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Gradualness, uncertainty and bipolarity: making sense of fuzzy sets. (English) Zbl 1238.03044

Summary: This paper discusses basic notions underlying fuzzy sets, especially gradualness, uncertainty, vagueness and bipolarity, in order to clarify the significance of using fuzzy sets in practice. Starting with the idea that a fuzzy set may represent either a precise gradual composite entity or an epistemic construction refereeing to an ill-known object, it is shown that each of this view suggests a different use of fuzzy sets. Then, it is argued that the usual phrase fuzzy number is ambiguous as it induces some confusion between gradual extensions of real numbers and gradual extensions of interval calculations. The distinction between degrees of truth that are compositional and degrees of belief that cannot be so is recalled. The truth-functional calculi of various extensions of fuzzy sets, motivated by the desire to handle ill-known membership grades, are shown to be of limited significance for handling this kind of uncertainty. Finally, the idea of a separate handling of membership and non-membership grades put forward by Atanassov is cast in the setting of reasoning about bipolar information. This intuition is different from the representation of ill-known membership functions and leads to combination rules differing from the ones proposed for handling uncertainty about membership grades.

MSC:

03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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[1] Zadeh, L., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
[2] Zadeh, L., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 1-28 (1978) · Zbl 0377.04002
[3] Lipski, W., On databases with incomplete information, Journal of ACM, 28, 1, 41-70 (1981) · Zbl 0464.68086
[4] Cacioppo, J. T.; Berntson, G. G., Relationship between attitudes and evaluative space: a critical review, with emphasis on the separability of positive and negative substrates, Psychological Bulletin, 115, 401-423 (1994)
[5] Zadeh, L., The concept of a linguistic variable and its application to approximate reasoning, Part I, Information Sciences, 8, 199-249 (1975) · Zbl 0397.68071
[6] Atanassov, K., Intuitionistic Fuzzy Sets (1999), Physica-Verlag: Physica-Verlag Heidelberg · Zbl 0597.03033
[7] Molodtsov, D., Soft set theory-first results, Computers & Mathematics with Applications, 37, 19-31 (1999) · Zbl 0936.03049
[8] Williamson, T., Vagueness (1994), Routledge: Routledge London
[9] Dubois, D.; Esteva, F.; Godo, L.; Prade, H., An information-based discussion of vagueness: six scenarios leading to vagueness, (Cohen, H.; Lefebvre, C., Handbook of Categorization in Cognitive Science (2005), Elsevier: Elsevier Amsterdam), 891-909, (Chapter 40)
[10] Dubois, D.; Prade, H., The three semantics of fuzzy sets, Fuzzy Sets and Systems, 90, 141-150 (1997) · Zbl 0919.04006
[11] Elkan, C., The paradoxical success of fuzzy logic, IEEE Expert, 9, 4, 3-8 (1994) · Zbl 1009.03517
[12] Kosko, B., Fuzzy entropy and conditioning, Information Sciences, 40, 2, 165-174 (1986) · Zbl 0623.94005
[13] Deschrijver, G.; Kerre, E. E., On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 133, 2, 227-235 (2003) · Zbl 1013.03065
[14] A. Herzig, J. Lang, P. Marquis, Action representation and partially observable planning using epistemic logic, in: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI-03), Acapulco, Morgan Kaufmann, San Francisco, 2003, pp. 1067-1072.; A. Herzig, J. Lang, P. Marquis, Action representation and partially observable planning using epistemic logic, in: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI-03), Acapulco, Morgan Kaufmann, San Francisco, 2003, pp. 1067-1072.
