey (1927), and later by Ayer (1936). A disquotational version of deflationism about truth is sometimes associated with Quine (1960, p. 24; 1970, pp. 1&3)' and Leeds (1978). The disquotational account of truth is further discussed by Field (1986, 1988) and in several recent books2 The minimalist theovy of truth defended by Paul Horwich (1990) is closely related to Ramsey's redundancy version of deflationism. Honvich claims that the totality of instances of the redundancy scheme RS yields something like the "whole truth about truth". For example, Horwich writes: ... our thesis is that it is possible to explain all the facts involving truth on the basis of the minimal theory. (Horwich 1990, p. 7) . . . the minimalist conception: i.e., the thesis that our theory of truth should contain nothing more than instances of the equivalence scheme. (Honvich 1990, p. 8) Two themes within deflationism about truth concern the logical function of the truth predicate. The idea is that the truth predicate provides a logical device, for a given object language L, of:
(i) "Disquoting" quotations ofL-expressions (and "dispropositionalizing" that-constructions) (ii) Finitely expressing certain infinitely long conjunctions and disjunctions (which would otherwise require a logical device of substitutional quantification). For the "disquotation effect" (i) we see that we are allowed to eliminate the truth predicate, by inferring from a truth-predication containing a quotation, (1) "snow is white" is true,
' Perhaps it is misleading to describe Quine as a deflationist, for he stresses that
in order to define truth, the Tarskian detour via satisfaction is necessary (Quine 1970, p. 13). Furthermore, he stresses the semantic paradoxes and Tarski's Indefinability Theorem, concluding that " [ ' x satisfies y'] is not a sentence of the object language . . . it is untranslatable foreign language" (Quine 1970, p. 45). E.g. David (1994) and Kirkham (1992). David introduces his book with the nice quip: "What is truth?" asked Pilate. "Truth is disquotation", replied Quine. (David 1994, p. 3)
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the quoted sentence itself, (2) snow is white.
As for the expression of infinitary conjunctions/disjunctions (ii), we are allowed to eliminate the truth predicate generally, by inferring from any open sentence,
(3) x is true, the "infinite disjunction",
(4) (x = "snow is white" and snow is white) or (x = "grass is green" and grass is green) or . . . . We shall represent both of these logical features in certain formalized deflationary theories of truth, which we shall discuss below in $2. 1.2Deflationism about mathematics According to one version of deflationism about mathematics, mathematical theories (such as arithmetic, analysis and set theory) are, despite redundant and dispensable appearances and arguments to the ~ontrary,~ from our scientific theorizing about the physical world: the idea perhaps is that we need to "deflate" the platonist notion that there is a realm of abstract mathematicalia-numbers, functions, sets, symplectic manifolds, Hilbert spaces, monstrous groups, fibre bundles, and so om---for mathematical (and physical) theories to talk about. There are no neat analogues of the redundancy or disquotation schemes associated with deflationism about mathematics. Perhaps the best worked-out deflationist approach to mathematics is Hartry Field's fictionalism, first set out in his Science Without Numbers (1980), which is a brilliant attempt to respond to the challenge laid down by Putnam (1971, Sc. V) to translate Newtonian gravitational physics into "nominalistic language". A central aspect of Field's strategy concerns a certain metalogical claim about the result of adding mathematical axioms M to a "mathematics-free" or "nominalistic" theory N. Briefly, Field claims that N u M is always a conservative extension of N (see Field 1980, appendix to Ch. 1). One way of formulating this is (A) The semantical conservativeness of mathematics 'The most famous argument to the contrary, namely that mathematics is integral and indispensable to our scientgc theorizing, was outlined by Quine (1948) and reinforced by Putnam (1971). It is known as the Quine-Putnam argument (for realisndplatonism). Quine's most recent summary is this: Science would be hopelessly crippled without abstract objects. We quantify over them. In the harder sciences, numbers and other abstract objects bid fair to steal the show. Mathematics subsists on them, and serious hard science without serious mathematics is hard to imagine. (Quine 1995, p. 40)
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Let N be a "mathematics-free" theory of the natural world, M be a standard mathematical theory4and cp be any mathematics-free assertion. Then, if cp is a logical consequence of N u M, then cp is a logical consequence of N. This metatheorem states that if N u M implies cp, then Nimplies cp (where cp is a nominalistic statement). Intuitively, this would mean that the addition of the mathematics M is "redundant" for deducing, or maybe even explaining, the phenomenon described by cp. But some of the problems with Field-style deflationism concern the fact that this result is slightly misleading. It can be argued that what should matter for the deflationist (with nominalistic inclinations) is that adding mathematical axioms is dispensable for theorem-proving. Indeed, as Field himself put it,
