Fuzzy sets and statistical data. (English) Zbl 0588.62002

Specific features of probability and possibility theories are discussed with emphasis on semantical aspects. Instead of putting forward the acknowledged usefulness of possibility theory for the non-statistical modelling of subjective categories, we try to figure out how statistical data and possibility theory could be matched. As a result, procedures for constructing weak possibilistic substitutes of probability measures, and for processing imprecise statistical data are outlined. They provide new insights on the relationship between fuzzy sets and probability theories.


62A01 Foundations and philosophical topics in statistics
60A99 Foundations of probability theory
Full Text: DOI


[1] Chanas, S; Kamburovski, J, The use of fuzzy variables in PERT, Fuzzy sets and systems, 5, 11-20, (1981)
[2] Dempster, A.P, Upper and lower probabilities induced by a multivalued mapping, Annals of mathematical statistics, 38, 325-339, (1967) · Zbl 0168.17501
[3] Dubois, D, Steps to a theory of qualitative possibility, ()
[4] Dubois, D; Prade, H, Algorithmes de plus courts chemins pour traiter des données floues, RAIRO, operations research series, 12, 213-227, (1978) · Zbl 0379.90046
[5] Dubois, D; Prade, H, Operations on fuzzy numbers, International journal on systems sciences, 9, 613-626, (1978) · Zbl 0383.94045
[6] Dubois, D; Prade, H, Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[7] Dubois, D; Prade, H, Unfair coins and necessity measures, towards a possibilistic interpretation of histograms, Fuzzy sets and systems, 10, 15-20, (1983) · Zbl 0515.60005
[8] Dubois, D; Prade, H, A class of fuzzy measures based on triangular norms, International journal on general systems, 8, 43-61, (1982) · Zbl 0473.94023
[9] Dubois, D; Prade, H, On several representations of an uncertain body of evidence, (), 167-181, North-Holland, Amsterdam
[10] Dubois, D; Prade, H, Upper and lower possibilistic expectations and some applications, () · Zbl 0562.94023
[11] Fine, T, Theories of probability, (1973), Academic Press New York
[12] Goodman, I.R, Fuzzy sets as equivalence classes of random sets, (), 327-343
[13] Höhle, U, A mathematical theory of uncertainty, (), 344-355
[14] Hugues, G.E; Cresswell, M.J, An introduction to modal logic, (1972), Methuen London
[15] Kaufmann, A, La simulation des ensembles flous, (), Lyon · Zbl 0302.02023
[16] Kendall, D.G, Foundations of a theory of random sets, (), 322-376 · Zbl 0275.60068
[17] Matheron, G, Random sets and integral geometry, (1975), John Wiley and Sons New York · Zbl 0321.60009
[18] MacVicar-Whelan, P.J, Fuzzy logic: an alternative approach, () · Zbl 0342.68057
[19] Moore, R, Interval analysis, (1966), Prentice-Hall Englewood Cliffs, NJ · Zbl 0176.13301
[20] Prade, H, Modal semantics and fuzzy set theory, (), 232-246
[21] Savage, L.J, The foundations of statistics, (1972), Dover New York · Zbl 0121.13603
[22] Shackle, G.L.S, Decision, order and time in human affairs, (1961), Cambridge University Press
[23] Shafer, G, A mathematical theory of evidence, (1976), Princeton University Press NJ · Zbl 0359.62002
[24] Shafer, G, Non-additive probabilities in the work of bernouilli and Lambert, Archive for history of exact sciences, 19, 309-370, (1978) · Zbl 0392.01010
[25] Sigal, G.E; Pritsker, A.A.B; Solberg, J, The stochastic route problem, Operations research, 28, 1122-1129, (1980) · Zbl 0451.90091
[26] Sugeno, M, Theory of fuzzy integrals and its applications, () · Zbl 0316.60005
[27] Wang, P.Z; Sanchez, E, Treating a fuzzy subset as a projectable random subset, (), 213-220, North-Holland, Amsterdam
[28] Wang, P.Z, From the fuzzy statistics to the falling random subsets, (), 81-96
[29] Yager, R.R, Level sets for membership evaluation of fuzzy subsets, (), 90-97
[30] Zadeh, L.A, Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606
[31] Zadeh, L.A, Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
[32] Zadeh, L.A, PRUF—A meaning representation language for natural languages, International journal of man-machine studies, 10, 395-460, (1978) · Zbl 0406.68063
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.