[15] Zadeh, L. A., PRUF - a meaning representation language for natural languages, International Journal of Man-Machine Studies, 10, 395-460 (1978) · Zbl 0406.68063
[16] Yager, R. R., Set-based representations of conjunctive and disjunctive knowledge, Information Sciences, 41, 1, 1-22 (1987) · Zbl 0626.68073
[17] Dubois, D.; Prade, H., Incomplete conjunctive information, Computers & Mathematics with Applications, 15, 797-810 (1988) · Zbl 0709.94588
[18] Banerjee, M.; Dubois, D., A simple modal logic for reasoning about revealed beliefs, (Sossai, C.; Chemello, G., ECSQARU, Lecture Notes in Computer Science, vol. 5590 (2009), Springer), 805-816, ISBN 978-3-642-02905-9 · Zbl 1245.03019
[19] Puri, M.; Ralescu, D., Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114, 409-422 (1986) · Zbl 0592.60004
[20] Dubois, D.; Prade, H., Possibility Theory (1988), Plenum Press: Plenum Press New York (NY) · Zbl 0645.68108
[21] Shackle, G. L., Decision, Order and Time in Human Affairs (1961), Cambridge University Press: Cambridge University Press UK
[22] Spohn, W., A general non-probabilistic theory of inductive reasoning, (Proceedings of UAI’90 (1990), Elsevier Science), 315-322
[23] De Finetti, B., Theory of Probability (1974), Wiley: Wiley New York
[24] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman & Hall · Zbl 0732.62004
[25] Dubois, D.; Prade, H., When upper probabilities are possibility measures, Fuzzy Sets and Systems, 49, 65-74 (1992) · Zbl 0754.60003
[26] Dubois, D.; Fargier, H.; Prade, H., Possibility theory in constraint satisfaction problems: handling priority, preference and uncertainty, Applied Intelligence, 6, 287-309 (1996) · Zbl 1028.91526
[27] Bistarelli, F. R.S.; Montanari, U., Semiring-based constraint solving and optimization, Journal of ACM, 44, 201-236 (1997) · Zbl 0890.68032
[28] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 100, 63-71 (1999), ISSN 0165-0114
[29] Bertoluzza, A. S.C.; Corral, N., On a new class of distances between fuzzy numbers, Mathware and Soft Computing, 2, 71-84 (1995) · Zbl 0887.04003
[30] Heilpern, S., Representation and application of fuzzy numbers, Fuzzy Sets and Systems, 91, 2, 259-268 (1997), ISSN 0165-0114 · Zbl 0920.04011
[31] Dubois, D.; Moral, S.; Prade, H., Belief change rules in ordinal and numerical uncertainty theories, (Dubois, D.; Prade, H., Belief Change (1998), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht), 311-392 · Zbl 0928.03018
[32] Diamond, P., Fuzzy least squares, Information Sciences, 46, 3, 141-157 (1988), ISSN 0020-0255 · Zbl 0663.65150
[33] Körner, R., On the variance of fuzzy random variables, Fuzzy Sets and Systems, 92, 83-93 (1997), ISSN 0165-0114 · Zbl 0936.60017
[34] Matheron, G., Random Sets and Integral Geometry (1975), J.Wiley & Sons: J.Wiley & Sons New York · Zbl 0321.60009
[35] Dempster, A. P., Upper and lower probabilities induced by a multivalued mapping, Annals of Mathematical Statistics, 38, 325-339 (1967) · Zbl 0168.17501
[36] Shafer, G., A Mathematical Theory of Evidence (1976), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0359.62002
[37] Kwakernaak, H., Fuzzy random variables—I. Definitions and theorems, Information Sciences, 15, 1, 1-29 (1978) · Zbl 0438.60004
[38] Kruse, R.; Meyer, K., Statistics with Vague Data (1987), D. Reidel Publishing Company: D. Reidel Publishing Company Dordrecht, The Netherlands · Zbl 0663.62010
[39] Couso, I.; Sánchez, L., Higher order models for fuzzy random variables, Fuzzy Sets and Systems, 159, 3, 237-258 (2008) · Zbl 1178.60004
[40] Biacino, L.; Lettieri, A., Equations with fuzzy numbers, Information Sciences, 47, 1, 63-76 (1989) · Zbl 0664.04006
[41] Hüllermeier, E., An Approach to modelling and simulation of uncertain dynamical systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 2, 117-138 (1997) · Zbl 1232.68131
