Abstract
In recent years there has been a major trend in uncertainty (more specifically, partial belief) modelling emphasizing the idea that the degree of confidence in an event is not totally determined by the confidence in the opposite event, as assumed in probability theory. Possibility theory belongs to this trend that describes partial belief in terms of certainty and plausibility, viewed as distinct concepts. The distinctive features of possibility theory are its computational simplicity, and its position as a bridge between numerical and symbolic theories of partial belief for practical reasoning. The name ‘possibility theory’ was coined by L. A. Zadeh in the late seventies [Zadeh, 1978a] as an approach to uncertainty induced by pieces of vague linguistic information, described by means of fuzzy sets [Zadeh, 1965]. Possibility theory offers a simple, non-additive modelling of partial belief, which contrasts with probability theory. As we shall see, it provides a potentially more qualitative treatment of partial belief since the operations ‘max’ and ‘min’ play a role somewhat analogous to the sum and the product in probability calculus.
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References
M. Akian. Theory of cost measures: Convergence of decision variables. INRIA Report, # 2611, INRIA, 78153 Rocquencourt, France, 1995.
V. Barnett. Comparative Statistical Inference. Wiley, New York, 1973.
R. E. Bellman and M. Giertz. On the analytic formalism of the theory of fuzzy sets. Information Science, 5, 149–157, 1973.
R. Bellman and L. A. Zadeh. Decision making in a fuzzy environment. Management Science, 17, B141 — B164, 1970.
S. Benferhat, D. Dubois and H. Prade. (1992) Representing default rules in possibilistic logic. In Proc. of the 3rd Inter. Conf. on Principles of Knowledge Representation and Reasoning (KR’92), Cambridge, MA, pp. 673–684, 1992.
L. Boldrin and C. Sossai. An algebraic semantics for possibilistic logic. In Proc. of the Uncertainty in Artificial Intelligence Conf, Montreal, pp. 27–35, 1995.
P. Bosc and H. Prade. (1997) An introduction to the fuzzy set and possibility theory-based treatment of soft queries and uncertain or imprecise databases. In Uncertainty Management in Information Systems: From Needs to Solutions, Ph. Smets and A. Motro, eds. pp. 285–324. Kluwer Academic Publ., 1997.
M. Cayrol, H. Farreny and H. Prade. Fuzzy pattern matching. Kybernetes, 11, 103–116, 1982.
L. J. Cohen. A note on inductive logic. The J. of Philosophy, LXX, 27–40, 1973.
L. M. De Campos, J. Gebhardt and R. Kruse. Axiomatic treatment of possibilistic independence. In Symbolic and Quantitative Approaches to Reasoning and Uncertainty, C. Froidevaux and J. Kohlas, eds. pp. 77–88. LNAI 946, Springer Verlag, Berlin, 1995.
L. M. De Campos, M. T. Lamata and S. Moral. The concept of conditional fuzzy measure. Int. J. of Intelligent Systems, 5, 237–246, 1990.
G. De Cooman. (1995) The formal analogy between possibility and probability theory. In: Foundations and Applications of Possibility Theory. In Proc. of the FAPT 95, Ghent, Belgium. G. de Cooman, D. Ruan and E. E. Kerre, eds. pp. 71–87. World Scientific, 1995.
G. De Cooman. Possibility theory Part I: Measure-and integral-theoretics groundwork; Part II: Conditional possibility; Part III: Possibilistic independence. Int. J. of General Systems, 25, 291–371, 1997.
G. De Cooman and D. Aeyels. (1996) On the coherence of supremum preserving upper previsions. In Proc. of the 6th Inter. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (1PMU’96), Granada, Spain, pp. 1405–1410, 1996.
G. De Cooman, D. Ruan and E. E. Kerre, eds. (1995) Foundations and Applications of Possibility Theory (Proc. of the FAPT 95, Ghent, Belgium, Dec. 13–15, 1995). World Scientific, 1995.
De Finetti, 19361 B. De Finetti. La logique de la probabilité. Actes du Congrès Inter. de Philosophie Scientifique, Paris, 1935, Hermann et Cie Editions, IV 1-IV9.
