Possibility Theory - Formalization of Possibility

Formalization of Possibility

For simplicity, assume that the universe of discourse Ω is a finite set, and assume that all subsets are measurable. A distribution of possibility is a function from to such that:

Axiom 1:

Axiom 2:

Axiom 3: for any disjoint subsets and .

It follows that, like probability, the possibility measure on finite set is determined by its behavior on singletons:

provided U is finite or countably infinite.

Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω.

Axiom 2 could be interpreted as the assumption that the evidence from which was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1.

Axiom 3 corresponds to the additivity axiom in probabilities. However there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1-3 imply that:

for any subsets and .

Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally:

When Ω is not finite, Axiom 3 can be replaced by:

For all index sets, if the subsets are pairwise disjoint,