[42] Bosc, P.; Pivert, O., About yes/no queries against possibilistic databases, International Journal of Intelligent Systems, 22, 7, 691-721 (2007) · Zbl 1121.68036
[43] Kasperski, A., Discrete Optimization with Interval Data: Minmax Regret and Fuzzy Approach, Studies in Fuzziness and Soft Computing, vol. 228 (2008), Springer: Springer Berlin · Zbl 1154.90017
[44] Dubois, D.; Prade, H., Gradual elements in a fuzzy set, Soft Computing, 12, 165-175 (2008) · Zbl 1133.03026
[45] Negoita, C. V.; Ralescu, D. A., Applications of Fuzzy Sets to Systems Analysis (1975), Birkhauser Verlag: Birkhauser Verlag Basel · Zbl 0326.94002
[46] Hutton, B., Normality in fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 8, 74-79 (1975) · Zbl 0297.54003
[47] Lowen, R., Fuzzy Real Numbers (1996), Kluwer: Kluwer Dordrecht
[48] Rodabaugh, S. E., Fuzzy addition in the L-fuzzy real line, Fuzzy Set and Systems, 8, 39-51 (1982) · Zbl 0508.54002
[49] Höhle, U., Fuzzy real numbers as Dededkind cuts with respect to a multiple-valued logic, Fuzzy Set and Systems, 24, 263-278 (1987) · Zbl 0638.03051
[50] Mizumoto, M.; Tanaka, K., The four operations of arithmetic on fuzzy numbers, Computer Systems Control, 7, 5, 73-81 (1976)
[51] Nahmias, S., Fuzzy variables, Fuzzy Sets and Systems, 1, 97-110 (1978) · Zbl 0383.03038
[52] Dubois, D.; Prade, H., Operations on fuzzy numbers, International Journal of Systems Science, 9, 613-626 (1978) · Zbl 0383.94045
[53] Fortin, J.; Fargier, H.; Dubois, D., Gradual numbers and their application to fuzzy interval analysis, IEEE Transactions on Fuzzy Systems, 16, 2, 388-402 (2008)
[54] Ogura, Y.; Li, S.-M.; Ralescu, D., Set defuzzification and Choquet integral, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9, 1, 1-12 (2001) · Zbl 1113.03342
[55] Dubois, D., On ignorance and contradiction considered as truth-values, Logic Journal of the IGPL, 16, 2, 195-216 (2008) · Zbl 1139.03013
[56] Dubois, D.; Prade, H., Possibility theory, probability theory and multiple-valued logics: a clarification, Annals of Mathematics and Artificial Intelligence, 32, 35-66 (2001) · Zbl 1314.68309
[57] Łukasiewicz, J., Philosophical remarks on many-valued systems of propositional logic (1930), (Borkowski, L., Selected Works (1970), North-Holland), 481-495
[58] Kleene, S. C., Introduction to Metamathematics (1952), Van Nostrand: Van Nostrand New York · Zbl 0047.00703
[59] Blamey, S., Partial logic, (Gabbay, D.; Guentner, F., Handbook of Philosophical Logic, second ed., vol. 5 (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands), 261-353
[60] van Fraassen, B. C., Singular terms, truth-value gaps, and free logic, Journal of Philosophy, 63, 481-495 (1966)
[61] Yao, Y. Y., Interval-set algebra for qualitative knowledge representation, (Abou-Rabia, O.; Chang, C. K.; Koczkodaj, W. W., ICCI (1993), IEEE Computer Society), 370-374, ISBN 0-8186-4212-2
[62] Pedrycz, W., Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 28, 1, 103-109 (1998)
[63] De Cooman, G., From possibilistic information to Kleene’s strong multi-valued logics, (Fuzzy Sets, Logics and Reasoning about Knowledge (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 315-323 · Zbl 1001.03520
[64] Fine, K., Vagueness, truth and logic, Synthese, 30, 265-300 (1975) · Zbl 0311.02011
[65] van Fraassen, B., Singular terms, truth-value gaps, and free logic, Journal of Philosophy, 63, 481-495 (1966)
[66] Williamson, T., Vagueness (1994), Routledge: Routledge London
[67] A. Orlov, Fuzzy and random sets, in: Prikladno Mnogomiernii Statisticheskii Analyz, Nauka, Moscow, 1978, pp. 262-280 (in Russian).; A. Orlov, Fuzzy and random sets, in: Prikladno Mnogomiernii Statisticheskii Analyz, Nauka, Moscow, 1978, pp. 262-280 (in Russian).