De Finetti, 1937] B. De Finetti. La prévision: Ses lois logiques, ses sources subjectives. Ann. Inst. Poincaré, 7,1–68, 1937. Translated in Studies in Subjective Probability,H. E. Kyburg and J. Smokier, eds. pp. 93–158. Wiley, New York, 1964.
M. Delgado and S. Moral. On the concept of possibility-probability consistency. Fuzzy Sets and Systems, 21, 311–318, 1987.
A. P. Dempster. Upper and lower probabilities induced by a multiple-valued mapping. Annals of Mathematical Statistics, 38, 325–339, 1967.
Di Nola et al.,1989] A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez. Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Kluwer Academic Publ., Dordrecht, 1989.
D. Dubois. Belief structures, possibility theory and decomposable measures on finite sets. Computers and Artificial Intelligence (Bratislava), 5, 403–416, 1986.
D. Dubois, E Dupin de Saint-Cyr and H. Prade. Updating, transition constraints and possibilistic Markov chains. In Advances in Intelligent ComputingIPMU’94, pp. 263–272, LNCS 945, Springer Verlag, Berlin, 1995.
D. Dubois, H. Fargier and H. Prade. Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty. Applied Intelligence, 6, 287–309, 1996.
D. Dubois, H. Fargier and H. Prade. Refinements of the maximin approach to decision-making in fuzzy environment. Fuzzy Sets and Systems, 81, 103–122, 1996.
Dubois et al.,1994] D. Dubois, L. Farinas del Cerro, A. Herzig and H. Prade. An ordinal view of independence with application to plausible reasoning. In Proc. of the 10th Conf. on Uncertainty in Artificial Intelligence,Seattle, WA, R. Lopez de Mantaras and D. Poole, eds. pp. 195–203.1994.
Dubois et al.,1996] D. Dubois, J. Fodor, H. Prade and M. Roubens. Aggregation of decomposable measures with application to utility theory. Theory and Decision,41 59–95,1996.
D. Dubois, J. Lang and H. Prade. Timed possibilistic logic. Fundamenta Informaticae, XV, 211–234, 1991.
D. Dubois, J. Lang and H. Prade. Dealing with multi-source information in possibilistic logic. In Proc. of the 10th Europ. Conf. on Artificial Intelligence (ECAI’92), Vienna, Austria, B. Neumann, ed. pp. 38–42. 1992.
D. Dubois, J. Lang and H. Prade. Automated reasoning using possibilistic logic: Semantics, belief revision and variable certainty weights. IEEE Trans. on Data and Knowledge Engineering, 6, 64–71, 1994.
D. Dubois, J. Lang and H. Prade. Possibilistic logic. In Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3. D. M. Gabbay, C. J. Hogger and J. A. Robinson, eds. pp. 439–513. Oxford University Press, 1994.
D. Dubois, S. Moral and H. Prade. A semantics for possibility theory based on likelihoods. In Proc. of the Inter. Joint Conf of the 4th IEEE Inter. Conf. on Fuzzy Systems (FUZZ-IEEE’95) and the 2nd Inter. Fuzzy Engineering Symp. (IFES’95), Yokohama, Japan, pp. 1597–1604, 1995. A revised and expanded version in J. of Mathematical Analysis and Applications, 205, 359–380, 1997.
D. Dubois, S. Moral and H. Prade. Belief change rules in ordinal and numerical uncertainty theories. In Handbook of Belief Change, Kluwer Academic Publ., to appear, 1997.
D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.
D. Dubois and H. Prade. A class of fuzzy measures based on triangular norms. Int. J. of General Systems, 8, 225–233, 1982.
D. Dubois and H. Prade. On several representations of an uncertain body of evidence. In Fuzzy Information and Decision Processes, M. M. Gupta and E. Sanchez, eds. pp. 167–181. North-Holland, Amsterdam, 1982.
