Abstract
The question of how to describe uncertainty and how to deal with it in a rigorous mathematical way has been discussed for many years (we only recall the dispute between objectivists and subjectivists in probability theory). More recently two attempts in the context of fuzzy sets (Zadeh [24]) have been made. In 1968 Zadeh [25] introduced fuzzy probability which led to an axiomatic study of this topic (Höhle [6], Klement [13, 15], Klement et al. [17]) in the sense of a valuation on a lattice, the so-called fuzzy σ-algebra (Klement [12, 14]), which also might be assumed to be non-distributive. Using a suitable complement terms such as discernibless and indiscernibless may be formulated (Höhle [9]). The second approach, again given by Zadeh [26] in 1978, was the introduction of possibility measures, mappings on a Boolean algebra where the addition is performed using the maximum instead of the sum.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aczel, J.: 1969, Lectures on FUNCTIONAL Equations and Their Applications, Academic Press, New York.
Bauer, H.: 1981, Probability Theory and Elements of Measure Theory, Academic Press, London.
3.Birkhoff, G.:. 1967, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. XXXV (Third Edition).
Frank, M.J.: 1979, ‘On the simultaneous associativity of <!-math-!>’, Aequationes Math. 19, 194–226.
Fuchs, L.: 1966, Teilweise geordnete algebraische Struk-turen, Vandenhoeck& Ruprecht, Göttingen.
Höhle, U.: 1976, ‘Maße aus unscharfen Mengen’, Z. Wahr cheinlichkeitstheorie verw. Gebiete 36, 179–188.
Höhle, U.: 1982, ‘A mathematical.theory of uncertainty’, in R.R. Yager (ed.). Fuzzy Set and Possibility Theory. Recent Developments, Pergamon Press, New York, pp. 344–355.
Höhle, U.: 1982, ‘Fuzzy plausibility measures’, in E.P. Element (ed.), Proceedings of the Third International Seminar on Fuzzy Set Theory, Linz, 1981, Johannes Kepler Universität, Linz, pp. 7–30.
Höhle U.: 1982, ‘Entropy with respect to plausibility measures’, in Proceedings, The Twelfth International Symposium on Multiple-valued Logic, Paris, 1982, IEEE, New York, pp. 167–169.
Jakobs, K.: 1978, Measure and Integral, Academic Press, New York.
Kappos, D.A.: 1969, Probability algebras and stochastic spaces, Academic Press, New York.
Klement, E.P.: 1980, ‘Fuzzy σ-algebras and fuzzy measurable functions’, Fuzzy Sets and Systems 4,83–93.
Klement, E.P.: 1980, ‘Characterization of finite fuzzy measures using Markoff-kernels’, J. Math. Anal. Appl. 75, 330–339.
Klement, E.P.: 1982, ‘Construction of fuzzy σ-algebras using triangular norms’, J. Math. Anal. Appl. 85, 543–565.
Klement, E.P.: 1982, ‘Characterization of fuzzy measures constructed by means of triangular norms’, J. Math. Anal. Appl. 86, 345–358.
Klement, E.P.: ‘Fuzzy measures assuming their values in the set of fuzzy numbers’, J. Math. Anal. Appl., in press.
Klement, E.P., Lowen, R. and Schwyhla, W.: 1981, ‘Fuzzy probability measures’, Fuzzy Sets and Systems 5, 21–30.
Klement, E.P. and Schwyhla, W.: 1982, ‘Correspondence between fuzzy measures and classical measures’, Fuzzy Sets and Systems 7, 57–70.
Łos, J.: 1955, ‘On the axiomatic treatment of probability’Colloq. Math. 3, 125–137.
Los, J.: 1960, ‘O ciatach zdarzeń i ich definicji w aksjomatycznej teorii prawdopodobieństwa’, Studia Lógica 9, 95–132. (‘Fields of events and their definitions in the axiomatic treatment of probability theory’, in Selected Translations of Mathematical Statistics and Probability, Vol. 7, Inst, of Math. Stat., Amer. Math. Soc., Providence, 1968, pp. 17–39)
Schweizer, B. and Sklar, A.: 1963, ‘Associative functions and abstract semigroups’, Publ. Math. Debrecen 10, 69–81.
Shafer, G.: 1976, A Mathematical Theory of Evidence, Princeton University Press, Princeton.
Shafer, G.: 1979, ‘Allocations of probability’, Annals of Prob. 7, 827–839.
Zadeh, L.A.: 1965, ‘Fuzzy Sets’, Inform. Control 8, 338–353.
Zadeh, L.A.: 1968,‘Probability measures of fuzzy events’, J. Math. Anal. Appl. 23, 421–427.
Zadeh, L.A.: 1978, ‘Fuzzy Sets as a basis for a theory of possibility’, Fuzzy Sets and Systems 1, 3–28.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 D. Reidel Publishing Company
About this chapter
Cite this chapter
Höhle, U., Klement, E.P. (1984). Plausibility Measures — A General Framework for Possibility and Fuzzy Probability Measures. In: Skala, H.J., Termini, S., Trillas, E. (eds) Aspects of Vagueness. Theory and Decision Library, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6309-2_3
Download citation
DOI: https://doi.org/10.1007/978-94-009-6309-2_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6311-5
Online ISBN: 978-94-009-6309-2
eBook Packages: Springer Book Archive