[68] Goodman, I. R., Some new results concerning random sets and fuzzy sets, Information Sciences, 34, 93-113 (1984) · Zbl 0552.60007
[69] Nguyen, H. T., On modeling of linguistic information using random sets, Information Sciences, 34, 265-274 (1984) · Zbl 0557.68066
[70] Lawry, J., Modelling and Reasoning with Vague Concepts, Studies in Computational Intelligence, vol. 12 (2006), Springer, ISBN 978-0-387-29056-0 · Zbl 1092.68095
[71] Pawlak, Z., Rough Sets—Theoretical Aspects of Reasoning about Data (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0758.68054
[72] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-209 (1990) · Zbl 0715.04006
[73] Radzikowska, A. M.; Kerre, E., A comparative study of fuzzy rough sets, Fuzzy Sets and Systems, 126, 2, 137-155 (2002) · Zbl 1004.03043
[74] Höhle, U., Many-valued equalities, singletons and fuzzy partitions, Soft Computing, 2, 3, 134-140 (1998)
[75] Klawonn, F., Fuzzy points, fuzzy relations and fuzzy functions, (Novak, V.; Perfilieva, I., Discovering the World with Fuzzy Logic (2000), Physica-Verlag GmbH: Physica-Verlag GmbH Heidelberg, Germany), 431-453 · Zbl 1010.03045
[76] Ruspini, E. H., On the semantics of fuzzy logic, International Journal of Approximate Reasoning, 5, 1, 45-88 (1991) · Zbl 0724.03018
[77] Godo, L.; Rodriguez, R. O., Logical approaches to fuzzy similarity-based reasoning: an overview, (Preferences and Similarities, CISM Courses and Lectures, vol. 504 (2008), Springer), 75-128 · Zbl 1187.68607
[78] Dubois, D.; Prade, H., Similarity versus preference in fuzzy set-based logics, (Orlowska, E., Modelling Incomplete Information: Rough Set Analysis, Studies in Fuzziness and Soft Computing (1998), Physica Verlag: Physica Verlag Heidelberg), 441-461
[79] Mendel, J. M., Advances in type-2 fuzzy sets and systems, Information Sciences, 177, 1, 84-110 (2007) · Zbl 1117.03060
[80] Goguen, J. A., \(L\)-fuzzy sets, Journal of Mathematical Analysis and Applications, 18, 145-174 (1967) · Zbl 0145.24404
[81] García, J. J.G.; Rodabaugh, S. E., Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic’’ sets, “intuitionistic’’ fuzzy sets and topologies, Fuzzy Sets and Systems, 156, 3, 445-484 (2005) · Zbl 1084.03042
[82] Gau, D. B.W. L., Vague sets, IEEE Transactions on Systems, Man and Cybernetics, 23, 610-614 (1993) · Zbl 0782.04008
[83] Cornelis, C.; Deschrijver, G.; Kerre, E. E., Advances and challenges in interval-valued fuzzy logic, Fuzzy Sets and Systems, 157, 5, 622-627 (2006) · Zbl 1098.03034
[84] Gasse, B. V.; Cornelis, C.; Deschrijver, G.; Kerre, E. E., Triangle algebras: a formal logic approach to interval-valued residuated lattices, Fuzzy Sets and Systems, 159, 9, 1042-1060 (2008) · Zbl 1174.03028
[85] Walker, C. L.; Walker, E. A., The algebra of fuzzy truth values, Fuzzy Sets and Systems, 149, 2, 309-347 (2005) · Zbl 1064.03020
[86] A.S. Narin’yani, Sub-definite set—New data type for knowledge representation, Memo 4-232, Computing Center, Novosibirsk, Russia, 1980 (in Russian, with an English summary).; A.S. Narin’yani, Sub-definite set—New data type for knowledge representation, Memo 4-232, Computing Center, Novosibirsk, Russia, 1980 (in Russian, with an English summary).
