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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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