Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and (Ideal) Paraconsistency

2026-01-13

Outstanding Contributions to Logic

AI Summary

This study investigates intermediate logics situated between degree-preserving Gödel fuzzy logic with involution and classical propositional logic, focusing on their paraconsistent behavior relative to involutive negation in the finite-valued setting. By introducing “saturated paraconsistency”—a notion strictly weaker than ideal paraconsistency—and combining tools from algebraic logic, the classification of intermediate logics, and finite-valued fuzzy logic, the work fully characterizes the boundaries of all ideal and saturated paraconsistent logics lying between the n-valued Gödel involutive logic and classical logic. Furthermore, it identifies a broad class of saturated paraconsistent logics within intermediate systems of finite-valued Łukasiewicz logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming
• 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
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In this paper we study intermediate logics between the degree preserving companion of Godel fuzzy logic with an involution and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts. Although these degree-preserving Godel logics are explosive with respect to Godel negation, they are paraconsistent with respect to the involutive negation. We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between the degree-preserving n-valued Godel fuzzy logic with an involution and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Lukasiewicz logics.

Problem

• intermediate logics
• degree-preserving logics
• paraconsistency
• Gödel logics
• involutive negation

Innovation

• degree-preserving logics
• Gödel logic with involution
• intermediate logics
• saturated paraconsistency
• ideal paraconsistency