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

Abstract

In principle, one can have a continuous functional dependence $y=f(x_1,\ldots ,x_n)$ for which, for each proper subset of $n+1$ variable $x_1,\ldots ,x_n,y$, 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 $x_i$; for example, for “and”-operations (t-norms) $y=f_ \& (x_1,x_2)$, we have $y\le x_1$; for “or”-operations (t-conorms) $y=f_\vee (x_1,x_2)$ we have $x_1\le y$, etc. In this paper, we prove a general mathematical explanation for this empirical fact.