Possibility Theory - Necessity

Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set, the necessity measure is defined by

In the above formula, denotes the complement of, that is the elements of that do not belong to . It is straightforward to show that:

for any

and that:

Note that contrary to probability theory, possibility is not self-dual. That is, for any event, we only have the inequality:

However, the following duality rule holds:

For any event, either, or

Accordingly, beliefs about an event can be represented by a number and a bit.