Is fuzzy set theory more general than probability theory?

Abstract

Fuzzy set theory and possibility theory are often described as extensions or generalizations of classical set theory and probability theory, respectively. Thus, for example, fuzzy sets are described as generalizations of the crisp sets used in classical set theory. Similarly, possibility theorists argue that the conventional definitions of fuzzy union and intersection (i.e. the max and min operations) are logical extensions of the corresponding operations in classical set theory. This paper shows that the operations of fuzzy union and intersection as conventionally defined can be viewed as special cases of probabilistic union and intersection. Under this interpretation, the fuzzy complement or negation is a generalization of the corresponding probabilistic operation, but fuzzy union and intersection are not. The relationships between fuzzy and probabilistic union and intersection also suggest a possible analogy to the Venn diagrams used to depict traditional set union and intersection. >