Editing Possibility theory (section)

==Interpretation==
There are four cases that can be interpreted as follows:

<math>N(U) = 1</math> means that <math>U</math> is necessary. <math>U</math> is certainly true. It implies that <math>\Pi(U) = 1</math>.

<math>\Pi(U) = 0</math> means that <math>U</math> is impossible. <math>U</math> is certainly false. It implies that <math>N(U) = 0</math>.

<math>\Pi(U) = 1</math> means that <math>U</math> is possible. I would not be surprised at all if <math>U</math> occurs. It leaves <math>N(U)</math> unconstrained.

<math>N(U) = 0</math> means that <math>U</math> is unnecessary. I would not be surprised at all if <math>U</math> does not occur. It leaves <math>\Pi(U)</math> unconstrained.

The intersection of the last two cases is <math>N(U) = 0</math> and <math>\Pi(U) = 1</math> meaning that I believe nothing at all about <math>U</math>. Because it allows for indeterminacy like this, possibility theory relates to the graduation of a [[many-valued logic]], such as [[intuitionistic logic]], rather than the classical [[bivalent logic|two-valued logic]].

Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example.

* Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle.
* Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full.