An early approach toward graded identity and graded membership in set theory (Q622026)

Fuzzy sets were introduced by \textit{L. A. Zadeh} in [Inf. Control 8, 338--353 (1965; Zbl 0139.24606)]. Independently and in the same year, German mathematician D. Klaua introduced a rather similar concept of a theory with graded membership predicates and graded equality relations. While Zadeh's idea has grown up enormously, Klaua's approach ran dry in a few years. This paper speculates why Klaua himself stopped to develop his theory. In the context of logics of left-continuous triangular norms (so-called MTL-logics), some properties of Klaua's graded relations are discussed. Moreover, simultaneous recursive definitions of these relations are compared with a rather similar approach introduced by Scott and Solovay in 1967.