# A Liar Paradox. Richard G. Heck, Jr. Brown University

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1 A Liar Paradox Richard G. Heck, Jr. Brown University It is widely supposed nowadays that, whatever the right theory of truth may be, it needs to satisfy a principle sometimes known as transparency : Any sentence S must be replacable, salva veritate, with S is true, and conversely. 1 But, whatever the merits of this principle, it is known to be incompatible with others we might also have wanted to affirm, for example, the law of excluded middle. For suppose there is a sentence Λ for which we have: (1) Λ T( Λ ) Then we can argue as follows. Suppose Λ. By transparency, T( Λ ); so by (1), Λ. If Λ Λ, however, then we have Λ either way, so Λ. By (1) again, T( Λ ); hence Λ, by transparency. So Λ Λ, a contradiction. It is not obvious, however, that this argument cannot be resisted. We might try rejecting the use of proof by cases 2 or look carefully at how we are allowing ourselves to reason with the biconditional (1). But there is a better form of the argument. It begins, not with a sentence like Λ, but with a term λ for which we have: (2) λ = T(λ) Now we can argue as follows: T(λ) T(λ) T( T(λ) ) T(λ) Premise Published in Thought 1 (2012), pp The theory of truth in Saul Kripke s Outline of a Theory of Truth (Kripke, 1975) was perhaps the first to satisfy this condition, which plays a crucial role in the more recent investigations of Hartry Field (2008). 2 This could be replaced by reductio we got Λ out of Λ, so we have Λ but then we don t need excluded middle. 1

2 2 T(λ) T(λ) T(λ) T( T(λ) ) T(λ) p p p T(λ) T(λ) + This argument uses very meagre logical resources. We are using Leibniz s Law, in a form allowing for the substitution of identicals; we are using the inference p p p; and we are using conjunction introduction. 3 We are not assuming anything about negation, other than that p p is contradictory. Of course, the argument depends crucially upon the existence of a term like λ. In the usual sort of setting, that is to say, where truththeories are developed as extensions of arithmetical theories, the argument depends upon the availability of what is sometimes called the strong form of the diagonal lemma. The strong form tells us that, for any formula A(x), there is a term g A such that we can prove: g A = A(g A ) and not just that there is a formula G A for which we can prove: G A A( G A ) But, as I have argued elsewhere (Heck, 2007), the strong form, though less well-known, is what we need if we want to capture the structure of the informal reasoning that leads to the Liar paradox. One typically begins with the assumption that there is a self-referential sentence, the Liar, that says of itself that it is not true. The weaker form of the diagonal lemma does not give us such a sentence. It only gives us a formula Λ that is provably equivalent to a sentence that says of Λ that it is not true. Neither Λ nor T( Λ ) refers to itself, and neither says of itself that it is not true. The strong form, on the other hand, does deliver a truly self-referential liar sentence. Since λ = T(λ), T(λ) is 3 We need conjunction introducion only to conclude that (1) and (2) imply a single sentence that is contradictory. If we say that a set of formulae is inconsistent if it implies both some sentence and its negation, we do not need it.

4 4 Whether gaining transparency is worth abandoning the law of noncontradiction is not an issue I can hope to resolve here. 6 But, for what it s worth, I don t myself regard transparency as non-negotiable it is of a piece with a sort of deflationism I find problematic (Heck, 2004) and some theories of truth reject it. For example, it follows from the foregoing that supervaluational accounts of truth, such as the one defended by Vann McGee (1990), must always be incompatible with transparency. As I said, however, maybe giving up transparency is worth saving the law of non-contradiction. Still, there are closely related principles about truth that such views must reject in fact, that any decent theory of truth must reject. Consider the following schemata: (3) (4) (S T( S )) ( S T( S )) These obviously follow from non-contradiction, given transparency, but they seem intuitively compelling to me, even absent transparency. It cannot be both that snow is white and that snow is not white is true; it cannot be both that grass is not green and that grass is not green is not true. Now consider the instance of (3) where S is replaced by T(λ): (5) (T(λ) T( T(λ) ))) By identity, (T(λ) T(λ)), so T(λ). So (5) implies T(λ). Taking S to be T(λ) in (4) gives us: (6) ( T(λ) T( T(λ) )) By identity, ( T(λ) T(λ)), so T(λ). Thus, (6) entails T(λ). So (5) and (6) together entail a contradiction: T(λ) T(λ). This argument depends only upon Leibniz s Law, conjunction introduction, and the appeal to the DeMorgan laws and double negation elimination to derive Λ Λ. But it seems clear we will need a lot more than we did with excluded middle. 6 It would nowadays be a common move to insist that we should not assert that the Liar is not both true and false, but only reject the claim that it is both true and false. But then we have also to reject the claim that the Liar is not both true and false, since that claim leads to paradox. This is not in itself contradictory, but it does point to the fact that rejection is a very weak attitude (Shapiro, 2004), if it is intelligible at all.

6 6 References Field, H. (2008). Saving Truth From Paradox. Oxford University Press. Friedman, H. and Sheard, M. (1987). An axiomatic approach to selfreferential truth, Annals of Pure and Applied Logic 33: (1988). The disjunction and existence properties for axiomatic systems of truth, Annals of Pure and Applied Logic 40: Heck, R. G. (2004). Truth and disquotation, Synthese 142: (2007). Self-reference and the languages of arithmetic, Philosophia Mathematica 15: Kripke, S. (1975). Outline of a theory of truth, Journal of Philosophy 72: McGee, V. (1990). Truth, Vagueness, and Paradox: An Essay on the Logic of Truth, Indianapolis, ed. Hackett. Shapiro, S. (2004). Simple truth, contradiction, and consistency, in G. Priest, et al. (eds.), The Law of Non-Contradiction. Oxford, Clarendon Press, the way the paper was organize.

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