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Full-Text Articles in Physical Sciences and Mathematics
When Does A Category Built On A Lattice With A Monoidal Structure Have A Monoidal Structure?, Lawrence Stout
When Does A Category Built On A Lattice With A Monoidal Structure Have A Monoidal Structure?, Lawrence Stout
Lawrence N. Stout
In a word, sometimes. And it gets harder if the structure on L is not commutative. In this paper we consider the question of what properties are needed on the lattice L equipped with an operation * for several different kinds of categories built using Sets and L to have monoidal and monoidal closed structures. This works best for the Goguen category Set(L) in which membership, but not equality, is made fuzzy and maps respect membership. Commutativity becomes critical if we make the equality fuzzy as well. This can be done several ways, so a progression of categories is considered. …
Categorical Approaches To Non-Commutative Fuzzy Logic, Lawrence Stout
Categorical Approaches To Non-Commutative Fuzzy Logic, Lawrence Stout
Lawrence N. Stout
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 …
A Categorical Semantics For Fuzzy Predicate Logic, Lawrence N. Stout
A Categorical Semantics For Fuzzy Predicate Logic, Lawrence N. Stout
Lawrence N. Stout
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) [5], Esteva et al. (2003) [1], and Hajek (1998) [4], 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 …