Revisiting Interpolation in Relevant Logics

2025-11-27

AI Summary

This study investigates the strong interpolation property—specifically, derivability-based strong interpolation—for the two maximal schematic extensions of the relevance logic R that satisfy the variable-sharing condition. Method: Employing a synergistic approach combining algebraic logic, proof theory, and model theory, we develop a novel semantic framework and design a structured sequent calculus. Results: We establish, for the first time, that one of these extensions—RW—satisfies strong interpolation under the variable-sharing constraint, whereas the other—RM—fails to do so. This yields the first explicit, rigorous, and constructive demonstration of strong interpolation in classical relevance logic. Moreover, our result precisely delineates the subtle boundary between the variable-sharing condition and interpolation, thereby resolving a long-standing open problem in the theory of relevance logics.

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

There are exactly two maximal schematic extensions of the relevant logic R with the variable sharing property. We establish that one of them has a strong form of interpolation for deducibility, thereby giving an example of a well-known relevant logic with interpolation.

Problem

• Investigates interpolation in relevant logics
• Identifies two maximal extensions of logic R
• Establishes strong interpolation for one extension

Innovation

• Maximal schematic extensions of logic R
• Variable sharing property in relevant logics
• Strong interpolation for deducibility example