Fuzzy Logic pp 363-385 | Cite as

Sources, Measurements and Models of Type 2 Fuzziness in the New Millennium

  • I. Burhan Türkşen
Conference paper
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 81)


There are two main sources of Type 2 fuzziness: (1) the acquisition of membership functions for linguistic values of linguistic variables, and (2) the combination of linguistic values with linguistic operators. First, it has been shown that acquisition of membership functions, whether (1) they are obtained by subjective measurement experiments, such as direct or reverse rating procedures or else (2) they are obtained with the application of fuzzy clustering methods, they all reveal a scatter plot, which ought to be captured with Type 2 membership functions. Type 1 membership functions are a result of reducing the information content and uncertainty embedded within scatter points via curve fitting or averaging techniques, which ignore the spread and hence, the content and uncertainty embedded within scatter points. Type 2 fuzziness can be represented either with interval-valued Type 2 or with “full” Type 2 membership functions, which specify gradations between the upper and lower bounds of the interval of the spread. Secondly, it has been shown that the combination of linguistic values with linguistic operators, “AND”, “OR”, “IMP”, etc., as opposed to crisp connectives that are known as t-norms and t-conorms and standard negation, lead to the generation of Fuzzy Disjunctive and Conjunctive Canonical Forms, FDCF and FCCF, respectively. It is noted that in current literature most authors start with Type 1 membership functions and end up with Type 1 membership functions in knowledge representation and approximate reasoning. This approach naturally ignores the spread of gradation both in knowledge representation and in approximate reasoning. Some researchers start out with Type 1 membership representation, but uses FDCF and FCCF and end up with interval-valued Type 2 reasoning results. More recently, a few researchers finally began to capture Type 2 knowledge representation and develop reasoning schemas that determine Type 2 consequences. It is to be forecasted that in the new millennium more and more researchers will attempt to capture Type 2 representation and end up with Type 2 conclusions that reveal the information content available in information granules, as well as expose the risk associated with the gradation between the lower and the upper membership degrees. In turn, this will entail more realistic system model developments, which will help explore computing with perceptions.


