A Measure of Average sensitivity for fuzzy Logics

Experts usually express their uncertainty by words of natural languages (like "perhaps", "for sure", etc). In the majority of expert systems and intelligent control systems, uncertainty of experts' statements is represented by a number from the interval [0, 1]. There are many different procedures that translate the words that experts use into numbers from [0, 1]. For one and the same word, different procedures can lead to different numbers. Some &– and ∨–operations are very sensitive to this difference in the sense that small changes in t(A) and t(B) can lead to absolutely different estimates for t(A&B) and t(A ∨ B). In view of that, it is reasonable to restrict ourselves to the operations that are the least sensitive to such changes. In this paper, we prove that ab and a + b − ab are "in the average" the least sensitive &– and ∨–operations. This result is in good accordance with the experimental data according to which in. many cases, these operations provide the best description of how experts actually think. We also show how this idea can be applied to other logical connectives (e.g., "not"), and to the choice of membership functions.