Measurement-theoretic frameworks for fuzzy set theory

  • Taner Bilgiç
  • I. Burhan Türkşen
Theoretical Developments
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1188)


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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  1. 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.Google Scholar
  2. Bilgiç, T. & Türkşen, I. (1995). Measurement-theoretic justification of fuzzy set connectives, Fuzzy Sets and Systems 76(3): 289–308. *∼bilgic/tanerpubl.htmlGoogle Scholar
  3. 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.Google Scholar
  4. Dubois, D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets, Computers and Artificial Intelligence 5(5): 403–416.Google Scholar
  5. 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.Google Scholar
  6. Dubois, D. & Prade, H. (1989). Fuzzy sets, probability and measurement, European Journal of Operational Research 40: 135–154.Google Scholar
  7. Fishburn, P. C. (1972). Mathematics of Decision Theory, Mouton, The Hague.Google Scholar
  8. Fishburn, P. C. (1985). Interval Orders and Interval Graphs: a study of partially ordered sets, John Wiley, New York. A Wiley-Interscience publication.Google Scholar
  9. Fodor, J. C. (1993). A new look at fuzzy connectives, Fuzzy Sets and Systems 57: 141–148.Google Scholar
  10. 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.Google Scholar
  11. Fuchs, L. (1963). Partially Ordered Algebraic Systems, Pergamon Press, London.Google Scholar
  12. 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.Google Scholar
  13. Kneale, W. C. (1962). The development of logic, Oxford, Clarendon Press, England.Google Scholar
  14. 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.Google Scholar
  15. Krantz, D. H., Luce, R. D., Suppes, P. & Tversky, A. (1971). Foundations of Measurement, Vol. 1, Academic Press, San Diego.Google Scholar
  16. Ling, C. H. (1965). Representation of associative functions, Publicationes Mathematicae Debrecen 12: 189–212.Google Scholar
  17. Luce, R., Krantz, D., Suppes, P. & Tversky, A. (1990). Foundations of Masurement, Vol. 3, Academic Press, San Diego, USA.Google Scholar
  18. Malinowski, G. (1993). Many-valued Logics, Vol. 25 of Oxford Logic Guides, Oxford University Press, England.Google Scholar
  19. 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].Google Scholar
  20. Narens, L. (1986). Abstract Measurement Theory, MIT Press, Cambridge, Mass.Google Scholar
  21. 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.Google Scholar
  22. 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.Google Scholar
  23. Palmer, F. (1981). Semantics, 2 edn, Cambridge University Press, New York.Google Scholar
  24. Roberts, F. (1979). Measurement Theory, Addison Wesley Pub. Co.Google Scholar
  25. Robinson, A. (1966). Non-standard Analysis, North-Holland Pub. Co., Amsterdam.Google Scholar
  26. Rosser, J. B. & Turquette, A. R. (1977). Many-Valued Logics, Greenwood Press, Westport Connecticut.Google Scholar
  27. Sapir, E. (1944). Grading: a study in semantics, Philosophy of Science 11: 93–116.Google Scholar
  28. Schweizer, B. & Sklar, A. (1983). Probabilistic Metric Spaces, North-Holland, Amsterdam.Google Scholar
  29. 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.Google Scholar
  30. Suppes, P. (1974). The measurement of belief, Journal of Royal Statistical Society Series B 36(2): 160–175.Google Scholar
  31. Suppes, P., Krantz, D., Luce, R. & Tversky, A. (1989). Foundations of Measurement, Vol. 2, Academic Press, San Diego.Google Scholar
  32. Türkşen, I. B. (1991). Measurement of membership functions and their assessment, Fuzzy Sets and Systems 40: 5–38.Google Scholar
  33. 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.Google Scholar
  34. Yager, R. R. (1979). A measurement-informational discussion of fuzzy union and intersection, International Journal of Man-Machine Studies 11: 189–200.Google Scholar
  35. Zadeh, L. A. (1965). Fuzzy sets, Information Control 8: 338–353.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Taner Bilgiç
    • 1
  • I. Burhan Türkşen
    • 1
  1. 1.University of TorontoTorontoCanada

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