Cutlike semantics for fuzzy logic and its applications

Each fuzzy set can be represented by a nested system of ordinary sets—its a-cuts. There is an extensive literature on fuzzy sets devoted to problems of the following kind: is it possible to reduce operations with fuzzy sets to operations with their a-cuts? Is it possible to reduce properties of fuzzy relations to properties of their a-cuts? More generally, can a fuzzy concept be represented by a collection of corresponding crisp concepts? Klir and Yuan (1995) speak of cutworthiness. We attempt to provide a general solution to this problem. The way we proceed is thus: a structure for fuzzy predicate logic can be represented by a nested system of crisp structures. The system of crisp structures can be used to define semantics of fuzzy predicate logic in an alternative way by using the nested structure and Boolean connectives only. Answers to the above questions are then obtained by simple application of the obtained general results; we present some examples (extension principle, properties of binary fuzzy relations, fuzzy automata).