SOME PROPERTIES AND A NECESSARY AND SUFFICIENT CONDITION FOR EXTENDEDKLEENE-STONE LOGIC FUNCTIONS

Recently, fuzzy logic which is a kind of infinite multiple-valued logi c has been studied to treat certain ambiguities, and its algebraic pro perties have been studied by the name of fuzzy logic functions. In ord er to treat modality (necessity, possibility) in fuzzy logic, which is an important concept of multiple-valued logic, the intuitionistic log ical negation is required in addition to operations of fuzzy logic. In finite multiple-valued logic functions introducing the intuitionistic logical negation into fuzzy logic functions are called Kleene-Stone lo gic functions, and they enable us to treat modality. The domain of mod ality in which Kleene-Stone logic functions can handle, however, is to o limited. We will define alpha-KS logic functions as infinite multipl e-valued logic functions using a unary operation instead of the intuit ionistic logical negation of Kleene-Stone logic functions. In alpha-KS logic functions, modality is closer to our feelings. In this paper we will show some algebraic properties of alpha-KS logic functions. In p articular we prove that any n-variable alpha-KS logic function is dete rmined uniquely by all inputs of 7 values which are 7 specific truth v alues of the original infinite truth values. This means that there is a bijection between the set of alpha-KS logic functions and the set of 7-valued alpha-KS logic functions which are restriction of alpha-KS l ogic functions to 7 specific truth values. Finally, we show a necessar y and sufficient condition for a 7-valued logic function to be a 7-val ued alpha-KS logic function.