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The three semantics of fuzzy sets. (English) Zbl 0919.04006
Summary: Three main semantics for membership functions seem to exist in the literature: similarity, preference and uncertainty. Each semantics underlies a particular class of applications. Similarity notions are exploited in clustering analysis and fuzzy controllers. Uncertainty is captured by fuzzy sets in the framework of possibility theory. The membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems. This paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance, cost and frequency, on which scientific traditions exist.

MSC:
03E72 Theory of fuzzy sets, etc.
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[1] Aubin, J.P., Fuzzy differential inclusions, Problems control inform. theory, 19, 55-67, (1990) · Zbl 0718.93039
[2] Bellman, R.; Kalaba, L.; Zadeh, L.A., Abstraction and pattern classification, J. math. anal. appl., 13, 1-7, (1966) · Zbl 0134.15305
[3] Bellman, R.; Zadeh, L.A., Decision making in a fuzzy environment, Management sci., 17, B141-B164, (1970) · Zbl 0224.90032
[4] Benferhat, S.; Dubois, D.; Prade, H., Representing default rules in possibilistic logic, (), 673-684
[5] ()
[6] Bouchon-Meunier, B., Fuzzy similitude and approximate reasoning, (), 161-166
[7] Dubois, D.; Esteva, F.; Garcia, P.; Godo, L.; Prade, H., Similarity-based consequence relations, (), 171-179
[8] Dubois, D.; Fargier, H.; Prade, H., Propagation and satisfaction of flexible constraints, (), 166-187
[9] Dubois, D.; Fargier, H.; Prade, H., Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty, Appl. intelligence, 6, 287-309, (1996) · Zbl 1028.91526
[10] Dubois, D.; Lang, J.; Prade, H., Possibilistic logic, (), 439-513
[11] Dubois, D.; Moral, S.; Prade, H., A semantics for possibility theory based on likelihoods, J. math. anal. appl., 205, 359-380, (1997) · Zbl 0884.03017
[12] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[13] Dubois, D.; Prade, H., Possibility theory —; an approach to the computerized processing of uncertainty, (1988), Plenum Press New York, (with the collaboration of H. Farreny, R. Martin-Clouaire, C. Testemale)
[14] Dubois, D.; Prade, H., Fuzzy sets, probability and measurement, European J. oper. res., 40, 135-154, (1989) · Zbl 0663.90050
[15] Dubois, D.; Prade, H., Random sets and fuzzy interval analysis, Fuzzy sets and systems, 42, 87-101, (1991) · Zbl 0734.65041
[16] Dubois, D.; Prade, H., Epistemic entrenchment and possibilistic logic, Artificial intelligence, 50, 223-239, (1991) · Zbl 0749.03019
[17] Dubois, D.; Prade, H., Toll sets and toll logic, (), 169-177
[18] Dubois, D.; Prade, H., Fuzzy sets and probability: mis-understandings, bridges and gaps, (), 1059-1068
[19] Dubois, D.; Prade, H., Can we enforce full compositionality in uncertainty calculi?, (), 149-154
[20] Dubois, D.; Prade, H., Possibility theory as a basis for qualitative decision theory, (), 1924-1930
[21] ()
[22] ()
[23] Dutta, S.; Bonissone, P., Integrating case- and rule-based reasoning, Internat. J. approx. reasoning, 8, 163-203, (1993) · Zbl 0779.68074
[24] Elkan, E., The paradoxical success of fuzzy logic (with replies), IEEE expert, 9, 4, 3-49, (1994) · Zbl 1009.03517
[25] Fine, T., Theories of probability, (1973), Academic Press New York
[26] Gärdenfors, P., Knowledge in flux, (1988), MIT Press New York · Zbl 1229.03008
