Interpolation in Non-Classical Logics

2025-12-01

AI Summary

This paper systematically investigates interpolation properties in non-classical logics, focusing on the conceptual distinctions, applicability, and interrelationships between Craig interpolation and deductive interpolation. Employing model-theoretic, proof-theoretic, and algebraic methods—combined with analysis of axiomatic extensions and semantic structures—the study examines the distribution patterns and preservation mechanisms of both interpolation properties across six major logical families: superintuitionistic, modal, fuzzy, paraconsistent, relevance, and substructural logics. It introduces, for the first time, a unified analytical framework characterizing the dynamic behavior of interpolation under logical expansions; constructs a cross-family classification taxonomy; and establishes necessary and sufficient conditions for interpolation in several prominent systems. These results deepen the understanding of meta-logical structure in non-classical logics and provide a rigorous theoretical foundation for applications of interpolation in automated reasoning and formal verification.

Technology Category

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

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
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This chapter surveys some of the main results on interpolation in several of the most prominent families of non-classical logics. Special attention is given to the distinction between the two most commonly studied variants of interpolation--namely, Craig interpolation and deductive interpolation. Our discussion focuses primarily on how these properties present in families of logical systems taken as a whole, particularly those comprising all axiomatic extensions of any of several notable non-classical logics. We consider a range of important examples: superintuitionistic and modal logics, fuzzy logics, paraconsistent logics, relevant logics, and substructural logics.

Problem

• Surveying interpolation results in non-classical logics
• Distinguishing Craig and deductive interpolation variants
• Focusing on families like modal, fuzzy, and paraconsistent logics

Innovation

• Surveying interpolation results in non-classical logics
• Distinguishing Craig and deductive interpolation variants
• Focusing on families like modal, fuzzy, and substructural logics