Intuitionistic modal logics: epistemic reasoning with distributed knowledge

2025-09-16

AI Summary

This paper addresses the limited expressive power of intuitionistic epistemic and doxastic logics by introducing distributed knowledge—formalized via a diamond modality—into an intuitionistic modal framework, thereby enabling distributed reasoning. Methodologically, it employs a parameterized box-based propositional language, develops a relational semantics, and systematically constructs a corresponding axiomatic system along with a rigorous truth definition. The main contributions are: (1) the first intuitionistic logic of distributed knowledge; (2) a proof of strong completeness of the system with respect to the proposed relational semantics; and (3) a significant extension of the theoretical boundaries of intuitionistic modal logic, providing a novel formal foundation for modeling distributed cognition and belief.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

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
• Security and Privacy: Blockchains and distributed ledgers

Abstract

In this article, we add a diamond to the parametrized box-based propositional language of intuitionistic doxastic logic and intuitionistic epistemic logic introduced by Artemov and Protopopescu. The main results of this article are the proofs of completeness with respect to their appropriate relational semantics of the resulting intuitionistic doxastic logic and intuitionistic epistemic logic with distributed knowledge.

Problem

• Extends intuitionistic doxastic and epistemic logic
• Adds diamond operator for distributed knowledge
• Proves completeness for relational semantics

Innovation

• Adding diamond operator to intuitionistic logic
• Completeness proofs for doxastic epistemic logic
• Relational semantics for distributed knowledge