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.
Keywords: Categoricity; Conservativeness; Deflationism; Isaacson's thesis; Reflection principles; Truth.
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