On a Class of Unary Operators in Continuous-Valued Logic

Abstract

The unary operators play an important role in continuous-valued logic and in artificial intelligence as well. Based on our previous results concerning these operators, we prove here that (a) the Pliant negation operator (also known as the Dombi form of negation); (b) the substantiating, weakening, modal and linguistic hedge operators; (c) the sharpness operator; and (d) the preference operator can all be written in a common form, which is called the kappa function. The kappa function is an operator class-dependent, universal operator. Here, a sufficient condition for the identity of two kappa functions is presented. Also, we provide the condition for which the conjunctive and disjunctive forms of the kappa function coincide. Next, we demonstrate that the inverse of a kappa function is a kappa function as well. Then, we show that for certain conditions, a set of kappa functions is closed under the composition and conjunctive (or disjunctive) operations. Also, we briefly describe two special cases of the kappa function: the product and the Dombi operator case; and we point out that its extended versions can be applied in various areas.