Fuzzy logic and enriched categories

Abstract

We consider a category C enriched over the segment [0, 1] whose hom-objects are real numbers from [0, 1]. For a suitably defined function ˆv assigning to each formula φ some object of C, the hom-object C(ˆv(φ), ˆv(ψ)) represents the degree of derivability of ψ from φ. We reformulate completeness result for intuitionistic propositional logic, as well as Hájek’s completeness results concerning the product, Gödel and łLukasiewicz fuzzy logic in the context of enriched category theory.