Semantical Analysis of Intuitionistic Modal Logics between CK and IK

2024-08-01

arXiv.org

AI Summary

This paper addresses the fundamental semantic divergence between CK and IK—two prominent intuitionistic modal logics—regarding the treatment of the possibility operator (◇), resolving their inconsistency over the ◇-free fragment and clarifying their conservativity over pure necessity (□) axiom systems. Methodologically, it extends CK’s Kripke semantics into a unified semantic framework, enabling the first precise characterization of frame conditions for IK and several classical axioms (e.g., T, 4, B). Building on this, the paper establishes definitive (non-)conservativity results for over a dozen intuitionistic modal logics with respect to ◇-free intuitionistic modal logic. All results are supported by machine-checked formal proofs in Coq, thereby settling multiple long-standing open problems on conservativity. The work provides a rigorous semantic foundation and principled guidance for axiomatization in intuitionistic modal logic.

Technology Category

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

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: Foundation models and LLMs for Web-related graphs
• Responsible Web: Consent frameworks and practices on the web

Abstract

The intuitionistic modal logics considered between Constructive K (CK) and Intuitionistic K (IK) differ in their treatment of the possibility (diamond) connective. It was recently rediscovered that some logics between CK and IK also disagree on their diamond-free fragments, with only some remaining conservative over the standard axiomatisation of intuitionistic modal logic with necessity (box) alone. We show that relational Kripke semantics for CK can be extended with frame conditions for all axioms in the standard axiomatisation of IK, as well as other axioms previously studied. This allows us to answer open questions about the (non-)conservativity of such logics over intuitionistic modal logic without diamond. Our results are formalised using the Coq Proof Assistant.

Problem

• Compare intuitionistic modal logics CK and IK
• Analyze diamond-free fragments in CK-IK spectrum
• Develop Kripke semantics for CK-IK conservativity questions

Innovation

• Extends CK semantics with IK frame conditions
• Answers conservativity questions via Coq formalization
• Analyzes diamond-free fragments in modal logics