. . . any inference from nominalistic premises to a nominalistic conclusion that can be made with the help of mathematics could be made (usually more long-windedly) without it. (Field 1980, p. x) That is, anything proved from mathematics-free premises by assuming in addition, say, set-theoretical axioms, could have been proved without the surplus. Thus, the required metatheorem would be: (B) The deductive conservativeness of mathematics If cp is a theorem of N u M, then cp is a theorem of N. Unfortunately, this metatheorem is simply not true, as Stewart Shapiro (1983) pointed out in 1983. Ironically, the basic Godelian idea is mentioned by Field himself in the final chapter of Science Without Numbers (after suggestions made by John Burgess and Yiannis Moschovakis). To cut a long story short, there is an ambiguity involved in "adding" one theory M, in an extended language, to another N when the "base theory" N contains axiom schemes (or is second-order). The ambiguity is whether (or not) to include formulas containing the new vocabulary in forming instances of the axiom schemes in N. Let us say (following Burgess and Rosen 1997, p. 194) that if the new vocabulary is not admitted, then the axiom scheme is treated as a list; if the new vocabulary is admitted, then the axiom scheme is treated as a rule. For example, consider extensions of axiomatic first-order Peano Arithmetic PA, which contains the Induction Scheme, (the universal closure of): 4Field (1982. pp. 5 6 7 ) points out that there are "non-standard" mathematical theories (e.g. set theory with the negation of the Axiom of Infinity) which are actually inconsistent with certain "nominalistic" theories (namely those that hold only in infinite models).
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"List-like" instances of this scheme are obtained by substituting for 0 any formula q(x, y,, . . .,y,) from L[O, s, +, x], the first-order language of arithmetic. In contrast, "rule-like" instances of the scheme are obtained by also including formulas containing any extra notation (say, E ) from the extended language. In the general case of relevance to nominalism, we initially have a mathematics-free notation L which is extended to a mathematical notation L - by adding, say, the membership predicate E (and perhaps a predicate U(x), meaning "x is an (non-mathematical) ur-element", to distinguish mathematical from non-mathematical entities). Applying mathematics in this instance simply means adding axioms from set theory (say, ZF set theory with ur-elements). For example, if the theory N talks about concrete Fs, then N u M may talk about the abstract set of Fs (or may talk about real numbers assigned as masses, co-ordinates, etc., associated with Fs). Suppose the formulation of the nominalistic theory N requires axiom schemes. This does occur when one tries to formulate spacetime theory mereologically, in terms of regions and space-time points: one required scheme is a completeness scheme which asserts the existence of a region R (i.e, aggregate) of points satisfying any definable condition on points. The question then is: what happens if one includes occurrences of E in formulas instantiating these schemes in N? It transpires that, if the nominalistic theory Nis consistent, then adding standard set theory with ur-elements (call it M ) to N yields a conservative extension, but only ifany axiom schemes within N are treated as lists. If schemes are treated as rules, then adding set theory may yield a non-conservative extension, contradicting Field's claim (see Burgess and Rosen 1997, pp. 194-6). Shapiro (1983) illustrated this in the context of Field's allegedly nominalistic theory Nof the gravitational field in flat Euclidean spacetime, presented by Field (1980). In fact, PA may be interpreted withinlY so Ncan be "godelized" (indeed, the syntax ofNcan be "geometrized", just as the syntax of PA can be "arithmetized"). Then, as Shapiro detailed, a Godel sentence Con(N), "saying" that Nis consistent, is expressible in L but is not a theorem ofN. However, Con(N) becomes a theorem of the extended theory N u M (roughly, because in the set theory M, one can prove that N has a model and is thus consistent). Hence, if N were true (which it is not, for Newton was wrong), Con(N) would be a mathematics-free truth about the geometrical structure of the world, but mathematics is required to prove it. In short, adding mathematics can add to the deductive power of a nonmathematical nominalistic (axiomatic) physical theory. In this sense, the incorporation of mathematical axioms within a scientific theory may be indispensable for the purposes of deducing (and presumably explaining) nominalistically-expressible assertions.
