Abstract
Fuzzy quantification is a linguistic granulation technique capable of expressing the global characteristics of a collection of individuals, or a relation between individuals, through meaningful linguistic summaries. However, existing approaches to fuzzy quantification fail to provide convincing results in the important case of two-place quantification (e.g. “many blondes are tall”). We develop an axiomatic framework for fuzzy quantification which complies with a large number of linguistically motivated adequacy criteria. In particular, we present the first models of fuzzy quantification which provide an adequate account of the “hard” cases of multiplace quantifiers, non-monotonic quantifiers, and non-quantitative quantifiers, and we show how the resulting operators can be efficiently implemented based on histogram computations.1
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References
J. Barwise and R. Cooper. Generalized quantifiers and natural language. Linguistics and Philosophy, 4: 159–219, 1981.
J. van Benthem. Questions about quantifiers. J. of Symb. Logic, 49, 1984.
I. Bloch. Information combination operators for data fusion: a comparative review with classification. IEEE Transactions on Systems, Man, and Cybernetics, 26 (l): 52–67, 1996.
P. Bosc and L. Lietard. Monotonie quantified statements and fuzzy integrals. In Proc. of the NAFIPS/IFI/NASA ‘84 Joint Conference, pages 8–12, San Antonio, Texas, 1994.
Z.P. Dienes. On an implication function in many-valued systems of logic. Journal of Symbolic Logic, 14: 95–97, 1949.
D. Dubois, H. Prade, and R.R. Yager, editors. Fuzzy Information Engineering. Wiley, New York, 1997.
B.R. Gaines. Foundations of fuzzy reasoning. Int. J. Man-Machine Studies, 8: 623–668, 1978.
I. Glöckner. Advances in DFS theory. TR2000–01, Technical Faculty, University Bielefeld, 33501 Bielefeld, Germany, 2000.
I. Glöckner. A broad class of standard DFSes. TR2000–02, Technical Faculty, University Bielefeld, 33501 Bielefeld, Germany, to appear.
I. Glöckner. DFS - an axiomatic approach to fuzzy quantification. TR97–06, Technical Faculty, University Bielefeld, 33501 Bielefeld, Germany, 1997.
I. Glöckner. A framework for evaluating approaches to fuzzy quantification. TR99–03, Technical Faculty, University Bielefeld, 33501 Bielefeld, Germany, 1999.
I. Glöckner and A. Knoll. Query evaluation and information fusion in a retrieval system for multimedia documents. In Proceedings of Fusion’99, pages 529–536, Sunnyvale, CA, 1999.
I. Glöckner and A. Knoll. Architecture and retrieval methods of a search assistant for scientific libraries. In W. Gaul and R. Decker, editors, Classification and Information Processing at the Turn of the Millennium. Springer, Heidelberg, 2000.
G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, NJ, 1995.
Y. Liu and E.E. Kerre. An overview of fuzzy quantifiers. (I), interpretations. Fuzzy Sets and Systems, 95: 1–21, 1998.
M. Mukaidono. On some properties of fuzzy logic. Syst.—Comput.—Control, 6 (2): 36–43, 1975.
A.L. Ralescu. A note on rule representation in expert systems. Information Sciences, 38: 193–203, 1986.
D. Ralescu. Cardinality, quantifiers, and the aggregation of fuzzy criteria. Fuzzy Sets and Systems, 69: 355–365, 1995.
D. Rasmussen and R. Yager. A fuzzy SQL summary language for data discovery. In Dubois et al. [6], pages 253–264.
B. Schweizer and A. Sklar. Probabilistic metric spaces. North-Holland, Amsterdam, 1983.
W. Silvert. Symmetric summation: A class of operations on fuzzy sets. IEEE Transactions on Systems, Man, and Cybernetics, 9: 657–659, 1979.
H. Thiele. On T-quantifiers and S-quantifiers. In The Twenty-Fourth International Symposium on Multiple-Valued Logic, pages 264–269, Boston, MA, 1994.
R.R. Yager. Quantified propositions in a linguistic logic. Int. J. Man-Machine Studies, 19: 195–227, 1983.
R.R. Yager. Approximate reasoning as a basis for rule-based expert systems. IEEE Trans, on Systems, Man, and Cybernetics, 14 (4): 636–643, Jul./Aug. 1984.
R.R. Yager. On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Trans, on Systems, Man, and Cybernetics, 18 (1): 183–190, 1988.
R.R. Yager. Connectives and quantifiers in fuzzy sets. Fuzzy Sets and Systems, 40: 39–75, 1991.
R.R. Yager. Counting the number of classes in a fuzzy set. IEEE Trans, on Systems, Man, and Cybernetics, 23 (l): 257–264, 1993.
R.R. Yager. Families of OWA operators. Fuzzy Sets and Systems, 59: 125–148, 1993.
L.A. Zadeh. The concept of a linguistic variable and its application to approximate reasoning. Information Sciences, 8,9:199–249, 301–357, 1975.
L.A. Zadeh. A theory of approximate reasoning. In J. Hayes, D. Michie, and L. Mikulich, editors, Mach. Intelligence, volume 9, pages 149–194. Halstead, New York, 1979.
L.A. Zadeh. A computational approach to fuzzy quantifiers in natural languages. Computers and Math. withAppl, 9: 149–184, 1983.
L.A. Zadeh. Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions. IEEE Trans, on Systems, Man, and Cybernetics, 15 (6): 754–763, 1985.
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Glöckner, I., Knoll, A. (2001). A Formal Theory of Fuzzy Natural Language Quantification and its Role in Granular Computing. In: Pedrycz, W. (eds) Granular Computing. Studies in Fuzziness and Soft Computing, vol 70. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1823-9_10
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DOI: https://doi.org/10.1007/978-3-7908-1823-9_10
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