D. Dubois and H. Prade. Fuzzy logics and the generalized modus ponens revisited. Int. J. of Cybernetics and Systems, 15, 293–331, 1984.
D. Dubois and H. Prade. Fuzzy sets and statistical data. European J. Operations Research, 25, 345–356, 1986.
D. Dubois and H. Prade. An alternative approach to the handling of subnormal possibility distributions A critical comment on a proposal by Yager. Fuzzy Sets and Systems, 24, 123–126, 1987.
D. Dubois and H. Prade. Fuzzy numbers: An overview. In The Analysis of Fuzzy Information Vol. 1: Mathematics and Logic, J. C. Bezdek, ed. pp. 3–39. CRC Press, Boca Raton, FL, 1987.
D. Dubois and H. Prade. Twofold fuzzy sets and rough sets Some issues in knowledge representation. Fuzzy Sets and Systems, 23, 3–18, 1987.
D. Dubois and H. Prade. An introduction to possibilistic and fuzzy logics. In Non-Standard Logics for Automated Reasoning, R. Smets, A. Mamdani, D. Dubois and H. Prade, eds. pp. 287–315. Academic Press, 1988. and Reply, pp. 321–326. Reprinted in Readings in Uncertain Reasoning, G. Shafer and J. Pearl, eds. pp. 742–761. Morgan Kaufmann, San Mateo, CA, 1990.
D. Dubois and H. Prade, (with the collaboration of H. Farreny, R. MartinClouaire and C. Testemale). Possibility Theory An Approach to Computerized Processing of Uncertainty. Plenum Press, New York, 1988.
D. Dubois and H. Prade. Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence, 4, 244–264, 1988.
D.Dubois and H. Prade. Fuzzy sets, probability and measurement. Europ. J. of Operations Research, 40, 135–154, 1989.
D.Dubois and H. Prade. Consonant approximations of belief functions. Int. J. Approximate Reasoning, 4, 419–449, 1990.
D.Dubois and H. Prade. Rough fuzzy sets and fuzzy rough sets. Int. J. of General Systems, 17, 191–209, 1990.
D.Dubois and H. Prade. Aggregation of possibility masures. In Multiperson Decision Making using Fuzzy Sets and Possibility Theory, J. Kacprzyk and M. Fedrizzi, eds. pp. 55–63. Kluwer, Dordrecht, 1990.
D.Dubois and H. Prade. Epistemic entrenchment and possibilistic logic. Artificial Intelligence, 50, 223–239, 1991.
D.Dubois and H. Prade. Fuzzy sets in approximate reasoning Part I: Inference with possibility distributions;, Part II (with J. Lang): Logical approaches. Fuzzy Sets and Systems, 40, Part I: 143–202; Part II: 203–244, 1991.
D.Dubois and H. Prade. Fuzzy rules in knowledge-based systems Modelling gradedness, uncertainty and preference. In An Introduction to Fuzzy Logic Applications in Intelligent Systems, R. R. Yager and L.A. Zadeh, eds. pp. 45–68. Kluwer Academic Publ., Dordrecht, 1992.
D.Dubois and H. Prade. When upper probabilities are possibility measures. Fuzzy Sets and Systems, 49, 65–74, 1992.
D. Dubois and H. Prade. Possibility theory as a basis for preference propagation in automated reasoning. In Proceedings of the International Conference on Fuzzy Systems (FUZZ-IEEE’92), San Diego, CA, pp. 821–832, 1992.
D.Dubois and H. Prade. Fuzzy sets and probability: misunderstandings, bridges and gaps. Proc. of the 2nd IEEE Inter. Conf. on Fuzzy Systems (FUZZ-IEEE’93), San Francisco, CA, pp. 1059–1068, 1993.
D. Dubois and H. Prade. Can we enforce full compositionality in uncertainty calculi? Proc. of the 12th National Conf. on Artificial Intelligence (AAAI’94), Seattle, WA, pp. 149–154, 1994.
D. Dubois and H. Prade. Possibility theory and data fusion in poorly informed environments. Control Engineering Practice, 2, 811–823, 1994.
