Abstract
Two different but related measurement problems are considered within the fuzzy set theory. The first problem is the membership measurement find the second is property ranking. These two measurement problems are combined and two axiomatizations of fuzzy set theory are obtained. In the first one, the indifference is transitive but in the second one this drawback is removed by utilizing interval orders.
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References
Bilgiç, T. (1995). Measurement-Theoretic Frameworks for Fuzzy Set Theory with Applications to Preference Modelling, PhD thesis, University of Toronto, Dept. of Industrial Engineering Toronto Ontario M5S 1A4 Canada.
Bilgiç, T. & Türkşen, I. (1995). Measurement-theoretic justification of fuzzy set connectives, Fuzzy Sets and Systems 76(3): 289–308. *http://analogy.ie.utoronto.ca/∼bilgic/tanerpubl.html
Bollmann-Sdorra, P., Wong, S. K. M. & Yao, Y. Y. (1993). A measurement-theoretic axiomatization of fuzzy sets, Fuzzy Sets and Systems 60(3): 295–307.
Dubois, D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets, Computers and Artificial Intelligence 5(5): 403–416.
Dubois, D. (1988). Possibility theory: Searching for normative foundations, in B. R. Munier (ed.), Risk, Decision and Rationality, D. Reidel Publishing Company, pp. 601–614.
Dubois, D. & Prade, H. (1989). Fuzzy sets, probability and measurement, European Journal of Operational Research 40: 135–154.
Fishburn, P. C. (1972). Mathematics of Decision Theory, Mouton, The Hague.
Fishburn, P. C. (1985). Interval Orders and Interval Graphs: a study of partially ordered sets, John Wiley, New York. A Wiley-Interscience publication.
Fodor, J. C. (1993). A new look at fuzzy connectives, Fuzzy Sets and Systems 57: 141–148.
French, S. (1984). Fuzzy decision analysis: Some criticisms, in H. Zimmermann, L. Zadeh & B. Gaines (eds), Fuzzy Sets and Decision Analysis, North Holland, pp. 29–44.
Fuchs, L. (1963). Partially Ordered Algebraic Systems, Pergamon Press, London.
Kamp, J. A. W. (1975). Two theories about adjectives, in E. L. Keenan (ed.), Formal Semantics of Natural Language, Cambridge University Press, London, pp. 123–155.
Kneale, W. C. (1962). The development of logic, Oxford, Clarendon Press, England.
Krantz, D. H. (1991). From indices to mappings: The representational approach to measurement, in D. R. Brown & J. E. K. Smith (eds), Frontiers of Mathematical Psychology: Essays in Honor of Clyde Coombs, Recent Research in Psychology, Springer-Verlag, Berlin, Germany, chapter 1.
Krantz, D. H., Luce, R. D., Suppes, P. & Tversky, A. (1971). Foundations of Measurement, Vol. 1, Academic Press, San Diego.
Ling, C. H. (1965). Representation of associative functions, Publicationes Mathematicae Debrecen 12: 189–212.
Luce, R., Krantz, D., Suppes, P. & Tversky, A. (1990). Foundations of Masurement, Vol. 3, Academic Press, San Diego, USA.
Malinowski, G. (1993). Many-valued Logics, Vol. 25 of Oxford Logic Guides, Oxford University Press, England.
McCall, S. & Ajdukiewicz, K. (eds) (1967). Polish logic, 1920–1939, Oxford, Clarendon P. papers by Ajdukiewicz [and others]; with, an introduction by Tadeusz Kotarbinski, edited by Storrs McCall, translated by B. Gruchman [and others].
Narens, L. (1986). Abstract Measurement Theory, MIT Press, Cambridge, Mass.
Norwich, A. M. & Türkşen, I. B. (1982). The fundamental measurement of fuzziness, in R. R. Yager (ed.), Fuzzy Sets and Possibility Theory: Recent Developments, Pergamon Press, New York, pp. 49–60.
Norwich, A. M. & Türkşen, I. B. (1984). A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Sets and Systems 12: 1–25.
Palmer, F. (1981). Semantics, 2 edn, Cambridge University Press, New York.
Roberts, F. (1979). Measurement Theory, Addison Wesley Pub. Co.
Robinson, A. (1966). Non-standard Analysis, North-Holland Pub. Co., Amsterdam.
Rosser, J. B. & Turquette, A. R. (1977). Many-Valued Logics, Greenwood Press, Westport Connecticut.
Sapir, E. (1944). Grading: a study in semantics, Philosophy of Science 11: 93–116.
Schweizer, B. & Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.
Scott, D. (1976). Does many-valued logic have any use?, in S. Körner (ed.), Philosophy of logic, Camelot Press, Southampton, Great Britain, chapter 2, pp. 64–95. with comments by T.J. Smiley, J.P. Cleave and R. Giles. Bristol Conference on Critical Philosophy, 3d, 1974.
Suppes, P. (1974). The measurement of belief, Journal of Royal Statistical Society Series B 36(2): 160–175.
Suppes, P., Krantz, D., Luce, R. & Tversky, A. (1989). Foundations of Measurement, Vol. 2, Academic Press, San Diego.
Türkşen, I. B. (1991). Measurement of membership functions and their assessment, Fuzzy Sets and Systems 40: 5–38.
Türkşen, I. B. & Bilgiç, T. (1995). Interval valued strict preference with zadeh triples. to appear in the special issue on fuzzy MCDM in Fuzzy Sets and Systems.
Yager, R. R. (1979). A measurement-informational discussion of fuzzy union and intersection, International Journal of Man-Machine Studies 11: 189–200.
Zadeh, L. A. (1965). Fuzzy sets, Information Control 8: 338–353.
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Bilgiç, T., Türkşen, I.B. (1997). Measurement-theoretic frameworks for fuzzy set theory. In: Martin, T.P., Ralescu, A.L. (eds) Fuzzy Logic in Artificial Intelligence Towards Intelligent Systems. FLAI 1995. Lecture Notes in Computer Science, vol 1188. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62474-0_19
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