, Volume 31, Issue 1, pp 1-27

Saving the Truth Schema from Paradox

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Abstract

The paper shows how we can add a truth predicate to arithmetic (or formalized syntactic theory), and keep the usual truth schema Tr(〈A〉)↔A (understood as the conjunction of Tr(〈A〉)→A and ATr(〈A〉)). We also keep the full intersubstitutivity of Tr(〈A〉)) with A in all contexts, even inside of an →. Keeping these things requires a weakening of classical logic; I suggest a logic based on the strong Kleene truth tables, but with → as an additional connective, and where the effect of classical logic is preserved in the arithmetic or formal syntax itself. Section 1 is an introduction to the problem and some of the difficulties that must be faced, in particular as to the logic of the →; Section 2 gives a construction of an arithmetically standard model of a truth theory; Section 3 investigates the logical laws that result from this; and Section 4 provides some philosophical commentary.