Possibility distributions are built on fuzzy sets (Dubois, Prade and Yager [Dubois et al., 1993]).

An expression such as `` X is F '', where X is a variable and F is a fuzzy set, can represent two kinds of situation:

• On the one hand, the expression ``X is F'' can appear in a situation where the value of X is really known and we estimate to what degree this value is compatible with label F (which meaning depends of course on the context).

In this case, we focus on the gradual aspect of the description given by `` X is F'' which can express a constraint.

For example, X can be a person of known age, and we wish to estimate how young he is. So, if Peter is 32, then, in a given context, a degree of occurrence of 0.8 can be assigned to the assertion ``Peter is young''.


• On the other hand, ``X is F'' can also mean that ``all we know about the value of X is that X is F''. In this case, we do not accurately know the value of X. It corresponds to a situation where information is incomplete (with lack of precision and certainty) and where the values of X can only be ordered according to their degree of plausibility or possibility.

When a fuzzy set is used to represent what is known about the value of a singly-valued variable, the degree related to a value expresses the degree of possibility that this value is the true value of the variable. Fuzzy set F is then seen to be a possibility distribution (Zadeh [Zadeh, 1978]) which expresses preferences for possible values of poorly-known variable X. Several distinct values can simultaneously have a possibility degree equal to 1.

In the case of incomplete information, we can compute to which point information ``X is F'' is strong with an assertion such as ``the value of X is in subset A''. Possibility measure expresses that. If A is a crisp subset, then is defined as the maximum of on A.