Nonclassical logics have played an important role in formal theories of truth. In fact, the development of many nonclassical logics has been motivated by the hope that they can facilitate a resolution of the semantic paradoxes. Strong Kleene logic and supervaluations and their use in the theory of truth have been mentioned already. Recently dialethic theories have somewhat superseded the partial approaches to truth: on the usual dialethic account, the liar sentence is both true and false. If the liar is accepted together with its negation, classical logic must be abandoned to avoid triviality and various paraconsistent logics have been proposed to block the derivation of arbitrary sentences from a contradiction. More recently, Field's book Saving Truth From Paradox (2008) has sparked an increased interest in nonclassical axiomatic truth theories.

Most of the axiomatic theories I have discussed in the previous parts of this book, however, are formulated in classical logic. The only exception is the system PKF, an axiomatization of Kripke's theory in Strong Kleene logic.

Given the extensive use of nonclassical logics in the literature on formal theories of truth, the reader might wonder why I do not dedicate more space to the analysis and discussion of nonclassical truth theories. Actually, a referee of an early version of this book proposed that it should be entitled Classical Axiomatic Theories of Truth, because nonclassical theories are largely ignored by it. So it seems that I need to defend myself for mainly considering theories of truth formulated in classical logic rather than in a paraconsistent or some other nonclassical logic.