Real-Valued Realizations of Boolean Algebras Are a Natural Frame for Consistent Fuzzy Logic

Fifty years ago, Lotfi Zadeh, as a visionary, realized a need for radically different mathematics to deal effectively with the systems which were generally orders of magnitude more complex than man-made systems [1]. Soon afterwards, he ingeniously introduced the notion of gradation in the theory of fuzzy sets [2] and in fuzzy logic in a wider sense. The use of fuzzy logic, based on the notion of gradation, first of all, enables a drastic reduction of the complexity immanent to the classical mathematical approaches in real problems. Thus, accordingly, fuzzy approaches are much more feasible due to their lower complexity compared to the classical approaches. Actually, a graded approach is much more descriptive than a classical two-valued (black and white) approach. From my point of view, the main drawback of the conventional fuzzy logics, in a wider sense, based on the truth functional principle, is the fact that they are not within the Boolean framework and hence these approaches are not Boolean consistent generalizations of the classical techniques.