Categorical approaches to non-commutative fuzzy logic

Abstract

In this paper we consider what it means for a logic to be non-commutative, how to generate examples of structures with a non-commutative operation * which have enough nice properties to serve as the truth values for a logic. Inference in the propositional logic is gotten from the categorical properties (products, coproducts, monoidal and closed structures, adjoint functors) of the categories of truth values. We then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type A with functors giving change of type and adjoints giving quantifiers. In the case where the semantics takes place in Set (L) (Goguen's category of L-fuzzy sets), the categories of predicates about A can be represented as internal category objects with the quantifiers as internal functors.