Membership Function Knowledge Representation Fuzzy Cluster Conjunctive Normal Form Fuzzy Logic System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bilgic, T., Türkşen, I.B. (2000) Measurement of Membership Functions: Theoretical and Empericial Work, in: Handbook of Fuzzy Theory, D. Dubois and H. Prade (eds.), 195–230.Google Scholar
  2. Bilgic, T., Türken, I.B. (1977) Measurement-Theoretical Frameworks in Fuzzy Theory, in: Fuzzy Logic in Artificial Intelligence, A. Ralescue, T. Martin (eds.), Springer-Verlag, Berlin, 552–565.Google Scholar
  3. Bilgic, T., Türken, I.B. (1997) Elicitation of Membership Functions: How Far Can Theory Take us?, Proceedings of Fuzzy-IEEE ‘87, July 1–5, Barcelona, Spain, Vol. III, 1321–1325.Google Scholar
  4. Bilgic, T., Türkşen, I.B. (1995) Measurement-Theoretical Justification of Connectives in Fuzzy Set Theory, Fuzzy Sets and Systems, 76, 289–307.MathSciNetMATHCrossRefGoogle Scholar
  5. Burillo, P., Bustince, H. (1996) Entropy on Intuitionistic Fuzzy Sets and on Fuzzy Sets and on Interval Valued Fuzzy Sets, Fuzzy Sets and Systems, 78, 305–316.MathSciNetMATHCrossRefGoogle Scholar
  6. Burillo, P., Bustince, H. (1995) Intuitionistic Fuzzy Relations Part I, Mathware Soft Comput. 2, 5–38.MathSciNetMATHGoogle Scholar
  7. Bustince, H., Burillo, P. (1996) Interval Valued Fuzzy Relations in a Set Structures, J. Fuzzy Math., 4, 765–785.MathSciNetMATHGoogle Scholar
  8. Gorzalczany, M.B. (1987) A method of Inference in Approximate Reasoning Based on Interval-valued, Fuzzy Sets Systems, 21, 1–17.MathSciNetMATHCrossRefGoogle Scholar
  9. Hisdal, E. (1981) The IF THEN ELSE Statement and Interval-valued Fuzzy Sets of Higher Type, Int’l Journal Man-Machine Studies, 15, 385–455.MathSciNetMATHCrossRefGoogle Scholar
  10. John, R.I. (1998) Type 2 Fuzzy Sets: An Appraisal of Theory and Applications, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.6, No. 6, December, 563–576.Google Scholar
  11. John, R.I., Innocent, P.R., Barns, M.R. (1998) Type 2 Fuzzy Sets and Neuro-fuzzy Clustering of Radiographic Tibia Images, 1998 IEEE International Conference on Fuzzy Systems, Anchorage, AK. May, 1373–1376.Google Scholar
  12. Karnik, N.N., Mendel, J.M. (1998) Type 2 Fuzzy Logic Systems: Type-Reduction, Presented at 1998 IEEE SMC Conference, San Diego, CA, October.Google Scholar
  13. Karnik, N.N., Mendel, J.M. (1998) Introduction to Type 2 Fuzzy Logic Systems, Presented at1998 IEEE FUZZ Conference, Anchorage, AK. May.Google Scholar
  14. Karnik, N.N., Mendel, J.M., Liang, Q. (1999) Type 2 Fuzzy Logic System“, IEEE Trans. Fuzzy Systems. Vol. 7, No. 5, October.Google Scholar
  15. Liang, Q., Mendel, J.M. (2000) Interval Type 2 Fuzzy Logic System, Proceedings of the 9` IEEE International Conference on Fuzzy Systems, 7–10 May San Antonio, Texas, 328–333.Google Scholar
  16. Liang, Q., Mendel, J.M. (2000) Decision Feedback Equalizers for Non-linear Time Varying Channels using Type 2 Fuzzy Adaptive Filters, Proceedings of the 9 th IEEE International Conference on Fuzzy Systems, 7–10 May, San Antonio, Texas, 883–888.Google Scholar
  17. Marinos, P.N. (1969) Fuzzy Logic and its Application to Switching Systems, IEEE-Trans Comput. 4, 343–348.CrossRefGoogle Scholar
  18. Menger, K. (1942) Statistical Metrics, in: Proceedings of the N. A. S, Vol. 28.Google Scholar
  19. Mizumoto, M, Tanaka, K. (1976) Some Properties of Fuzzy Sets of Type 2, Information and Control, 31, 312–340.MathSciNetMATHCrossRefGoogle Scholar
  20. Norwich, A.M., Türkşen, I.B. (1982) The Fundamental Measurement of Fuzziness, in: Fuzzy Sets and Possibility Theory, R.R. Yager (ed.), Pergamon Press, New York, 49–60.Google Scholar
  21. Norwich, A.M., Türkşen, I.B. (1982) The Construction of Membership Functions, in: Fuzzy Sets and Possibility Theory, R.R. Yager (ed.), Pergamon Press, New York, 61–67.Google Scholar
  22. Norwich, A.M., Türkşen, I.B. (1982) Stochastic Fuzziness, in: Fuzzy Information and Decision Processes, M.M. Gupta and E.E. Sanches (eds.), North Holland, Amsterdam, 13–22.Google Scholar
  23. 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.MathSciNetMATHCrossRefGoogle Scholar
  24. Resconi, G., Türk,şen, I.B. (2001) Canonical Forms of Fuzzy Truthoods by Meta-Theory Based Upon Modal Logic, Information Sciences, 131, 157–194.MathSciNetMATHGoogle Scholar
  25. Roy, M.K., Biswas, R. (1992) I-v Fuzzy Relations and Sanchez’s Approach for Medical Diagnosis, Fuzzy Sets Systems, 47, 35–38.MathSciNetMATHCrossRefGoogle Scholar