[87] Dubois, D.; Gottwald, S.; Hájek, P.; Kacprzyk, J.; Prade, H., Terminological difficulties in fuzzy set theory—the case of “Intuitionistic Fuzzy Sets’’, Fuzzy Sets and Systems, 156, 3, 485-491 (2005) · Zbl 1098.03061
[88] Cattaneo, G.; Ciucci, D., Basic intuitionistic principles in fuzzy set theories and its extensions (A terminological debate on Atanassov IFS), Fuzzy Sets and Systems, 157, 24, 3198-3219 (2006) · Zbl 1112.03050
[89] de Cooman, G., A behavioural model for vague probability assessments, Fuzzy Sets and Systems, 154, 3, 305-358 (2005) · Zbl 1123.62006
[90] Dubois, D.; Prade, H., Twofold fuzzy sets and rough sets. Some issues in knowledge representation, Fuzzy Sets and Systems, 23, 3-18 (1987) · Zbl 0633.68099
[91] Bellman, R. E.; Zadeh, L. A., Local and fuzzy logics, (Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh (1996), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ, USA), 283-335, ISBN 981-02-2422-2 · Zbl 0382.03017
[92] Prade, H., A computational approach to approximate and plausible reasoning with applications to expert systems, IEEE Transactions on Pattern Analysis and Machine Intelligence, 7, 260-283 (1985), (correction: 747-748) · Zbl 0565.68089
[93] Zadeh, L., Fuzzy logic and approximate reasoning, Synthese, 30, 407-428 (1975) · Zbl 0319.02016
[94] Dubois, D.; Esteva, F.; Godo, L.; Prade, H., Fuzzy-set based logics—an history-oriented presentation of their main developments, (Gabbay, D.; Woods, J., The Many-valued and Nonmonotonic Turn in Logic, Handbook of The History of Logic, vol. 8 (2007), Elsevier), \(325-449, \langle\) http://www.elsevier.com/\( \rangle \)
[95] Hájek, P., The Metamathematics of Fuzzy Logics (1998), Kluwer Academic: Kluwer Academic Dordrecht
[96] Lehmke, S., Degrees of truth and degrees of validity, (Discovering the World with Fuzzy Logic (2001), Physica Verlag: Physica Verlag Heidelberg), 192-237 · Zbl 1005.03029
[97] Hähnle, R., Proof theory of many-valued logic-linear optimization-logic design: connections and interactions, Soft Computing, 1, 3, 107-119 (1997)
[98] Hájek, P., On vagueness, truth-values and fuzzy logics, Studia Logica, 91, 3, 367-382 (2009) · Zbl 1173.03023
[99] Osgood, C. E.; Suci, G.; Tannenbaum, P. H., The Measurement of Meaning (1957), University of Illinois Press: University of Illinois Press Chicago
[100] Cacioppo, J. T.; Gardner, W. L.; Berntson, G. G., Beyond bipolar conceptualizations and measures: the case of attitudes and evaluative space, Personality and Social Psychology Review, 1, 3-25 (1997)
[101] Grabisch, M., The Moebius transform on symmetric ordered structures and its application to capacities on finite sets, Discrete Mathematics, 28, 1-3, 17-34 (2004) · Zbl 1054.06008
[102] Dubois, D.; Prade, H., An introduction to bipolar representations of information and preference, International Journal of Intelligent Systems, 23, 8, 866-877 (2008) · Zbl 1147.68708
[103] Tversky, A.; Kahneman, D., Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5, 297-323 (1992) · Zbl 0775.90106
[104] Amgoud, L.; Prade, H., Using arguments for making and explaining decisions, Artificial Intelligence, 173, 3-4, 413-436 (2009) · Zbl 1343.68219
[105] Dubois, D.; Prade, H., Formal representations of uncertainty, (Bouyssou, D.; Dubois, D.; Pirlot, M.; Prade, H., Decision-making—Concepts and Methods (2009), Wiley), \(85-156, \langle\) http://www.wiley.com/\( \rangle\) Chapter 3 · Zbl 0582.03040
[106] B. Liu, Uncertainty theory, second ed., Studies in Fuzziness and Soft Computing, vol. 154, Springer, Berlin, 2007.; B. Liu, Uncertainty theory, second ed., Studies in Fuzziness and Soft Computing, vol. 154, Springer, Berlin, 2007.
