Abstract
Deflationists argue that ‘true’ is merely a logico-linguistic device for expressing blind ascriptions and infinite generalisations. For this reason, some authors have argued that deflationary truth must be conservative, i.e. that a deflationary theory of truth for a theory S (that interprets a sufficient amount of mathematics, or syntax) must not entail sentences in S’s language that are not already entailed by S. However, it has been forcefully argued that any adequate theory of truth for S must be non-conservative and that, for this reason, truth cannot be deflationary (Shapiro in J Philos XCVI(10):493–521, 1998; Ketland in Mind 108(429):69–94, 1999). We consider two defences of conservative deflationism, respectively proposed by Waxman (Mind 126(502):429–463, 2017) and Tennant (Mind 111(443):551–582, 2002), and argue that they are both unsuccessful. In Waxman’s hands, deflationists are committed either to a non-purely expressive notion of truth, or to a conception of mathematics that does not allow them to justifiably exclude non-conservative theories of truth. Tennant’s conservative deflationism fares no better: if deflationist truth must be conservative over arithmetic, it can be shown to collapse into a non-conservative variety of deflationism.
Highlights
Technical preliminariesFollowing Waxman, we define the following two notions of conservativeness of a theory Sþ with language LSþ over a theory S with language LS, where LS LSþ : Definition 1 (Syntactic conservativeness) Sþ is a syntactic conservative extension of S if, for every u 2 LS, if Sþ ‘ u, S ‘ u
Deflationists argue that ‘true’ is merely a logico-linguistic device for expressing blind ascriptions and infinite generalisations
As for Tennant, he suggests that standard pieces of mathematical knowledge such as GPA and ConðPAÞ can be recaptured via prooftheoretic means, without resorting to truth-theoretic resources
Summary
Following Waxman, we define the following two notions of conservativeness of a theory Sþ with language LSþ over a theory S with language LS, where LS LSþ : Definition 1 (Syntactic conservativeness) Sþ is a syntactic conservative extension of S if, for every u 2 LS, if Sþ ‘ u, S ‘ u. Still following Waxman, we let (first-order) PA be our base theory, i.e. the theory to which we add the principles governing the truth predicate, and LPA be its language. We can distinguish between two versions of the compositional theory: PATr, which is given by PA together with the compositional axioms for Tr; and PAþTr, which is given by adding to PATr all the instances of the Induction Schema that in which the truth predicate occurs.. We can distinguish between two versions of the compositional theory: PATr, which is given by PA together with the compositional axioms for Tr; and PAþTr, which is given by adding to PATr all the instances of the Induction Schema that in which the truth predicate occurs.8 It is well-known that that while PATr is syntactically conservative over PA, PAþTr isn’t. Deflationary truth can be both conservative and adequate
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