Document Type : Research Paper

**Author**

Dept. of Mathematical and Statistical Sciences, University of Colorado Denver, Colorado, USA

**Abstract**

This presentation outlines from a quantitative point of view, the relationships between probability theory, possibility theory, and generalized uncertainty theory, and the role that fuzzy set theory plays in the context of these theories. Fuzzy sets, possibility, and probability entities are de
ned in terms of a function. In each case, these three functions map the real numbers to the interval [0,1]. However, each of these functions are de
ned with di¤erent properties. There are generalizations associated with these three theories that lead to intervals (sets of connected real numbers bounded by two points) and interval functions (sets of functions that are bounded by known upper and lower functions). An interval or interval function encodes the fact that it is unknown which of the points or functions is the point or function in questions, that is, the numerical value or real-valued function is unknown, it is uncertain. For generalizations given by pairs of numbers or functions, a case is made for a particular type of generalized uncertainty theory, interval-valued probability measures, as a way to unify the generalizations of probability, possibility theory, as well as other generalized probability theories via fuzzy intervals and fuzzy interval functions. This presentation brings a new understanding of quantitative fuzzy set theory, possibility theory, probability theory, and generalized uncertainty and gleans from existing research with the intent to organize and further clarify existing approaches.

**Keywords**

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[3] Dubois, D. (2010), The role of fuzzy sets in decision sciences: old techniques and new directions," Fuzzy Sets and Systems, 184:1, pp. 3{28.

[4] Dubois, D. (2014). Uncertainty Theories: A Uni ed View", Plenary talk at FLINS and CBSF III 2014, Jo~ao Pessoa, Brazil.

[5] Dubois, D., Kerre, E., Mesiar,R., and Prade, H. (2000). Chapter 10, Fuzzy Interval Analysis", in Didier Dubois and Henri Prade, (editors) Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Boston.

[6] Dubois, D., Nguyen, H. T., and Prade, H. (2000). Chapter 7, Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps", in Didier Dubois and Henri Prade, (editors) Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Boston.

[7] Dubois, D. and Prade, H. (1982). On several representations of an uncertainty body of evidence", in M. M. Gupta and E. Sanchez, editors. Fuzzy Information and Decision Processes, North-Holland, Amsterdam, pp. 167-181.

[8] Dubois, D. and Prade, H. (1988). Possibility Theory, Plenum Press, New York.

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[10] Dubois, D. and Prade H., editors (2000). Fundamentals of Fuzzy Sets. Kluwer Academic Press.

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[15] Ferson, S., Kreinovich, V., Ginzburg, R., Sentz, K., and Myers, D. S., (2003). Constructing Probability Boxes and Dempster-Shafer Structures. Sandia National Laboratories, Technical Report SAND2002-4015, Albuquerque, New Mexico.

[16] Jamison, K. D., Lodwick, W. A. (2004), Interval-Valued Probability Measures,"UCD/CCM Report No. 213, March 2004.

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[18] Kaufmann, A. and Gupta, M. M. (1985) . Introduction to Fuzzy Arithmetic -Theory and Applications, Van Nostrand Reinhold.

[2] Dubois, D. (2005) On the links between probability and possibility theories", Plenary talk at IFSA 2005, Beijing, China.

[3] Dubois, D. (2010), The role of fuzzy sets in decision sciences: old techniques and new directions," Fuzzy Sets and Systems, 184:1, pp. 3{28.

[4] Dubois, D. (2014). Uncertainty Theories: A Uni ed View", Plenary talk at FLINS and CBSF III 2014, Jo~ao Pessoa, Brazil.

[5] Dubois, D., Kerre, E., Mesiar,R., and Prade, H. (2000). Chapter 10, Fuzzy Interval Analysis", in Didier Dubois and Henri Prade, (editors) Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Boston.

[6] Dubois, D., Nguyen, H. T., and Prade, H. (2000). Chapter 7, Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps", in Didier Dubois and Henri Prade, (editors) Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, Boston.

[7] Dubois, D. and Prade, H. (1982). On several representations of an uncertainty body of evidence", in M. M. Gupta and E. Sanchez, editors. Fuzzy Information and Decision Processes, North-Holland, Amsterdam, pp. 167-181.

[8] Dubois, D. and Prade, H. (1988). Possibility Theory, Plenum Press, New York.

[9] Dubois, D. and Prade, H. (1992). When upper probabilities are possibility measures", Fuzzy Sets and Systems,49, pp. 65-74.

[10] Dubois, D. and Prade H., editors (2000). Fundamentals of Fuzzy Sets. Kluwer Academic Press.