  26. Türk,şen, I.B. (1979) Measurement of Linguistic Variables, Proceedings of the 23 rd Annual North American Meeting, Society for General Systems Research. January 3–6, Houston, Texas, 278–284.Google Scholar
  27. Türkşen, I.B., Norwich, A.M. (1981) Measurement of Fuzziness, Proceedings of the International Conference on Policy Analysis and Information Systems, August 19–22, Taipei, Taiwan, Meadea Enterprises Co., Ltd. Taipei, Taiwan, 745–754.Google Scholar
  28. Türkşen, I.B. (1982) Stochastic and Fuzzy Sets, Proceedings of the 2’’ World Conference on Mathematics at the Service of Man. June 28-July 3, Las Palmas, Canary Islands, Spain, 649–654.Google Scholar
  29. Türkşen, I.B., Yao, D.D. (1982) Bounds on Fuzzy Inference, Proceedings of the Sixth European Meeting on Cybernetics and System Research, Vienna, April 13–16, 729–734.Google Scholar
  30. Türkşen, I.B. (1983) Inference Regions for Fuzzy Propositions, in: Advances in Fuzzy Sets, Possibility Theory and Applications, P.P.Wang (ed), Plenum Press, New York, 137–148.Google Scholar
  31. Türk,şen, I.B. (1986) Interval-valued Fuzzy Sets Based on Normal Forms“, Fuzzy Sets Systems, 20, 191–210.CrossRefGoogle Scholar
  32. Türkşen, I.B., Zhong, Z. (1990) An Approximate Analogical Reasoning Schema Based on Similarity Measures and Interval-valued Fuzzy Sets“, Fuzzy Sets Systems, 34, 323–346.CrossRefGoogle Scholar
  33. Türkşen, I.B. (1992) Interval-valued Fuzzy Sets and `Compensatory AND“, Fuzzy Sets Systems, 51, 295–307.CrossRefGoogle Scholar
  34. Türkşen, I.B. (1983) Measurement of Fuzziness: An Interpretation of the Azioms of Measurement, Proceedings of the IFAC Symposium, Marseill, France, July 19–21, 97–102.Google Scholar
  35. Türkşen, I.B. (1991) Measurement of Membership Functions and Their Acquisitions, Fuzzy Sets and Systems, 40, 5–38.MathSciNetCrossRefGoogle Scholar
  36. Türk,şen, I.B. (1995) Fuzzy Normal Forms, Fuzzy Sets and Systems, 69, 319–346.MathSciNetMATHCrossRefGoogle Scholar
  37. Türkşen, I.B., Bilgic, T. (1996) Interval-valued Strict Preference with Zadeh Triples, Fuzzy Sets Systems, 78, 183–195.MATHCrossRefGoogle Scholar
  38. Türkyen, I.B., Kandel A., Zhang, Y.Q. (1998) Universal Truth Tables and Normal Forms, IEEE- Trans. On Fuzzy Systems, 6, 2, 295–303.CrossRefGoogle Scholar
  39. Türkşen, I.B., Kandel, A., Zhang, Y.Q. (1999) Normal Forms of Fuzzy Middle and Fuzzy Contradiction, IEEE-SMC, 29, Part B (Cybernetics) 2, 237–253CrossRefGoogle Scholar
  40. Türkşen, I.B. (1999) Theories of Set and Logic with Crisp or Fuzzy Information Granules, Journal of Advance Computational Intelligence, 3, 4, 264–273.Google Scholar
  41. Wagenknecht, M., Haitmann, K. (1988) Application of Fuzzy Sets of Type 2 to The Solution of Fuzzy Equation Systems, Fuzzy Sets and Systems, 25, 183–190.MathSciNetMATHCrossRefGoogle Scholar
  42. Gerhrk, M., Walker, C., Walker, E. (2000) Fuzzy Normal Forms and Truth Tables, Proceedings of JCIS-2000, Feb. 27-March 3, Atlantic City, New Jersey, 211–214.Google Scholar
  43. Yager, R.R. (1980) Fuzzy Subsets of Type II in Decisions, J. of Cybernetics, 10, 137–159.MathSciNetCrossRefGoogle Scholar
  44. Zadeh, L.A. (1965) Fuzzy Sets, Information and Control Systems, 8, 338–353.MathSciNetMATHCrossRefGoogle Scholar
  45. Zadeh, L.A. (1973) Outline of a New Approach to The Analysis of Complex Systems and Decision Processes Interval-valued Fuzzy Sets“, IEEE Trans. Syst. Man Cybernet. 3 (1973) 28–44.MathSciNetMATHGoogle Scholar
  46. Zadeh, L.A. (1975) The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Part II, Information Sci., 8, 301–357.MathSciNetMATHCrossRefGoogle Scholar
  47. Zadeh, L.A. (1996) Fuzzy Logic=Computing With Words, IEEE-Trans on Fuzzy Systems, 4, 2, 103–111.MathSciNetCrossRefGoogle Scholar
  48. Zadeh, L.A. (2000) Computing with Perceptions, Keynote Address, IEEE-Fuzzy Theory Conference, San Antonio, May 7–10.Google Scholar
  49. Zimmermann, H.J., Zysno, P. (1980), Latent Connectives in Human Decision Making, Fuzzy Sets and Systems, 4, 37–51.MATHCrossRefGoogle Scholar
  50. Zuo, Q., Türkşen, I.B., Nguyen, H.T., et al. (1995) In Expert Systems, Even If We Fix AND/OR Operations, A Natural Answer to a Composite Query is The Interval of Possible Degrees of Belief, Reliable Computing, Supplement (Extended Abstracts of APIC ‘85: International Workshop on Applications of Interval Computations), El Paso, TX, February 23–25, 236–240.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • I. Burhan Türkşen
    • 1
  1. 1.Information / Intelligent Systems Laboratory Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

Personalised recommendations