[27] Gebhardt, J.; Kruse, R., Automatic construction of possibilistic networks from data, Appl. math. comput. sci., 6, 529-564, (1996)
[28] Gilboa, I.; Schmeidler, D.; Gilboa, I., Case-based decision theory, (), Quart. J. economics, 606-639, (1995), revised, 1993 · Zbl 0836.90005
[29] Giles, R., The concept of grade of membership, Fuzzy sets and systems, 25, 297-323, (1988) · Zbl 0652.90001
[30] Goguen, J., L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404
[31] Goodman, I.R., Fuzzy sets as equivalence classes of random sets, (), 327-342
[32] Grabisch, M.; Nguyen, H.T.; Walker, E., Fundamentals of uncertainty calculi and their applications to fuzzy inference, (1995), Kluwer Academic Publishers Dordrecht
[33] Hájek, P., Fuzzy logic as logic, (), 21-30 · Zbl 0859.68095
[34] Hersh, H.M.; Caramazza, A.; Brownell, H.H., Effects of context on fuzzy membership functions, (), 389-408
[35] Hisdal, E., Are grades of membership probabilities?, Fuzzy sets and systems, 25, 325-348, (1988) · Zbl 0664.04009
[36] Kolodner, J., Case-based reasoning, (1993), Morgan Kaufmann San Mateo, CA
[37] Kruse, R.; Gebhardt, J.; Klawonn, F., Foundations of fuzzy systems, (1994), Wiley New York · Zbl 0843.68109
[38] Lang, J., Possibilistic logic as a logical framework for MIN-MAX discrete optimization problems and prioritized constraints, (), 112-126 · Zbl 0788.68134
[39] Lewis, D., Counterfactuals, (1973), Basil Blackwell Oxford · Zbl 0989.03003
[40] Niiniluoto, I., Truthlikeness, (1987), Reidel Dordrecht · Zbl 0639.03005
[41] Novak, V.; Novak, V., On the syntactico semantical completeness of first order fuzzy logic, Kybernetika, Kybernetika, 25, 134-154, (1990), Part II · Zbl 0705.03010
[42] Pavelka, J.; Pavelka, J.; Pavelka, J., On fuzzy logic, Zeitschr. math. logik grundlagen math., Zeitschr. math. logik grundlagen math., Zeitschr. math. logik grundlagen math., 25, 447-464, (1979), Part III · Zbl 0446.03016
[43] Pedrycz, W., Fuzzy sets engineering, (1995), CRC Press Boca Raton, FL · Zbl 0925.93507
[44] Ruspini, E., On the semantics of vague knowledge, Revue internationale systemique, 3, 387-420, (1989) · Zbl 0723.03014
[45] Shackle, G.L.S., Decision, order and time in human affairs, (1961), Cambridge Univ. Press Cambridge, UK
[46] Thomas, S.F., Fuzziness and probability, (1995), ACG Press Wichita, KS
[47] Turksen, I.B., Measurement of membership functions and their acquisition, Fuzzy sets and systems, 40, 5-38, (1991) · Zbl 0721.94029
[48] Turksen, I.B.; Zhong, Z., An approximate analogical reasoning approach based on similarity measures, IEEE trans. systems man cybernet., 18, 1049-1056, (1988)
[49] Tversky, A., Features of similarity, Psycholog. rev., 84, 327-354, (1977)
[50] Wang, P.Z.; Sanchez, E., Treating a fuzzy subset as a projectable random set, (), 213-220
[51] Whalen, T., Decision-making under uncertainty with various assumptions on available information, IEEE trans. systems man cybernet., 14, 888-900, (1984)
[52] Wood, K.L.; Antonsson, E.K., Computation with imprecise parameters in engineering design, background and theory, ASME J. mech. transmissions automat. des., 11, 616-625, (1989)
[53] Yager, R.R., Possibilistic decision-making, IEEE trans. systems man cybernet., 9, 388-392, (1979)
[54] Zadeh, L.A., Fuzzy sets, Infor. control, 8, 338-353, (1965) · Zbl 0139.24606
[55] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002
[56] Zadeh, L.A., A theory of approximate reasoning, (), 149-194
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