Each Realistic Continuous Functional Dependence Implies a Relation Between Some Variables: A Theoretical Explanation of a Fuzzy-Related Empirical Phenomenon

Authors

Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-1-2021

Comments

Technical Report: UTEP-CS-21-10b

Published in Julia Rayz, Victor Raskin, Scott Dick, and Vladik Kreinovich (eds.), Explainable AI and Other Applications of Fuzzy Techniques,Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS'2021, West Lafayette, Indiana, June 7-9, 2021, Springer, Cham, Switzerland, 2022, pp. 196-202.

Abstract

In principle, one can have a continuous functional dependence y=f(x1,...,x_n) for which, for each proper subset of n+1 variable x1,...,x_n,y, there is no relation: i.e., for each selection of n variables out of these n+1, all combinations of these n values are possible. However, for fuzzy operations, there is always some non-trivial relation between y and one of the inputs xi; for example, for "and"-operations (t-norms) y=f(x1,x2), we have y ≤ x1; for "or"-operations (t-conorms) y=f(x1,x2) we have x1 ≤ y, etc. In this paper, we prove a general mathematical explanation for this empirical fact.