ON CLASSICAL DETERMINATE TRUTH

Abstract The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both classical), and feature a defined determinateness predicate satisfying desirable and widely agreed principles. The theories capture a conception of truth and determinateness according to which the generalizing power associated with the classicality and full compositionality of truth is combined with the identification of a natural class of sentences—the determinate ones—for which clear-cut semantic rules are available. Our theories can also be seen as the classical closures of Kripke–Feferman truth: their $\omega $ -models, which we precisely pin down, result from including in the extension of the truth predicate the sentences that are satisfied by a Kripkean closed-off fixed-point model. The theories compare to recent theories proposed by Fujimoto and Halbach, featuring a primitive determinateness predicate. In the paper we show that our theories entail all principles of Fujimoto and Halbach’s theories, and are proof-theoretically equivalent to Fujimoto and Halbach’s $\mathsf {CD}^{+}$ . We also show establish some negative results on Fujimoto and Halbach’s theories: such results show that, unlike what happens in our theories, the primitive determinateness predicate prevents one from establishing clear and unrestricted semantic rules for the language with type-free truth.