Membership Functions and Probability Measures of Fuzzy Sets

The notion of fuzzy sets has proven useful in the context of control theory, pattern recognition, and medical diagnosis. However, it has also spawned the view that classical probability theory is unable to deal with uncertainties in natural language and machine learning, so that alternatives to probability are needed. One such alternative is what is known as “possibility theory.” Such alternatives have come into being because past attempts at making fuzzy set theory and probability theory work in concert have been unsuccessful. The purpose of this article is to develop a line of argument that demonstrates that probability theory has a sufficiently rich structure for incorporating fuzzy sets within its framework. Thus probabilities of fuzzy events can be logically induced. The philosophical underpinnings that make this happen are a subjectivistic interpretation of probability, an introduction of Laplace’s famous genie, and the mathematics of encoding expert testimony. The benefit of making probability theory work in concert with fuzzy set theory is an ability to deal with different kinds of uncertainties that may arise within the same problem.