On Classical Determinate Truth
Luca Castaldo, Carlo Nicolai
TL;DR
The paper develops a family of type-free, fully classical theories of truth with a defined determinateness predicate, achieving full compositionality and Kripke–Feferman–style semantics via a classical closure. Key results show that the determinateness class can be captured without primitive notions by defining determinateness through truth—specifically using $\mathrm{D}x\;\equiv\;\mathrm{TT}x \lor \mathrm{TF}x$—and that the resulting systems (notably $\mathsf{CD}^+_{\rm T}$ and its classical closure $\mathsf{CKF}_{\mathsf{cs}}$) are $\mathbb{N}$-categorical, mutually interpretable, and align with the Kripke fixed-point closures of consistent theories. The work compares these to Fujimoto–Halbach’s $\mathsf{CD}^+$ and $\mathsf{KF}$-based approaches, showing that while traditional primitive-determinateness predicates complicate semantic analysis, a truth-derived determinateness supports both robust semantic rules and strong generalization power. It also analyzes complete, symmetric, and mixed fixed-points, locating CKF-style variants as robust, well-behaved closures and identifying open questions about mixed-model viability and extensions via reflection principles. Overall, the framework provides a coherent, classical account of truth and determinateness that preserves compositionality, yields transparent deep theories, and produces precise model-theoretic characterizations anchored in Kripke–Feferman semantics.
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 \emph{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 \emph{classical closures} of Kripke-Feferman truth: their $ω$-models, which we precisely pinned 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 $\cdplus$. {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.