Are grades of membership probabilities?

Abstract

A small sample is given of the difficulties with present-day fuzzy set theory with respect to the requirements of a scientific theory; and a summary of some probabilistic interpretations of grades of membership to be found in the literature. The ‘TEE model’ for grades of membership is then suggested. Instead of starting out from mathematical postulates, such as the assumed max and min operations for or and and, the TEE model uses a semantic, physio-logical, psycho-logical starting point by investigating the possibilities which a person has for assigning linguistic labels and partial grades of membership in a meaningful way. The assignment of partial grades of membership is explained by the assumption that humans have the ability to take errors of observation and intersubject differences into consideration. A membership value μ λ ( u ex) is identified with the subject's estimate of the probability that the label λ (e.g. λ = tall) would be assigned to an object of exact attribute value u ex in a natural language situation. This situation being a ‘labeling’ or a ‘yes-no’ one in which at least one of three possible sources of uncertainty is present. Almost all the accepted formulas of fuzzy set theory, as well as the shapes of the grade of membership curves, are then derived instead of having to be postulated. However, the derived formulas for or and and reduce to the max and min operations only in a number of special cases. In another special case they reduce to the summation and the ‘zero operation’ respectively. The generally valid formulas are presented.