Relations between possibility measures and necessity measures

Relations between Possibility Measures and Necessity Measures

The properties of possibility measures and necessity measures suggest that they are associated.

The value of a necessity measure N can be obtained from a possibility measure $\Pi$ , through the complement $\complement_A$ of any part A of $\Omega$ :

$\begin{displaymath}\forall A \in \mathcal{P}(\Omega), N(A) = 1 - \Pi\left(\complement_A\right)\end{displaymath}$

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$\Pi(A)$ measures the degree to which event A is likely to happen.
N(A) indicates the degree of certainty which can be assigned to this occurrence.

The more an event A is assigned a great necessity value, the less the complement event $\complement_A$ is possible, therefore the more we are certain of the occurrence of A.

The occurrence of event A is certain (N(A) = 1) if and only if the occurrence of its complement $\complement_A$ is impossible ( $\Pi\left(\complement_A\right) = 0$ , therefore $\Pi(A) = 1$ ).

If possibility measure $\Pi$ is defined from possibility distribution $\pi$ , necessity measure N, dual of $\Pi$ , can be defined as:

$\begin{displaymath}\forall A \in \mathcal{P}(\Omega), N(A) = \inf_{x \notin A}(1 - \pi(x))\end{displaymath}$

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The duality of $\Pi$ and N still appears in the following relations, for any part A of $\Omega$ :

$\textstyle \Pi(A) \geqslant N(A)$

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$\textstyle \max(\Pi(A), 1-N(A)) = 1$

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These relations impose that, if $N(A) \neq 0$ , then $\Pi(A) = 1$ , which means that any event about which we are certain, at least a little, is completely possible.

In the same way, if $\Pi(A) \neq 1$ , then N(A) = 0. This is interpreted by saying that we have no certainty about an event which is only relatively possible.

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