Deflationism and the Gödel Phenomena: Reply to Cieśliński

We clarify how the requirement of conservative extension features in the thinking of various deflationists, and how this relates to another litmus claim, that the truth-predicate stands for a real, substantial property. We discuss how the deflationist can accommodate the result, to which Cieslinski draws attention, that non-conservativeness attends even the generalization that all logical theorems in the language of arithmetic are true. Finally we provide a four-fold categorization of various forms of deflationism, by reference to the two claims of conservativeness and substantiality. This helps to clarify the various possible positions in the deflationism debate. In offering the following considerations on behalf of deflationism, the present author is continuing to act as an amicus curiae, rather than as a contending party.1 For he is a substantialist about truth, believing that the truth of a sentence consists in the existence of a truth-maker (and that falsity ∗For helpful comments on an earlier draft, the author is grateful to Salvatore Florio, Robert Kraut, Patrick Reeder, Kevin Scharp and Stewart Shapiro. Paul Horwich was kind enough to clarify aspect of his position in correspondence. Remaining defects are the author’s sole responsibility. For the original brief, see Tennant 2002. consists in the existence of a falsity-maker). Those philosophical convictions, however, do not turn on the issue of whether a deflationary construal of the truth-predicate is possible in light of the Gödel phenomena. 1 The assumption of conservative extension For the reader not familiar with the debate at hand, it is worth emphasizing an assumption common to both parties: that it is criterial, for a theory of truth to be deflationist, that it be conservative over one’s non-semantic theorizing. That is, no unprovable assertion or underivable inference in the language L of one’s non-semantical theorizing becomes provable or derivable, respectively, upon the adjunction of one’s theory of truth for L. This assumption of conservative extension (on behalf of deflationism) made its first explicit appearance in Horsten 1995, which examined Paul Horwich’s minimalist theory of truth (Horwich 1990), ‘widely regarded as the deflationist theory that has been developed in most detail and that has been given the clearest expression to date’ (p. 175). Horsten writes at p. 183: The minimalist theory entails that a truth predicate should be conservative over a given theory that is stated without the truth predicate (or any other semantical notions). He adds immediately (fn. 15): Maybe I am wrong here. But if I am, then I do not see what the neutrality of the notion of truth according to deflationism amounts to. In any case, this is an issue which I think should be addressed by defenders of deflationist theories of truth. Later, in discussing Feferman’s axiomatic system KF (which seeks to govern the behaviour of a truth predicate added to the language of Peano arithmetic), Horsten writes (p. 184): . . . there are purely arithmetical sentences that are provable in KF, but not in Peano arithmetic. The notion of truth inflates the mathematical theory to which it is applied, and this seems to accord ill with any version of the deflationary theory of truth. Shapiro 1998 appears to share the view that the deflationist ought to be making the assumption of conservative extension (p. 497):