Intuitionistic Justification Logic, Semantically

2026-06-30

AI Summary

This study addresses intuitionistic justification logic with satisfaction operators by introducing, for the first time, two distinct semantic frameworks: basic modular models and modular models incorporating Kripke semantics. These constructions provide a rigorous semantic foundation that transcends the limitations of prior syntax-driven approaches. By integrating modular semantic structures with possible-worlds semantics, the work establishes a realization theorem linking this justification logic to its corresponding intuitionistic modal logic at the semantic level. The research not only proves soundness and completeness for both model classes but also achieves the first semantic correspondence between intuitionistic justification logic with satisfaction operators and intuitionistic modal logic, thereby filling a significant gap in the semantic understanding of this logical system.

Abstract

Justification logics are explicit versions of modal logic. In the classical setting, this means boxes are refined with explicit proof terms and interact with each other through proof operations. This exercise was extended to intuitionistic modal logic with native diamonds. In this setting, diamonds are refined to satisfier terms and come equipped with additional operations. Justification logic enjoys a connection to its corresponding modal logic through a realisation theorem. In the classical setting, this is achieved through either proof-theoretic or semantic methodology. So far, intuitionistic justification logic with satisfiers has only been presented syntactically with a proof-theoretic realisation theorem. We present two classes of semantics for intuitionistic justification logic with soundness and completeness results: basic modular models, which extend possible world semantics for intuitionistic propositional logic; modular models which contain Kripke-style machinery to promote "backwards compatibility" to modal logic. Using modular models, we present a realisation theorem to establish a connection between intuitionistic justification logic and its corresponding intuitionistic modal logic.

Problem

• Intuitionistic Justification Logic
• Satisfier Terms
• Realisation Theorem
• Semantic Models
• Intuitionistic Modal Logic

Innovation

• intuitionistic justification logic
• satisfier terms
• modular models
• realisation theorem
• semantic methodology