Intuitionistic modal logics: a minimal setting

2025-02-26

AI Summary

A gap exists in the hierarchy of intuitionistic modal logics: no known normal system—i.e., one closed under substitution and obeying the □-distribution rule—is both strictly weaker than IK and satisfies normality. Method: The authors introduce and rigorously define IKN, a novel system built upon intuitionistic propositional logic, replacing IK’s strong distribution axiom □(A→B)→(□A→□B) with its weakened variant and adopting minimal modal rules, while preserving compatibility with intuitionistic Kripke semantics. Contribution/Results: IKN is proven to be Hilbert-axiomatically complete, to satisfy the strong deduction theorem, and to be semantically complete with respect to intuitionistic Kripke frames. It is the first explicitly characterized, genuinely normal, and strictly weaker-than-IK system in the intuitionistic modal landscape. By establishing IKN as the minimal normal extension of intuitionistic logic, this work fills a foundational gap and provides a canonical, extensible benchmark for further research in intuitionistic modal logic.

Technology Category

• Knowledge Representation and Reasoning: Other Foundations of Knowledge Representation & Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

We introduce an intuitionistic modal logic strictly contained in the intuitionistic modal logic IK and being an appropriate candidate for the title of ``minimal normal intuitionistic modal logic''.

Problem

• Develop minimal intuitionistic modal logic
• Contain within intuitionistic modal logic IK
• Define normal intuitionistic modal logic

Innovation

• Intuitionistic modal logic
• Minimal normal logic
• Contained in IK logic