Publication Date



Technical Report: UTEP-CS-19-109


In the traditional fuzzy logic, experts' degrees of confidence are described by numbers from the interval [0,1]. Clearly, not all the numbers from this interval are needed: in the whole history of the Universe, there will be only countably many statements and thus, only countably many possible degree, while the interval [0,1] is uncountable. It is therefore interesting to analyze what is the set S of actually used values. The answer depends on the choice of "and"-operations (t-norms) and "or"-operations (t-conorms). For the simplest pair of min and max, any finite set will do -- as long as it is closed under negation 1 &minus a. For the next simplest pair -- of algebraic product and algebraic sum -- we prove that for a finitely generated set, if the "and"-operation is exact, then the "or"-operation is almost always approximate, and vice versa. For other "and"- and "or"-operations, the situation can be more complex.