# Saving the Truth Schema from Paradox

## Authors

DOI: 10.1023/A:1015063620612

- Cite this article as:
- Field, H. Journal of Philosophical Logic (2002) 31: 1. doi:10.1023/A:1015063620612

## 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 *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 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.