How to Take All Available Information into Account in Fuzzy Decision Making: Beyond Traditional “And” – and “Or” – Operations
Vladik Kreinovich and Olga Kosheleva
In the traditional fuzzy logic, we can use “and”-operations (also known as t-norms) to estimate the expert’s degree of confidence in a composite statement A& B based on his/her degrees of confidence d(A) and d(B) in the corresponding basic statements A and B. But what if we want to estimate the degree of confidence in A & B & C in situations when, in addition to the degrees of estimate d(A), d(B), and d(C) of the basic statements, we also know the expert’s degrees of confidence in the pairs d(A & B), d(A & C), and d(B & C)? Traditional “and”-operations can provide such an estimate – but only by ignoring some of the available information. In this paper, we show that, by going beyond the traditional “and”- and “or”-operations, we can find a natural estimate that takes all available information into account – and thus, hopefully, leads to a more accurate estimate.
Keywords: Fuzzy logic, degree of certainty, “and”-operation, t-norm, “or”- operation, t-conorm, maximum entropy approach, probabilistic uncertainty