Fuzzy Models of Linear Logic

There has long been a perception among fuzzy set theorists that the negation based on the operation a 7→ 1 − a is the “correct” negation. It is easy to see why this is so. In the model of fuzzy sets based on the unit interval [0, 1], the classical complement of a subset is a destructive operation. For example, if A0 is a fuzzy subset of a crisp set A , then in the classical complement of A0 every element has degree of membership either 0 or 1. Any element that has any positive degree of membership in A0 is entirely out of the complement. This operation is simply too destructive to be useful. It turns out that if the logic is weakened to be linear that the “linear” negation based on a 7→ 1− a works. As is expected in linear logic the model uses not the category of morphisms, but of relations. A word about notation. We follow the practice, standard in computer science, of denoting the composite of arrows f :X −→ Y and g:Y −→ Z by f ; g:X −→ Z . In order to avoid excess parentheses, we also put functions to the right of the argument, also delimited by semicolons, so that we write x; f ; g instead of (f ; g)(x). This is based on the perception that there is no difference between thinking of elements of x ∈ X and functions x: 1 −→ X . Thus x; f ; g can be read indifferently as the element x to which the composite f ; g is applied or the element x; f to which the function g is applied. Of course, this notation is abandoned for binary operations that it is customary to infix.