Conditioning in possibility theory with strict order norms.

*(English)*Zbl 0985.28015Summary: The general order-norm-based approach of defining the conditional possibility \(\Pi(A\mid B)\) as the greatest solution of the equation \({\mathcal T}(x,\Pi(B))= \Pi(A\cap B)\), with \({\mathcal T}\) a t-norm, and of defining the conditional necessity \(N(A\mid B)\) as the smallest solution of the equation
\[
{\mathcal S}(x, N(\text{co }B))= N(A\cup\text{co } B),
\]
with \({\mathcal S}\) a t-conorm, is carefully studied. In particular, it is investigated under which conditions the conditional possibilities (resp. necessities) again establish a possibility (resp. necessity) measure. Due to the new characterization of strict t-norms (resp. t-conorms) presented in this paper, it is shown that, in general, only a strict t-norm (resp. t-conorm) can be used, or, in other words, a transformation of the algebraic product (resp. probabilistic sum) by means of an order preserving permutation of the unit interval. This indicates that the algebraic product not only has a probabilistic, but also a surprising possibilistic nature.

##### MSC:

28E10 | Fuzzy measure theory |

03E72 | Theory of fuzzy sets, etc. |

60A99 | Foundations of probability theory |

##### Keywords:

strict order norm; conditional possibility; conditional necessity; algebraic product; probabilistic sum
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\textit{B. De Baets} et al., Fuzzy Sets Syst. 106, No. 2, 221--229 (1999; Zbl 0985.28015)

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##### References:

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