Why Product "And"-Operation Is Often Efficient: One More Argument

It is an empirical fact that the algebraic product is one the most efficient “and”-operations in fuzzy logic. In this paper, we provide one of the possible explanations of this empirical phenomenon. 1 Formulation of the Problem Fuzzy logic and “and”-operations (t-norms): a brief reminder. To describe the experts’ uncertainty in their statements, Lotfi Zadeh proposed a special formalism of fuzzy logic, in which for each statement, the expert’s degree of certainty in this statement is described by a number from the interval [0, 1]. In this description, 1 means that the expert is absolutely sure that the corresponding statement is true, 0 means that the expert is absolutely sure that the statement is false, and values between 0 and 1 correspond to intermediate degree of confidence; see, e.g., [1, 2, 3]. To make a decision, an expert often uses several statements. For example, he or she may use a rule according to which a certain action need to be taken if two conditions are satisfied, i.e., if the first condition A is satisfied and the second condition B is satisfied. It is therefore desirable to find out how confident is the expert that the corresponding “and”-statement A&B holds, or, more generally, that the “and”-combination A1 & . . . &An holds. Ideally, we should extract these degrees from the experts. However, for n statements, we have 2 − 1 possible “and”-combinations. Already for n = 100, we thus get an astronomical number of combinations, there is no way to ask the expert about each of these combinations. In situations when we cannot explicitly ask an expert about his/her degree of certainty in an “and”-combination A&B, we need to estimate this degree based on the known degrees of certainty a and b in statements A and B. Let us denote this estimate by f&(a, b). This function is known as an “and”-operation, or, alternatively, a t-norm.