Paracomplete truth theory with KFS-definable determinateness

Abstract

Abstract One way to deal with the liar paradox is the paracomplete approach to theories of truth, which 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 $\textit {KFS}$ due to Saul Kripke. At face value, this theory suffers from the problem that it cannot say anything about the liar paradox, so a defender of this theory cannot explain their rejection of the liar sentence within the language of $\textit {KFS}$. This was one of the motivations for Hartry Field to extend $\textit {KFS}$ with a conditional that is not definable within $\textit {KFS}$. 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. Field’s determinateness operator can be transfinitely iterated to create stronger notions of determinateness required to explain the rejection of paradoxical sentences involving the determinateness operator. In this paper, we show that Field’s complex extension of $\textit {KFS}$ 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 $\textit {KFS}$. This allows for Field’s philosophically appealing treatment of the liar sentence, the truth-teller and strengthenings of the liar sentence to be reproducible within the theory $\textit {KFS}$, which is semantically much simpler than Field’s extension of $\textit {KFS}$ with a conditional.