Supervaluations, truth, and intuitionistic logic

The supervaluationist approach to fixed-point semantics is, arguably, the most celebrated and studied competitor to the Strong Kleene approach within Kripkean truth. In this paper, we show how to obtain a supervaluationist fixed-point theory of truth for intuitionistic logic. In particular, we show how to do supervaluations over Kripke structures for intuitionistic logic, and we obtain the corresponding theories of truth, both semantic and axiomatic. Furthermore, we show how the theory of truth changes when the Kripke structures over which the supervaluations are defined change. Finally, we advance a supervaluationist theory of truth that, unlike the classical case or any of the other intuitionistic, supervaluationist theories, is compositional in nature.