Validity as a primitive

Analysis 72 (3):421-430 (2012)
A number of recent works consider treating validity as a primitive notion rather than one defined in some standard manner. There seem to have been three motivations. First, to understand how truth and validity interact in potentially paradoxical settings. Second, to argue that validity is in fact afflicted with paradoxes analogous to the semantic paradoxes. Third, to develop a ‘deflationary’ conception of validity or consequence. This article treats the notion of validity as a primitive notion and shows how to provide a consistent theory of classical validity (and self-applicative truth), conservative over Peano arithmetic
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Lionel Shapiro (2013). Validity Curry Strengthened. Thought: A Journal of Philosophy 2 (1):100-107.
Julien Murzi (2012). On Heck's New Liar. Thought: A Journal of Philosophy 1 (2):258-269.

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Christopher Gauker (1990). Semantics Without Reference. Notre Dame Journal of Formal Logic 31 (3):437-461.
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