Fuzzy sets, rough set and probability

A rough membership function uses counting probability (ratio of cardinal numbers) to define a membership. An extension, called granular membership function (GMF), generalizes the counting probability to a general set function (GSF), such as probability, possibility, belief function, etc. have been investigated previously. The "set theoretical operations" (STO) of GMF are induced naturally from the operations of GSF. In particular, probabilistic GMF (PGMF) are defined according to the rules of probability; their operations depend not only on the numerical grades but also on the events. This is often expressed as "STO are not truth functional." On the other hand, STO on traditional fuzzy sets are truth functional. This phenomenon led us to conclude the grade of traditional fuzzy sets can not be interpreted as a probability.