Intuitionistic Monotone Modal Logic: Proof Theory and Semantics

2026-06-30

AI Summary

This work addresses the long-standing lack of a systematic semantic characterization and effective proof system for intuitionistic monotonic modal logic (IM). It introduces, for the first time, a constructive neighborhood semantics that provides exact models for IM and its natural extensions. Building upon the structured sequent calculus for classical monotonic modal logic M, the paper develops a tailored structured sequent calculus for IM. Through proof-theoretic analysis, the calculus is shown to be sound, complete, and decidable. Furthermore, the study uncovers a deep analogy between IM and intuitionistic modal logic K, thereby establishing IM as the faithful intuitionistic counterpart of M.

Abstract

We study the recently introduced intuitionistic monotone modal logic IM. We first provide a semantic characterisation for a family of natural extensions of IM in terms of constructive neighbourhood models. We then present a calculus for IM and its extensions, obtained by adapting a structured calculus for the classical monotone modal logic M. Based on the calculus, we prove some preliminary results for IM, including its decidability. Our calculus also reveals an interesting analogy between constructive and intuitionistic variants of M and the corresponding variants of K, thereby further justifying IM as a faithful intuitionistic variant of M.

Problem

• Intuitionistic Modal Logic
• Monotone Modal Logic
• Constructive Semantics
• Proof Theory
• Neighbourhood Models

Innovation

• intuitionistic monotone modal logic
• constructive neighbourhood semantics
• structured sequent calculus
• decidability
• modal logic analogy