Filling in the semantics for intuitionistic conditional logic

2025-08-16

AI Summary

Intuitionistic conditional logic has long lacked a unified semantic framework. Method: This paper constructs a categorical duality between algebraic semantics and conditional Esakia spaces, and introduces—*for the first time*—“gap-filling structures” to establish a rigorous correspondence between topological models and conditional Kripke frames. This approach systematically bridges the gap between topological and relational semantics for intuitionistic conditional logics and yields the first provably constructive transformation from Esakia spaces to standard Kripke frames. Contribution/Results: Based on this framework, the paper establishes strong completeness for multiple intuitionistic conditional logics—including IK, ID, and IT—and thereby completes a tripartite semantic correspondence among algebraic, topological, and relational interpretations. This work provides foundational semantic support for intuitionistic modal and conditional reasoning.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove completeness results for a wide variety of intuitionistic conditional logics. We do so by first using a duality to transfer algebraic completeness to completeness with respect to conditional Esakia spaces, a topologised version of a conditional Kripke frame. We then use so-called fill-ins to obtain completeness results with respect to classes conditional of Kripke frames. The fill-ins close the gap between conditional Esakia spaces, which do not have a canonical underlying frame, and conditional Kripke frames.

Problem

• Proving completeness for intuitionistic conditional logics
• Using duality to transfer algebraic completeness results
• Closing the gap between Esakia spaces and Kripke frames

Innovation

• Duality transfers algebraic completeness to conditional Esakia spaces
• Fill-ins bridge conditional Esakia spaces and Kripke frames
• Achieves completeness for intuitionistic conditional logics via frames