Supervaluations, truth, and intuitionistic logic
Pablo Dopico
TL;DR
This paper develops supervaluational fixed-point theories of truth for intuitionistic logic by extending Kripke-style semantics to intuitionistic Kripke ω-structures and defining a universal supervaluation framework. It constructs both semantic (svi-based) and axiomatic (ISV, IVB/IVF/IMC, CSV) theories, and demonstrates fixed-point existence and compositionality for disjunction and existential quantification, addressing a key limitation of classical supervaluationism. By comparing the fixed-frame and embedding-based approaches, the work clarifies the benefits of ranging over all structures and demonstrates that a highly compositional, intuitionistically valid truth theory can be obtained. The CSV framework further yields a fully compositional truth theory, with proof-theoretic strength matching ID^i_1, offering a constructive truth theory aligned with intuitionistic principles and the disjunction/existence properties of HA.
Abstract
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 supervaluationist fixed-point theories 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 technically compare our approach with previous attempts to formalize Kripkean supervaluational theories over possible-world semantics. Finally, we show that, unlike in the case of classical logic, a fixed-point theory over intuitionistic logic can be given that is both supervaluational and highly compositional in nature.