Fuzzy inequational logic

TL;DR

This paper introduces a fuzzy inequational logic that extends traditional inequational logic to handle graded inequalities and partial truths using residuated lattices, providing a Pavelka-style completeness proof.

Contribution

It develops a novel fuzzy inequational logic framework with semantic and syntactic degrees of entailment, generalizing classical logic for reasoning with graded inequalities.

Findings

Proves Pavelka-style completeness of the logic

Defines semantic and syntactic degrees using residuated lattices

Introduces a logic for graded if-then rules as a special case

Abstract

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if-then rules which is obtained as particular case of the general result.