Fuzzy Logic and Possibility Theory

Consider an arbitrary proposition in the canonical form x is P, where x is an object from some universal set X and P is a predicate relevant to the object. To qualify for a treatment by classical logic, the proposition must be devoid of any uncertainty that is, it must be possible to determine whether the proposition is true or false. Any proposition that does not satisfy this requirement, due to some inherent uncertainty in it, is thus not admissible in classical logic. This is overly restrictive since uncertainty-free propositions are rather rare in human affairs. 

The second type of uncertainty in propositions of the given type results from information deficiency regarding the object x. While the predicate P is in this case defined precisely, information about x is insufficient to determine whether or not x satisfies P. The proposition is in this case either true or false, but its actual truth status cannot be determined. However, it is useful to assign a number in the unit interval [0,1] to the proposition to express the degree of evidence that the proposition is true. Assigning degrees of evidence to relevant propositions is a topic dealt with in measure theory. 

Propositions may also contain uncertainties of both types. To deal with information contained in such propositions, a measure-theoretic counterpart of fuzzy set theory was introduced by Zadeh under the name possibility theory.The following are basic notions of the theory. 

In this interpretation, the possibility degree that the value of is in any given crisp set A is equal to the greatest possibility degree for all x A. That is, given a possibility distribution function rp, the associated possibility measure Pos is defined for all crisp subsets A of X via the formula 

From these basic properties of possibility measures, the full calculus of possibility theory, analogous to the calculus of probability theory, has been developed. Its primary role is to deal with incomplete information expressed in terms of fuzzy propositions. Due to limited space, it is not possible to cover here details of this calculus. 

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