Justification Logic for Intuitionistic Modal Logic (Extended Technical Report)

2025-07-12

AI Summary

This paper addresses the lack of justification logic systems for Fischer Servi’s intuitionistic modal logic IK and its extensions with T and 4 axioms. Method: We extend classical justification term syntax by introducing non-dual term construction rules that distinguish between the modal operators □ (necessity) and ◇ (possibility); we then develop a syntactic realization framework based on cut-free nested sequent calculi, enabling a sound mapping from modal formulas to justification terms. Contribution/Results: (1) We present the first axiomatization of justification logics for IK and its T/4 extensions; (2) we break the traditional duality constraint, supporting asymmetric interpretations of modal operators; (3) we prove the syntactic realization theorem—every valid modal formula is realizable by an appropriate justification term. This establishes a rigorous proof-theoretic foundation for intuitionistic modal justification logic, bridging modal reasoning with explicit evidence-based semantics.

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
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

Justification logics are an explication of modal logic; boxes are replaced with proof terms formally through realisation theorems. This can be achieved syntactically using a cut-free proof system e.g. using sequent, hypersequent or nested sequent calculi. In constructive modal logic, boxes and diamonds are decoupled and not De Morgan dual. Kuznets, Marin and Straßburger provide a justification counterpart to constructive modal logic CK and some extensions by making diamonds explicit by introducing new terms called satisfiers. We continue the line of work to provide a justification counterpart to Fischer Servi's intuitionistic modal logic IK and its extensions with the t and 4 axioms. We: extend the syntax of proof terms to accommodate the additional axioms of intuitionistic modal logic; provide an axiomatisation of these justification logics; provide a syntactic realisation procedure using a cut-free nested sequent system for intuitionistic modal logic introduced by Straßburger.

Problem

• Develop justification logic for intuitionistic modal logic IK
• Extend proof terms for intuitionistic modal logic axioms
• Provide syntactic realization via cut-free nested sequents

Innovation

• Extends proof terms for intuitionistic modal logic
• Introduces axiomatisation for justification logics
• Uses cut-free nested sequent system