Dubois and Prade, 1995a1 D. Dubois and H. Prade. Conditional objects, possibility theory and default rules. In Conditionals: From Philosophy to Computer Sciences, G. Crocco, L. Farinas del Cerro and A. Herzig, eds. pp. 301–336. Oxford University Press, Oxford, 1995.
D. Dubois and H. Prade. Fuzzy relation equations and abductive reasoning. Fuzzy Sets and Systems, 75, 119–134, 1995.
D. Dubois and H. Prade. Possibility theory as a basis for qualitative decision theory. In Proceedings of the the 14th Inter. Joint Con! on Artificial Intelligence (IJCAI’95), Montréal, pp. 1924–1930. Morgan Kaufmann, 1995.
D. Dubois and H. Prade. Belief revision with uncertain inputs in the possibilistic setting. Proc. of the 12th Conf. on Uncertainty in Artificial Intelligence, Portland, Oregon, E. Horvitz and E Jensen, eds. pp. 236–243. Morgan Kaufmann, San Mateo, CA, 1996.
Dubois and Prade, 1996b1 D. Dubois and H. Prade. Focusing vs. revision in possibility theory. Proc. of the 5th IEEE Inter. Conf. on Fuzzy Systems (FUZZ-IEEE’96), New Orleans, LO, pp. 1700–1705, 1996.
Dubois and Prade, 19971 D. Dubois and H. Prade. An overview of ordinal and numerical approaches to diagnostic problem-solving. In Handbook of Abductive Reasoning and Learning, D. Gabbay and R. Kruse, eds. to appear, 1997.
D. Dubois, H. Prade and S. Sandri. On possibility/probability transformations. In Fuzzy Logic: State of the Art, R. Lowen and M. Lowen, eds. pp. 103–112. Kluwer Academic Publ., 1993.
D. Dubois, H. Prade and C. Testemale. Weighted fuzzy pattern matching. Fuzzy Sets and Systems, 28, 313–332, 1988.
A. W. E. Edwards. Likelihood. Cambridge University Press, Cambridge, UK, 1972. [Elkan, 1993 ] C. Elkan. The paradoxical success of fuzzy logic. Proc. of the National Conf. on Artificial Intelligence (AAAI’93), Washington, DC, pp. 698–703, 1993.
R. Fagin and J. Y. Halpern. A new approach to updating beliefs. Research Report RJ 7222, IBM, Research Division, San Jose, CA, 1989.
H. Fargier. Fuzzy scheduling: principles and experiments. In Fuzzy Information Engineering: A Guided Tour of Applications, D. Dubois, H. Prade and R. R. Yager, eds. pp. 655–668. Wiley, New York, 1997.
L. Farinas del Cerro and A. Herzig. A modal analysis of possibility theory. Proc. of the Inter. Workshop on Fundamentals of Artificial Intelligence Research (FAIR’91), Smolenice Castle, Czechoslovakia, Ph. Jorrand and J. Kelemen, eds. pp. 11–18. Lecture Notes in Computer Sciences, Vol. 535, Springer Verlag, Berlin, 1991.
P. Fonck. Conditional independence in possibility theory. Proc. of the 10th Conf. on Uncertainty in Artificial Intelligence, pp. 221–226, 1994.
Y. Friedman and U. Sandler. Evolution of systems under fuzzy dynamic laws. Preprint No. AM-001. 94, Jerusalem College of Technology, Israel, 1994.
R. Gärdenfors.Knowledge in Flux Modeling the Dynamics ofEpistemic States. The MIT Press, Cambridge, MA, 1988.
J. Gebhardt and R. Kruse. The context model: An integrating view of vagueness and uncertainty. Int. J. of Approximate Reasoning, 9, 283–314, 1993.
J. Gebhardt and R. Kruse. POSSINFER: A software tool for possibilistic inference. In Fuzzy Information Engineering: A Guided Tour of Applications, D. Dubois, H. Prade and R. R. Yager, eds. Wiley, New York, 1997.
