On the fundamentals of fuzzy set theory
In the following considerations we restrict our interest to foundation-theoretically oriented aspects of fuzzy set theory (resp. fuzzy logic), especially to their mathematical semantics. We distinguish two mainstreams: \(\bullet\) Plausibility theory (including possibility and probability theory). \(\bullet\) Non-classical model theory. In both cases lattice-valued (resp. \([0,1]\)-valued) maps play a dominant role. The purpose of this paper is twofold: On the one hand, we give for the first time a coherent presentation of some of the most important results of these disciplines; on the other hand, we apply these methods to long-standing open problems of fuzzy set theory. A typical situation is fuzzy control theory. Starting from such fundamental phenomena as indistinguishability and local extent of existence we develop in terms of singletons and many-valued equalities the mathematical semantics of the so-called fuzzy control situation. In particular, we give a mathematically sound solution of the fuzzification and defuzzification problem [\textit{C. C. Lee}, IEEE Trans. Syst. Man Cybern. 20, 404-418, 419-435 (1990; Zbl 0707.93036, Zbl 0707.93037)].
scientific article; zbMATH DE number 930223