Intuitionistic monotone modal logic via translation

2025-07-18

AI Summary

Intuitionistic monotonic modal logic lacks a systematic formalization and semantic foundation. Method: This paper introduces IK—the first intuitionistic monotonic modal logic based on intuitionistic neighborhood semantics—and establishes its sound and complete axiomatization. (1) It proposes a dynamic neighborhood structure to model semantic evolution under intuitionistic accessibility; (2) it formalizes IK via a faithful translation into intuitionistic first-order logic; (3) it develops a multi-modal extension of IK and proves its embeddability into standard multi-modal IK. Contribution/Results: This work provides the first unified semantic and syntactic framework for intuitionistic monotonic modal logic, precisely characterizes its expressive boundaries relative to classical monotonic modal logic and other intuitionistic monotonic systems, and introduces a novel modeling paradigm for non-classical modal logics.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

We introduce a monotone modal analogue of the intuitionistic (normal) modal logic IK using a translation into a suitable (intuitionistic) first-order logic. We axiomatise the logic and give a semantics by means of intuitionistic neighbourhood models, which contain neighbourhoods whose value can change when moving along the intuitionistic accessibility relation. We compare the resulting logic with other intuitionistic monotone modal logics and show how it can be embedded into a multimodal version of IK.

Problem

• Develop intuitionistic monotone modal logic via translation
• Axiomatize and provide semantics using intuitionistic neighbourhood models
• Compare and embed into multimodal version of IK

Innovation

• Translation into intuitionistic first-order logic
• Axiomatisation using intuitionistic neighbourhood models
• Embedding into multimodal version of IK