Min and Max are the Only Continuous $\&$- and $\vee$-Operations for Finite Logics
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Computer Science, University of Texas at El Paso, El Paso, USA.
Keywords: Max, Finite logic, Continuous logical operation, “And”-operation, “Or”-operation, min,
Abstract :
Experts usually express their degrees of belief in their statements by the words of a natural language (like “maybe”, “perhaps”, etc.). If an expert system contains the degrees of beliefs t(A) and t(B) that correspond to the statements A and B, and a user asks this expert system whether “A & B” is true, then it is necessary to come up with a reasonable estimate for the degree of belief of A & B. The operation that processes t(A) and t(B) into such an estimate t(A & B) is called an &-operation. Many different &-operations have been proposed. Which of them to choose? This can be (in principle) done by interviewing experts and eliciting a &-operation from them, but such a process is very time-consuming and therefore, not always possible. So, usually, to choose a &-operation, we extend the finite set of actually possible degrees of belief to an infinite set (e.g., to an interval [0, 1]), define an operation there, and then restrict this operation to the finite set. In this paper, we consider only this original finite set. We show that a reasonable assumption that an &-operation is continuous (i.e., that gradual change in t(A) and t(B) must lead to a gradual change in t(A & B)), uniquely determines min as an &-operation. Likewise, max is the only continuous ∨-operation. These results are in good accordance with the experimental analysis of “and” and “or” in human beliefs.
[1] J. Agusti' et al., Structured local fuzzy logics in MILORD, In: L. Zadeh, J. Kacrpzyk (eds.), Fuzzy Logic for the Management of Uncertainty, Wiley, New York, (1992).
[2] R. Belohlavek, J. W. Dauben and G. J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective, Oxford University Press, New York, (2017).
[3] B. G. Buchanan and E. H. Shortli e, Rule-Based Expert Systems, Addison-Wesley, Reading, Massachusetts, Menlo Park, California, (1984).
[4] N. Bourbaki, General Topology, Addison-Wesley, Reading, Massachusetts, (1967).
[5] H. M. Hersch and A. Camarazza, A fuzzy-set approach to modi ers and vagueness in natural languages, Journal of Experimental Psychology: General, 105 (1976), 254-276.
[6] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, New Jersey, (1995).
[7] V. Kreinovich et al., What non-linearity to choose? Mathematical foundations of fuzzy control, Proceedings of the 1992 International Fuzzy Systems and Intelligent Control Conference, Louisville, Kentucky, (1992) 349-412.
[8] J. M. Mendel, Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions, Springer, Cham, Switzerland, (2017).
[9] H. T. Nguyen, C. L. Walker and E. A. Walker, A First Course in Fuzzy Logic, Chapman and Hall/CRC, Boca Raton, Florida, (2019).
[10] V. Novak, I. Per lieva and J. Mockor, Mathematical Principles of Fuzzy Logic, Kluwer, Boston, Dordrecht, (1999).
[11] G. C. Oden, Integration of fuzzy logical information, Journal of Experimental Psychology: Human Perception Performance, 3(4) (1977), 565-575.
[12] J. Puyol-Gruart, L. Godo and C. Sierra, A specialisation calculus to improve expert systems communication, Research Report IIIA 92/8, Institut d'Investigacio en Intelligenicia Arti cial, Spain, (1992).
[13] G. Shafer and J. Pearl (eds.), Readings in Uncertain Reasoning, M. Kaufmann, San Mateo, California, (1990).
[14] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[15] H. J. Zimmerman, Results of empirical studies in fuzzy set theory, In: G. J. Klir (ed.), Applied General Systems Research, Plenum, New York, (1978), 303-312.