R. Giles. Foundations for a theory of possibility. In Fuzzy Information and Decision Processes, M. M. Gupta and E. Sanchez, eds. pp. 183–195. North-Holland, 1982.
I. Hacking. All kinds of possibility. Philosophical Review, 84, 321–347, 1975. [Hisdal, 19781 E. Hisdal. Conditional possibilities Independence and non-interactivity. Fuzzy Sets and Systems, 1, 283–297, 1978.
M. Inuiguchi, H. Ichihashi and H. Tanaka. Possibilistic linear programming with measurable multiattribute value functions. ORSA J. on Computing, 1, 146–158, 1989.
M. Inuiguchi and Y. Kume. Necessity measures defined by level set inclusions: Nine kinds of necessity measures and their properties. lnt. J. of General Systems, 22, 245–275, 1994.
J. Y. Jaffray. Bayesian updating and belief functions. IEEE Trans. on Systems, Man and Cybernetics, 22, 1144–1152, 1992.
Kampé de Fériet. Interpretation of membership functions of fuzzy sets in terms of plausibility and belief. In Fuzzy Information and Decision Processes, M. M. Gupta and E. Sanchez, eds. pp. 93–98. North-Holland, Amsterdam, 1982.
A. Kaufmann and M. M. Gupta. Introduction to Fuzzy Arithmetic Theory and Applications. Van Nostrand Reinhold Publ., New York, 1985.
Klir and Folger, 19881 G. J. Klir and T. Folger. Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, NJ, 1988.
G. J. Klir and B. Parviz. Probability-possibility transformation: A comparison. Int. J. of General Systems, 21, 291–310, 1992.
G. J. Klir and Bo Yuan. Fuzzy Sets and Fuzzy Logic Theory and Applications. Prentice Hall, Upper Saddle River, NJ, 1995.
S. Kraus, D. Lehmann and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207, 1990.
Kruse, Gebhardt and Klawonn, 19941 R. Kruse, J. Gebhardt and E. Klawonn. Foundations of Fuzzy
Systems. John Wiley, Chichester, West Sussex, 1994.
H. E. Kyburg, Jr. Bayesian and non-Bayesian evidential updating. Artificial Intelligence, 31, 271–293, 1987.
D. Lehmann and M. Magidor. What does a conditional knowledge base entail? Artificial Intelligence, 55, 1–60, 1992.
D. K. Lewis. Counterfactuals and comparative possibility. J. Philosophical Logic, 2, 1973.
Lewis, 1973b1 D. K. Lewis. Counterfactuals. Basil Blackwell, Oxford, 1973. 2nd edition, Billing and Sons Ltd., Worcester, UK, 1986.
E. Mamdani. Application of fuzzy logic to approximate reasoning using linguistic systems. IEEE Trans. on Computer, 26, 1182–1191, 1977.
R. Mesiar. On the integral representation of fuzzy possibility measures. Int. J. of General Systems, 23, 109–121, 1995.
Moulin, 1988] H. Moulin. Axioms of Cooperative Decision-Making,Wiley, New York, 1988. [Mundici, 1992] D. Mundici. (1992) The logic of Ulam games with lies. In Knowledge, Belief and
Strategic Interaction,C. Bicchieri and M. Dalla Chiara, eds. pp. 275–284. Cambridge University
Press, 1992.
Narin’yani, 1980] A. S. Narin’yani. Sub-definite sets: New data-type for knowledge representation. (In Russian) Memo no 4–232, Computing Center, Novosibirsk, 1980.
B. Natvig. Possibility versus probability. Fuzzy Sets and Systems, 10, 31–36, 1983.
II. T. Nguyen. On conditional possibility distributions. Fuzzy Sets and Systems, 1, 299–309, 1978.
A. M. Norwich and I. B. Turksen. The fundamental measurement of fuzziness. In Fuzzy Sets and Possibility Theory: Recent Developments, R. R. Yager, ed. pp. 49–60. Pergamon Press, 1982.
Pawlak, 19821 Z. Pawlak. Rough sets. Int. J. of Computer and Information Sciences, 11, 341–356, 1982.
