Minimization of Average Sensitivity as a Method of Selecting Fuzzy Functions and Operations: Successes and Limitations

Fuzzy logic is an extension of the standard 2-valued logic – with two possible truth values 0 (“false”) and (“true”) – to values (degrees of certainty) represented by arbitrary numbers from the interval [0, 1]. One of the main challenges in fuzzy logic is that we need to extend the usual logical operations from the set {0, 1} to the entire interval, and there are many possible extensions. One promising technique for selecting a reasonable extension is to take into account that the fuzzy degrees of certainty are themselves only known with uncertainty; so, it makes sense to select an operation which is, on average, the least sensitive to the corresponding uncertainty. This technique has successfully worked in selecting unary and binary operations and in selecting membership functions. In this paper, we show, however, that this minimization technique does not work well for selecting ternary operations, and that in the discrete case, the results of applying this technique are somewhat counterintuitive. c ⃝2014 World Academic Press, UK. All rights reserved.