Four intuitionistic modal connectives

2026-06-05

AI Summary

This study addresses the lack of a unified treatment for four classes of modal connectives in intuitionistic modal logic by systematically integrating the Prenosil-style and Wijesekera-style pairs of connectives, thereby constructing a coherent syntactic and semantic framework. Employing model-theoretic methods, frame semantics, and axiomatic system construction techniques, the work clarifies the semantic relationships among these four connective classes and establishes a unified analytical framework. The main contributions include proving that the minimal intuitionistic modal logics determined by the corresponding frame classes are all decidable and providing effective axiomatizations for several important frame classes, thus resolving their semantic definability and strong completeness problems.

Abstract

We introduce the syntax and the semantics of intuitionistic modal logics based on a diamond connective à la Prenosil, its dual box connective, a diamond connective à la Wijesekera and its dual box connective. We analyze the modal definability of some elementary classes of frames. We study the complete axiomatizability of the sets of valid formulas determined by these classes of frames. We prove the decidability of the minimal intuitionistic modal logic determined by the class of all frames.

Problem

• intuitionistic modal logic
• modal definability
• complete axiomatizability
• decidability
• frame classes

Innovation

• intuitionistic modal logic
• modal connectives
• frame definability
• complete axiomatizability
• decidability