Z. Pawlak. Rough Sets Theoretical Aspects of Reasoning about Data. Kluwer Academic Publ., Dordrecht, 1991.
Pearl, 19901 J. Pearl. System Z: A natural ordering of defaults with tractable applications to default reasoning. Proc. of the 3rd Con!. on the Theoretical Aspects of Reasonig About Knowledge (TARK’90), pp. 121–135. Morgan Kaufmann, 1990.
H. Prade. Nomenclature of fuzzy measures. Proc. of the 1st Inter. Seminar on Theory of Fuzzy Sets, Johannes Kepler Univ., Linz, Austria, 9–25, 1979.
H. Prade. Modal semantics and fuzzy set theory. In Recent Developments in Fuzzy set and Possibility Theory, R. R. Yager, ed. pp. 232–246. Pergamon Press, New York, 1982.
H. Prade and R. R. Yager. Estimations of expectedness and potential surprise in possibility theory. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 2, 417–428, 1994.
A. Ramer. Concepts of fuzzy information measures on continuous domains. Int. J. of General Systems, 17, 241–248, 1990.
N. Rescher. Plausible Reasoning. Van Gorcum, Amsterdam, 1976.
E. Ruspini. On the semantics of fuzzy logic. Int. J. of Approximate Reasoning, 5, 4588, 1991.
Sanchez, 19781 E. Sanchez. On possibility-qualification in natural languages. Information Sciences, 15, 45–76, 1978.
Savage, 1954] L. J. Savage. The Foundations of Statistics. Wiley, New York, 1954.2nd edition, Dover Publications Inc., New York, 1972.
Shackle, 19611 G. L. S. Shackle. Decision, Order and Time, in Human Affairs. 2nd edition, Cambridge University Press, Cambridge, UK, 1961.
Shafer, 19761 G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, 1976.
G. Shafer. Belief functions and possibility measures. In Analysis of Fuzzy Information Vol. 1: Mathematics and Logic, J. Bezdek, ed. pp. 51–84. CRC Press, Boca Raton, FL, 1987.
P. P. Shenoy. Using possibility theory in expert systems. Fuzzy Sets and Systems, 52, 129–142, 1992.
N. Shilkret. (1971) Maxitive measure and integration. Indag. Math., 33, 109–116, 1971.
Y. Shoham. Reasoning About Change. The MIT Press, Cambridge, MA, 1988.
R. Slowinski and J. Teghem, eds. Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty. Kluwer Academic Publ., Dordrecht, 1990.
P. Smets. Possibilistic inference from statistical data. Proc. of the 2nd World Con!. On Mathematics at the Service of Man, Las Palmas (Canary Island), Spain, pp. 611–613, 1982.
Smets, 1990] P. Smets. Constructing the pignistic probability function in a context of uncertainty. In Uncertainty in Artificial Intelligence, 5 ,M. Henrion et al.,eds. pp. 29–39. North-Holland, Amsterdam,1990.
P. Smets and R. Kennes. The transferable belief model. Artificial Intelligence, 66, 191–234, 1994.
Smith, 19611 C. A. B. Smith. Consistency in statistical inference and decision. J. Royal Statist. Soc., B-23, 1–37, 1961.
W. Spohn. Ordinal conditional functions: a dynamic theory of epistemic states. In Causation in Decision, Belief Change and Statistics, W. Harper and B. Skyrms, eds. pp. 105–134,1988. [Stallings, 1977] W. Stallings. Fuzzy set theory versus Bayesian statistics. IEEE Trans. on Systems, Man and Cybernetics, pp. 216–219, 1977.
M. Sugeno. (1977) Fuzzy measures and fuzzy integrals A survey. In Fuzzy Automata and Decision Processes, M. M. Gupta, G. N. Saridis and B. R. Gaines, eds. pp. 89–102. North-Holland, Amsterdam, 1977.
S. F. Thomas. A theory of semantics and possible inference with application to decision analysis. PhD Thesis, University of Toronto, Canada, 1979.