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Hellman (1989) discusses the relation of these logical facts to the Quine-Putnam Indispensability Argument,
. . . one of the uses of mathematics is to prove theorems that may be statable lower down . . .. If higher mathematics is indispensable in this sense . . . then surely that is indispensability enough for purposes of the usual pro-platonist arguments. In no way is its force weakened by the quite different consideration of semantic conservativeness. (Hellman 1989, p. 139)
2. The conservativeness of deflations y theories of truth Let L be any standard formalized first-order language. L will be the "object level" or "base" language for which we shall examine some (also formalized) truth theories. (We shall use the symbol ",L" to refer to the set of L-formulas with n free variables.) I will concentrate on three deflationary theories of truth for L: (i) MT: the minimalist theory of truth for (propositions expressible in) L (ii) DT: the disquotational theory of truth for L (iii) SDT: the substitutional disquotation theory of truth for L. The minimalist theory MT (c.f. Honvich 1990) is formulated in a metalanguage L+,which is an extension of L (so, L c L') obtained by adding a monadic predicate Tr(x) and a term-forming operator which operates only on L-sentences to form L+-terms. If cp 6 & (i.e. cp is an L-sentence), then is an L+-term(we assume that is injective: if cp, and cp, are distinct L-expressions, then so are and ). Intuitively speaking, means "the proposition that . . .". Let Ramsey(cp) be the L+-sentenceTr() t,cp, where cp E &. The axioms of MT are then all these L+-sentences, Ramsey(cp). Thus, the axioms of MT are analogous to: (1) The proposition that snow is white is true if and only if snow is white5. 'For an example of an apparent rejection of the redundancy scheme, consider the following passage from Spinoza's On the Irnpi-overnent of the Understanding: . . . if anyone asserts, for instance, that Peter exists, without knowing whether Peter really exists or not, the assertion, as far as its asserter is concerned, is false, or not true, even though Peter actually does exist. The assertion that Peter exists is true only with regard to he who knows for certain that Peter does exist. (Spinoza 1677, p. 26, emphasis added) Seemingly, Spinoza didn't much improve the understanding of the concept of truth!