[11] Dubois, D. and Prade, H. (2009), Formal representations of uncertainty," Chapter 3 in. D. Bouyssou, D. Dubois, H. Prade, Editors, Decision-Making Process, ISTE, London, UK & Wiley, Hoboken, N.J., USA.

[12] Dubois, D. and Prade, H. (2011). Mathware&Soft Computing Magazine 18:1, December 2011, pp. 18-71.

[13] Dubois, D., and Prade, H. (2012). Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets." Fuzzy Sets and Systems, 192, pp. 3-24.

[14] Dubois, D. and Prade, H. (2015). Possibility theory and its applications: Where do we stand?" in Kacprzyk, J. and Pedrycz, W., (editors) Handbook of Computational Intelligence, Springer, pp. 31-60.

[15] Ferson, S., Kreinovich, V., Ginzburg, R., Sentz, K., and Myers, D. S., (2003). Constructing Probability Boxes and Dempster-Shafer Structures. Sandia National Laboratories, Technical Report SAND2002-4015, Albuquerque, New Mexico.

[16] Jamison, K. D., Lodwick, W. A. (2004), Interval-Valued Probability Measures,"UCD/CCM Report No. 213, March 2004.

[17] Jamison, K. D. and Lodwick, W. A. (2020) A new approach to interval-valued probability measures, A formal method for consolidating the languages of information de ciency: Foundations", Information Sciences, 507, pp. 86-107.

[18] Kaufmann, A. and Gupta, M. M. (1985) . Introduction to Fuzzy Arithmetic -Theory and Applications, Van Nostrand Reinhold.

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[21] Lodwick, W. A., Neumaier, A. and Newman, F. D., Optimization under uncertainty: Methods and applications in radiation therapy," Proceedings 10th IEEE International Conference on Fuzzy Systems 2001, 3, pp. 1219-1222.

[22] Lodwick, W. A. and Sales Neto, L. (2021). Flexible and Generalized Uncertainty Optimization. Springer-Verlag.

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[25] Klir, G. J., and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, New Jersey.

[26] Moore, R. E., Kearfott R. B. , and Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.

[27] Puri, M. L., and Ralescu, D. (1982). A possibility measure is not a fuzzy measure", Fuzzy Sets and Systems, 7, pp. 311-313.

[28] Shafer G., (1976). A Mathematical Theory of Evidence. Princeton University Press. Princeton N.J.

[29] Sugeno M., (1974). Theory of fuzzy integrals and its applications," Ph.D. Thesis, Tokyo Institute of Technology.

[30] Weichselberger, K. (2000), The theory of interval-probability as a unifying concept for uncertainty," International Journal of Approximate Reasoning 24, pp. 149-170.

[31] Zadeh, L. (1978). Fuzzy sets as a basis for a theory of possibility", Fuzzy Sets and Systems, 1(1), pp. 3-28.

[20] Lodwick, W. A., and Jenkins, O. (2013), Constrained intervals and interval spaces," Soft Computing, 17: 8, pp. 1393-1402.

[21] Lodwick, W. A., Neumaier, A. and Newman, F. D., Optimization under uncertainty: Methods and applications in radiation therapy," Proceedings 10th IEEE International Conference on Fuzzy Systems 2001, 3, pp. 1219-1222.

[22] Lodwick, W. A. and Sales Neto, L. (2021). Flexible and Generalized Uncertainty Optimization. Springer-Verlag.

[23] Klir, G. J., (1989). Is there more to uncertainty than some probability theorists might have us believe?" International Journal of General Systems, 15, pp. 347-378.

[24] Klir, G. J., (1995). Principles of uncertainty: What are they? Why do we need them?" Fuzzy Sets and Systems, 74, pp.15-31.

[25] Klir, G. J., and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, New Jersey.

[26] Moore, R. E., Kearfott R. B. , and Cloud, M. J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.

[27] Puri, M. L., and Ralescu, D. (1982). A possibility measure is not a fuzzy measure", Fuzzy Sets and Systems, 7, pp. 311-313.

[28] Shafer G., (1976). A Mathematical Theory of Evidence. Princeton University Press. Princeton N.J.

[29] Sugeno M., (1974). Theory of fuzzy integrals and its applications," Ph.D. Thesis, Tokyo Institute of Technology.

[30] Weichselberger, K. (2000), The theory of interval-probability as a unifying concept for uncertainty," International Journal of Approximate Reasoning 24, pp. 149-170.

[31] Zadeh, L. (1978). Fuzzy sets as a basis for a theory of possibility", Fuzzy Sets and Systems, 1(1), pp. 3-28.

Special Issue Dedicated to Professor M. Radjabalipour on the occasion of his 75th birthday.

October 2021Pages 73-101

**Receive Date:**12 August 2021**Revise Date:**04 September 2021**Accept Date:**13 September 2021