Saving the Truth Schema from Paradox

Hartry Field
2002 Journal of Philosophical Logic  
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 A → Tr ( 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
more » ... al 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.
doi:10.1023/a:1015063620612 fatcat:hwltsocl7zhrjkmiyx3a53cm5e