[107] Dubois, D.; Prade, H., An overview of the asymmetric bipolar representation of positive and negative information in possibility theory, Fuzzy Sets and Systems, 160, 1355-1366 (2009) · Zbl 1187.68602
[108] Mamdani, E., Application of fuzzy logic to approximate reasoning using linguistic systems, IEEE Transactions on Computers, 26, 1182-1191 (1977) · Zbl 0397.94025
[109] Dubois, D.; Hüllermeier, E.; Prade, H., Fuzzy set-based methods in instance-based reasoning, IEEE Transactions on Fuzzy Systems, 10, 3, 322-332 (2002)
[110] Dubois, D.; Prade, H., Possibility theory: qualitative and quantitative aspects, (Smets, P., Handbook on Defeasible Reasoning and Uncertainty Management Systems—Volume 1: Quantified Representation of Uncertainty and Imprecision (1998), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht, The Netherlands), 169-226 · Zbl 0924.68182
[111] de Cooman, G.; Aeyels, D., Supremum preserving upper probabilities, Information Sciences, 118, 173-212 (1999) · Zbl 0952.60009
[112] Dubois, D.; Hajek, P.; Prade, H., Knowledge-driven versus data-driven logics, Journal of Logic, Language, and Information, 9, 65-89 (2000) · Zbl 0942.03023
[113] Neumaier, A., Clouds, fuzzy sets and probability intervals, Reliable Computing, 10, 249-272 (2004) · Zbl 1055.65062
[114] Destercke, S.; Dubois, D.; Chojnacki, E., Unifying practical uncertainty representations: II. Clouds, International Journal of Approximate Reasoning, 49, 664-677 (2008) · Zbl 1184.68504
[115] Wang, G.-J.; Liu, H.-W., Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research, 179, 1, 220-233 (2007) · Zbl 1163.90558
[116] Grabisch, M.; Greco, S.; Pirlot, M., Bipolar and bivariate models in multicriteria decision analysis: descriptive and constructive approaches, International Journal of Intelligent Systems, 23, 9, 930-969 (2008) · Zbl 1158.68043
[117] Fortemps, P.; Slowinski, R., A graded quadrivalent logic for preference modelling: Loyola-like approach, Fuzzy Optimization and Decision Making, 1, 93-111 (2002) · Zbl 1091.91504
[118] Öztürk, M.; Tsoukiàs, A., Bipolar preference modeling and aggregation in decision support, International Journal of Intelligent Systems, 23, 9, 970-984 (2008) · Zbl 1153.68474
[119] Dubois, D.; Fargier, H.; Bonnefon, J.-F., On the qualitative comparison of decisions having positive and negative features, Journal of Artificial Intelligence Research, 32, 385-417 (2008) · Zbl 1183.68573
[120] Benferhat, S.; Dubois, D.; Kaci, S.; Prade, H., Bipolar possibility theory in preference modeling: representation, fusion and optimal solutions, Information Fusion, 7, 135-150 (2006)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.