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 $hat{v}$  assigning to each formula $varphi$ some object of $C$, the hom-object $C(hat{v} (varphi),hat{v}(psi))$ represents the degree of derivability of $psi$ from $varphi$. We reformulate completeness result for intuitionistic propositional logic, as well as H' ajek's completeness results concerning the product, G" odel and L ukasiewicz fuzzy logic in the context of enriched category theory.