Nonclassical theories of truth

This chapter attempts to give a brief overview of nonclassical (-logic) theories of truth. Due to space limitations, we follow a victory-through-sacrifice policy: sacrifice details in exchange for clarity of big-picture ideas. This policy results in our giving all-too-brief treatment to certain topics that have dominated discussion in the non-classical-logic area of truth studies. (This is particularly so of the ‘suitable conditoinal’ issue: §4.3.) Still, we present enough representative ideas that one may fruitfully turn from this essay to the more-detailed cited works for further study. Throughout – again, due to space – we focus only on the most central motivation for standard non-classical-logic-based truth theories: namely, truth-theoretic paradox (specifically, due to space, the liar paradox). Our discussion is structured as follows. We first set some terminology concerning theories and logics; this terminology allows us to frame the discussion in a broad-but-clean fashion. (On the logic side, we present a very basic sequent system for truth and negation – and nothing more.) We then present a strippeddown version of the liar paradox. The paradox, as we set it up, turns on four basic rules (not including the truth rules; it’s the job of our target non-classical truth theories to preserve these in unrestricted form): two rules governing negation’s behavior, and two rules governing the ‘structure’ of the validity relation itself. These four rules serve as choice points for the four basic theoretical directions that we sketch. While details, as warned above, are sacrificed for space and big-picture clarity, we hope that the discussion nonetheless charts the main directions of non-classical response to basic truth-theoretic paradox.