Possibility Theory,Probability Theory,and Fuzzy Set Theory

Abstract

Since L. Zadeh proposed the concept of a fuzzy set in 1965, the relationships between probability theory and fuzzy set theory have been further discussed. Both theories seem to be similar in the sense that both are concerned with some type of uncertainty and both use the [0, 1] interval for their measures as the range of their respective functions (At least as long as one considers normalized fuzzy sets only!). Other uncertainty measures, which were already mentioned in chapter 4, also focus on uncertainty and could therefore be included in such a discussion. The comparison between probability theory and fuzzy set theory is difficult primarily for two reasons:

• 1. The comparison could be made on very different levels, that is, mathematically, semantically, linguistically, and so on.

• 2. Fuzzy set theory is not or is no longer a uniquely defined mathematical structure, such as Boolean algebra or dual logic. It is rather a very general family of theories (consider, for instance, all the possible operations defined in chapter 3 or the different types of membership functions). In this respect, fuzzy set theory could rather be compared with the different existing theories of multivalued logic.