E. Trillas and L. Valverde. On mode and implication in approximate reasoning. In Approximate Reasoning in Expert Systems, M. M. Gupta, A. Kandel, W. Bandler and J. B. Kiszka, eds. pp. 157–166. North-Holland, Amsterdam, 1985.
P. Walley. Statistical Inference with Imprecise Probabilities. Chapman and Hall, London, 1991.
R. Walley. Measures of uncertainty in expert systems. Artificial Intelligence, 83, 1–58, 1996.
P. Walley and T. Fine. Towards a frequentist theory of upper and lower probability. The Annals of Statistics, 10, 741–761, 1982.
Wang, 19831 R Z. Wang. From the Fuzzy Statistics to the Falling Random Subsets. In Advances in Fuzzy Sets, Possibility Theory and Applications, P. P. Wang, ed. pp. 81–96. Plenum Press, New York, 1983.
R. Z. Wang and E. Sanchez. Treating a fuzzy subset as a projectable random subset. In Fuzzy Information and Decision Processes, M. M. Gupta and E. Sanchez, eds. pp. 213219. North-Holland, Amsterdam, 1982.
S. Weber. 1—decomposable measures and integrals for Archimedean t-conorms 1. J. of Math. Anal. Appl., 101, 114–138, 1984.
T. Weston. Approximate truth. J. Philos. Logic, 16, 203–227, 1987.
T. Whalen. Decision making under uncertainty with various assumptions about available information. IEEE Trans. on Systems, Man and Cybernetics, 14, 888–900, 1984.
K. L. Wood, K. N. Otto and E. K. Antonsson. Engineering design calculations with fuzzy parameters. Fuzzy Sets and Systems, 52, 1–20, 1992.
R. R. Yager. Possibilistic decision making. IEEE Trans. on Systems, Man and Cybernetics, 9, 388–392, 1979.
R. R. Yager. An introduction to applications of possibility theory. Human Systems Management, 3, 246–269, 1983.
R. R. Yager. Aggregating evidence using quantified statements. Information Sciences, 36, 179–206, 1985.
R. R. Yager. A modification of the certainty measure to handle subnormal distributions. Fuzzy Sets and Systems, 20, 317–324, 1986.
R. R. Yager. On the specificity of a possibility distribution. Fuzzy Sets and Systems, 50, 279–292, 1992.
L. A. Zadeh. Fuzzy sets. Information and Control, 8, 338–353, 1965.
L. A. Zadeh. Probability measures of fuzzy events. J. Math. Anal. Appl., 23, 421–427, 1968.
Zadeh, 1973] L. A. Zadeh. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems, Man and Cybernetics,3 28–44, 1973.
L. A. Zadeh. Calculus of fuzzy restrictions. In Fuzzy Sets and their Applications to Cognitive and Decision Processes, L. A. Zadeh, K. S. Fu, K. Tanaka and M. Shimura, eds. pp. 139. Academic Press, New York, 1975.
L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28, 1978.
L. A. Zadeh. PRUF A meaning representation language for natural languages. Int. J. of Man-Machine Studies, 10, 395–460, 1978.
Zadeh, 1979a1 L. A. Zadeh. A theory of approximate reasoning. In Machine Intelligence, Vol. 9. J. E. Hayes, D. Michie and L. I. Mikulich, eds. pp. 149–194. Elsevier, New York, 1979.
Zadeh, 1979b] L. A. Zadeh. Fuzzy sets and information granularity. In Advances in Fuzzy Set Theory and Applications,M. M. Gupta, R. Ragade and R. R. Yager, eds. pp. 3–18. North-Holland, Amsterdam,1979.
L. A. Zadeh. Test score semantics for natural languages and meaning representation via PRUF. In Empirical Semantics, Vol. 1. 13. B. Rieger, ed. Brockmeyer, Bochum, 1982.
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Dubois, D., Prade, H. (1998). Possibility Theory: Qualitative and Quantitative Aspects. In: Smets, P. (eds) Quantified Representation of Uncertainty and Imprecision. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1735-9_6
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