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The disquotational theory DT is formulated in a metalanguage L', which is an extension of L (again, L C L')obtained by adding a monadic predicate Tr(x) and a term-forming operator . So, if E is any L-expression, then <E> is an L+-term(again, we assume injectiveness: if and s2are distinct, then so are <E,> and ). Intuitively speaking, is the quota. ~ will say that an interpretation I+of L+ is quotationtion ~ p e r a t o rWe normal if, for any L-expression &, I+[] = &. (Such quotation-normal interpretations just invert quotation.) Let Tarski(cp) be the L+-sentenceTr(< cp>) t,cp, where cp E ,L. The axioms of D T are then all these L+-sentences,Tarski(cp). Thus, the axioms of DT are analogous to:
(2) The sentence "snow is white" is true if and only if snow is white. ' From a formal point of view, the truth theories MT and DT are identical, as I have hinted in the construction above.8 The substitutional disquotational theory SDT is formulated in a containing , the monadic predicate Tr(x), strengthened metalanguage L',,, 6Thequotation operator is not extensional. A context C(. . .) is extensional in an = then interpretation Ijust in case, for any expressions and E ~ if, I[C(E,)]= I[C(&,)]. Clearly, a sufficient condition for this is that there is a function fc such that, for any E,I[C(E)] =fc(l[&]). Let I+be an interpretation ofLt and let E, and E, be L-expressions. Even if s, and E, are co-extensive, that is, if I'[E,] = I'[E,], it certainly needn't follow that = I'[<E~>].Indeed, intuitively, if I' "respects quotation", then p [ < ~ > = ] E, SO if and c2 are distinct but co-extensive expressions, then # I+[]. 'This instance of the disquotation scheme is Tarski's famous example. Putnam calls the disquotation scheme the "equivalence principle". He insists further that, . . . the equivalence principle is philosophically neutral, and so is Tarski's work. On any theory of truth, "snow is white" is equivalent to "snow is white" is true. (Putnam 1981, p. 129) he construction described above is aimed to avoid the Liar Paradox. The important restriction is that the set D T must not contain formulas Tr() ++cp, where cp itself contains Tr. The reason, briefly, is that when a "base theory" T is introduced, it may be strong enough to satisfy the Diagonal Lemma (or Fixed Point Theorem). If so, then for any formula P(x), there will be a closed "fixed point" formula cp such that T t P() t,cp. Taking P(x) as iTr(x), this means that T b iTr() t,cp. Then, T u D T would be inconsistent. Some authors (e.g. David 1994, pp. 107-10) express concern that theories of truth like MT and DTare in$nitely axiomatized and thus not "finitely statable". Is this a problem? These authors have finitely defined the theory. (It is like saying we cannot consider the set of reals between 0 and 1, because we cannot finitely list them all.) Almost any interesting theory is not finitely (first-order) axiomatizable, e.g. first-order PA with the infinity of induction axioms, the first-order theory of order-complete fields (i.e. real analysis) with an infinity of completeness axioms and ZFC with an infinity of separation and replacement axioms. Indeed, take almost any interesting infinite mathematical structure A (e.g. (N, ) ++ ~ c pAlso, . D T t iTr() ++ l c p . Hence, DT t Tr() t,~Tr(), for any fosmula cp E &.
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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(), for each cp, but does not imply VxP(x). We may summarize the findings of the last three sub-sections thus: Theorem 3: D T is "a-incomplete". Proof: D T implies all the "instances" of NEG, but does not imply NEG. rn This is analogous to the axiom system known as Robinson Arithmetic Q. If n and m are any natural numbers, then, Q 1 g + rzz = 122 + 12. But Q does not prove the general law of commutativity for addition. That is, Q does not imply YxVj)(x + y = y + x). In brief, Q is good at finite arithmetical sums, but bad at algebra.I6
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,() t,cp (for each i), we see that DT, u DT2 t Tr,() t,Tr,(), for each cp E &. But it does not follow that Tr, and Tr, are coextensive in DTl u DT,, as an argument using the Beth Definability Theorem" shows.
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() t,cp. By conservativeness, we infer that, for each closed L-formula cp, T 1 Y() t,cp. This contradicts Tarski's Indefinability Theorem.
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() v iTr(neg()), for any L-sentence cp. He then points out that the universal closure of this formula, Vx(iTr(x) v iTr(neg(x))), which he calls the Law of Non-Contradiction, is not a theorem of DT. He discusses this along with a proposed "Rule of Infinite Induction", or an o-rule (Tarski 1936, pp. 25741).
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() are somehow "equivalent". I think this is correct (indexicals aside), and if a truth theory satisfies Convention Tthen it proves the equivalence. But we must go further. Any adequate theory of truth should be able to prove the "equivalence" of a (possibly infinitely axiomatized) theory T and its "truth" True(7) (that is, the metalanguage formula Vx(Prov(x) -+ Tr(x)). And Tarski's theory comes up trumps. When T is an axiomatic theory, it is possible to show (Tarski 1936, Theorem 5) that Tarski's theory TS satisfies the following: T u Tarski's theory of satisfaction 1 True(7). However, the preceding arguments indicate that the deflationary theories are too weak:
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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.