Possibility Theory, Probability and Fuzzy Sets Misunderstandings, Bridges and Gaps

Possibility theory was coined by L.A. Zadeh in the late seventies as an approach to model flexible restrictions constructed from vague pieces of information, described by means of fuzzy sets. Possibility theory is also a basic non-classical theory of uncertainty, different from but related to probability theory. This chapter discusses the basic elements of the theory: possibility and necessity measures (as well as two other set functions associated with a possibility distribution), the minimal specificity principle which underlies the whole theory, the notions of possibilitic conditioning and possibilistic independence, the combination, and projection of joint possibility distributions, as well as the possibilistic counterparts to integration. The relations and differences between this approach and other uncertainty frameworks, and especially probability theory, are pointed out. The difference between probability theory and fuzzy set theory is thus tentatively clarified. Lastly, decision-theoretic justifications of possibility theory are given.