Possibility theory

There are four cases that can be interpreted as follows:


${\displaystyle N(U)=1}$ means that ${\displaystyle U}$ is necessary. ${\displaystyle U}$ is certainly true. It implies that ${\displaystyle \Pi (U)=1}$.


${\displaystyle \Pi (U)=0}$ means that ${\displaystyle U}$ is impossible. ${\displaystyle U}$ is certainly false. It implies that ${\displaystyle N(U)=0}$.


${\displaystyle \Pi (U)=1}$ means that ${\displaystyle U}$ is possible. I would not be surprised at all if ${\displaystyle U}$ occurs. It leaves ${\displaystyle N(U)}$ unconstrained.


${\displaystyle N(U)=0}$ means that ${\displaystyle U}$ is unnecessary. I would not be surprised at all if ${\displaystyle U}$ does not occur. It leaves ${\displaystyle \Pi (U)}$ unconstrained.


The intersection of the last two cases is ${\displaystyle N(U)=0}$ and ${\displaystyle \Pi (U)=1}$ meaning that I believe nothing at all about ${\displaystyle U}$. 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 classic example.

Necessity logic[edit]

We call generalized possibility every function satisfying Axiom 1 and Axiom 3. We call generalized necessity the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called necessity logic. In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual modus ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).