Hartry Field, Saving Truth from Paradox

Abstract

This is a difficult, challenging, thick book (approximately four hundred pages) consisting of five parts and 27 chapters. A substantial amount of it consists of the author's original contributions, but it also offers a detailed comparison with different competing approaches. The influential work in the area of formal semantics, at least the one closer to Field's own views (Tarski, Kripke, Friedman and Sheard, Priest) is thoroughly considered, but not in the spirit of a systematic neutral attitude: in the author's own words, "the book is a survey, but an opinionated one, of philosophical work on paradoxes, with occasional side steps into issues like vagueness, validity, incompleteness theorems". According to Field, there is a coherent concept of full truth after all, but in dealing with it one must allow for a generalization of classical logic. Indeed, the central result of the book is a new model for self-referential truth. In particular, the unrestricted truth schema T( 'A]) A (for A arbitrary) is maintained, and the object language includes its own metalanguage (Here [A] is a fixed effective Godei numbering of A; henceforth we simply write TA instead of the proper T( [A] )). More generally, the intersubstitutivity principle holds: if D results from Cby replacing (in transparent contexts) some occurrence of A in C with TA, then D implies C and conversely. Due to these features, the author's solution can be summarized as follows: (1) the application of classical logic in particular tertium non datur TND is restricted to conditions that do not involve the truth predicate or, more generally, predication, or instantiation; (2) a non-standard implication, which reduces to full classical reasoning in "ordinary contexts", including standard set theory, is defined by means of a non-trivial construction. Feature (1) corresponds to the so called paracomplete solutions; these are in contrast with (and in a sense dual to) the so-called