Outline of an Intensional Theory of Truth

Abstract

We expand on the fixed point semantic approach of Kripke via the addition of two unary intensional operators: a paradoxicality operator Π where $\Pi (\mathrm{\Phi })$ is true at a fixed point if and only if Φ is paradoxical (i.e., if and only if Φ receives the third, non-classical value on all fixed points that extend the current fixed point), and an unbounded truth operator ${\mathrm{{\rm Y}}}_{\top }$ where ${\mathrm{{\rm Y}}}_{\top }(\mathrm{\Phi })$ is true at a fixed point if and only if any fixed point extending the current fixed point can be extended to one on which Φ receives the value true. We prove a generalized version of Kripke’s fixed point theorem guaranteeing the existence of models of this new language, as well as an expressive completeness result. We conclude with an exploration of the significant improvements in expressive power that result from the addition of these new operators, and we precisely identify what still cannot be said on this intensional extension of the Kripkean framework.