zbMATH — the first resource for mathematics

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.

28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
60A99 Foundations of probability theory
Full Text: DOI
[1] De Baets, B., Solving fuzzy relational equations: an order-theoretic approach, (), 389, (in Dutch)
[2] De Baets, B., Model implicators and their characterization, (), A42-A49
[3] B. De Baets, Coimplicators, the forgotten connectives, Tatra Mountains Math. Publications, to appear. · Zbl 0954.03029
[4] De Baets, B.; Kerre, E., A primer on solving fuzzy relational equations on the unit interval, Int. J. uncertainty, fuzziness and knowledge-based systems, 2, 205-225, (1994) · Zbl 1232.03039
[5] G. De Cooman, Possibility theory II: conditional possibility, Int. J. General Systems, to appear. · Zbl 0955.28013
[6] De Cooman, G.; Kerre, E., Order norms on bounded partially ordered sets, The J. fuzzy math., 2, 281-310, (1994) · Zbl 0814.04005
[7] Dubois, D.; Prade, H., Théorie des possibilités, (1985), Masson Paris
[8] Dubois, D.; Prade, H., Possibility theory — an approach to computerized processing of uncertainty, (1988), Plenum Press New York
[9] Dubois, D.; Prade, H., The logical view of conditioning and its application to possibility and evidence theories, Int. J. approximate reasoning, 4, 23-46, (1990) · Zbl 0696.03006
[10] Hisdal, E., Conditional possibilities: independence and non-interaction, Fuzzy sets and systems, 1, 283-297, (1987) · Zbl 0393.94050
[11] E.-P. Klement, R. Mesiar, E. Pap, Triangular Norms, in preparation. · Zbl 0972.03002
[12] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier New York · Zbl 0546.60010
[13] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press Princeton · Zbl 0359.62002
[14] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.