Fuzzy inequational logic
This work addresses a foundational issue in fuzzy logic and universal algebra, providing a formal framework for graded reasoning, but it is incremental as it builds on existing Pavelka-style approaches.
The paper tackles the problem of reasoning about graded inequalities by generalizing ordinary inequational logic to a fuzzy setting, proving that the logic is Pavelka-style complete and extending it to graded if-then rules.
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.