Generated Connectives in Many Valued Logic

Abstract

Many valued logic is based on the truth values range and the corresponding connectives of conjunction, disjunction, negation, implication, etc. Depending on the corresponding required properties, several relationships between MV connectives occur. So, e.g., de Morgan rules brings together conjunction, disjunction and negation; residuation relates conjunction and implication; negation is often required to be consistent with the corresponding implication; etc. On the other hand, up to the negation, all other mentioned connectives are binary operations and their real evaluation may be rather time consuming. Therefore a representation by means of (one argument) functions which are called generators is considered. Recall such a wellknown representation of Archimedean continuous t-norms (as conjunctions on [0,1]) or t-conorms (as disjunctions on [0,1]) which is due to Ling [7] and in its general form was shown already by Mostert and Shields [9]. Note that the choice of an additive generator (which is strictly decreasing) of a continuous Archimedean t-norm T is unique up to a positive multiplicative constant. Further, if a continuous Archimedean t-norm T is generated by an additive generator f, then the dual t-conorm S, S(x, y) = 1 − T(1 − x, 1 − y) is generated by an additive generator g = f(1 − x). More, the corresponding residual implicator I is also defined by means of f, I(x, y) = f ⁽⁻¹⁾(max(0, f(y) − f(x))). See [2].