Field’s logic of truth

Abstract

Saving Truth from Paradox is a re-exciting development. The 70s and 80s were a time of excitement among people working on the semantic paradoxes. There were continual formal developments, with the constant hope that these results would yield deep insights. The enthusiasm wore off, however, as people became more cognizant of the disparity between what they had accomplished, impressive as it was, and what they had hoped to accomplish. They moved onto other problems that they hoped would prove more yielding. That, at least, was how it seemed to me, so I was delighted to see a dramatically new formal development that is likely to rekindle our enthusiasm. Field didn't build from scratch, of course. Since Thaies, no one has. A construction upon which he particularly relied was given by Saul Kripke (1975), who applied the methods of first-order positive inductive definitions1 to produce fixed points for a language with truth-value gaps, evaluations in which a sentence2 (p always receives the same semantic status - true, false,3 or unsettled - as the sentence TOV1). Truth-value gaps were handled by the strong 3-valued semantics of Kleene (1952b, §54), so that a disjunction counts as true iff one or both disjuncts are true and as false iff both disjuncts are false, and similarly for the other connectives.