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. Interpretation There are four cases that can be interpreted as follows: means that is necessary. is certainly true. It implies that . means that is impossible. is certainly false. It implies that . means that is possible. I would not be surprised at all if occurs. It leaves unconstrained. means that is unnecessary. I would not be surprised at all if does not occur. It leaves unconstrained. The intersection of the last two cases is and meaning that I believe nothing at all about . 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 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 classical 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. Possibility theory as an imprecise probability theory There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. A possibility measure can be seen as a consonant plausibility measure in Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set set of admissible probability distributions by This allows one to study possibility theory using the tools of imprecise probabilities. Necessity logic We call generalized