The object of this study is to look at categorical approaches to many valued logic, both propositional and predicate, to see how different logical properties result from different parts of the situation. In particular, the relationship between the categorical fabric I introduced at Linz in 2004 and the Fuzzy Logics studied by Hajek (2003) , Esteva et al. (2003) , and Hajek (1998) , comes from restricting the kind of structures used for truth values. We see how the structure of the various kinds of algebras shows up in the categorical logic, giving a variant on natural deduction for these logics. Quantification typically needs more completeness than is present in the algebras used in Hajek (1998) , hence the need for safe interpretations. The categorical setting gives a predicate logic without variables. The language in the more traditional sense comes from a structure built on a particular freely generated Cartesian category. Formulas have a clear meaning in that more restricted context. Interpretation of the language in other categorical fabrics is given by application of a product preserving functor. Traditional completeness results relate to this kind of interpretation. Completeness can also be understood as showing that the derivable truths in the general fabric are the necessary truths: those which are true in all of the possible worlds.
Stout, Lawrence, "A Categorical Semantics for Fuzzy Predicate Logic" (2010). Scholarship. 77.