Saving the truth schema from paradox

Journal of Philosophical Logic 31 (1):1-27 (2002)
The paper shows how we can add a truth predicate to arithmetic (or formalized syntactic theory), and keep the usual truth schema Tr( ) ↔ A (understood as the conjunction of Tr( ) → A and A → Tr( )). 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
Keywords conditionals  law of excluded middle  paradoxes  truth
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DOI 10.1023/A:1015063620612
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Douglas Patterson (2009). Inconsistency Theories of Semantic Paradox. Philosophy and Phenomenological Research 79 (2):387 - 422.
Tore Fjetland Øgaard (2016). Paths to Triviality. Journal of Philosophical Logic 45 (3):237-276.

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