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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))[left right arrow]A (understood as the conjunction of Tr((A))[arrow right]A and A[arrow right]Tr((A))). We also keep the full intersubstitutivity of Tr((A))) with A in all contexts, even inside of an [arrow right]. Keeping these things requires a weakening of classical logic; I suggest a logic based on the strong Kleene truth tables, but with [arrow right] 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 [arrow right]; 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.