Conservatively extending classical logic with transparent truth

Abstract. This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth-involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system features admissible Cut, but the other does not.) §1. Introduction. Adding a truth predicate to a language governed by classical logic is not easy. It is particularly tricky when the truth predicate is intended to be transparent— such that T 〈A 〉 and A are everywhere intersubstitutable. The trouble, as is well-known, comes from such paradoxes as the liar; because of them, theories of truth typically either use a nontransparent truth predicate (Halbach, 2011; Maudlin, 2004) or a logic weaker tha