Basic fuzzy set theory

Abstract

It is an old problem of fuzzy set theory, already stressed in Zadeh (1965), that we do not have really convincing arguments for something like a “right choice” of our connectives, i.e. of our operations describing the union and intersection of fuzzy sets. Of course, the problem in the past has been discussed from different points of view, cf. e.g. Bellman/Giertz (1973), Yager (1979), Gottwald (1979a) and Giles (1976), (1979), Thole/Zimmer-mann/Zysno (1979), Zimmermann/Zysno (1980), Hamacher (1978) too. Besides those motivational discussions for the choice of “right” connectives for fuzzy set operations there is another mainstream in the current research work: to accept a broad class of possible candidates for fuzzy set operations, to discuss them all in parallel, and to choose concrete ones for concrete applications. In this sense different families of operations have been discussed by for example Yager (1980), Dombi (1982), Weber (1983), all of which proved to be special cases of the t-norms of Schweizer/Sklar (1961) which we discussed extensively in section 1.2. Hence, if we do not change the set [0,1] of generalized membership degrees, these t-norms and their duals, the t-conorms (which are sometimes called s-norms too), seem to require attention.