Paracomplete Truth Theory with a Definable Hierarchy of Determinateness Operators

Abstract

One way to deal with the Liar paradox is the paracomplete approach to theories of truth that gives up proofs by contradiction and the Law of the Excluded Middle. This allows one to reject both the Liar sentence and its negation. The simplest paracomplete theory of truth is KFS due to Saul Kripke. At face value, this theory suffers from the problem that it cannot say anything about the Liar paradox, so a de-fender of this theory cannot explain their rejection of the Liar sentence within the language of KFS. This was one of the motivations for Hartry Field to extend KFS with a conditional that is not definable within KFS. With the help of this conditional, Field defines a determinateness operator that can be used to explain one’s rejection of the Liar sentence within the object language of his theory. The determinateness operator can be transfinitely iterated to create stronger notions of determinate-ness required to explain the rejection of paradoxical sentences involving the determinateness operator. In this paper, we show that Field’s complex extension of KFS is not required in order to express rejection of paradoxical sentences like the Liar sentence. Instead one can work with a transfinite hierarchy of determinateness operators that are definable in KFS. After defining this hierarchy of determinateness operators, we compare their properties with the transfinitely iterable determinateness operator due to Field.