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Conditioning in possibility theory with strict order norms. (English) Zbl 0985.28015
Summary: 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
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##### References:
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