On Łukasiewicz’ Infinie-Valued Logic and Fuzzy _L

Abstract

An algebra of Łukasiewicz’ logic Ł$_{\aleph}$ is considered without the extra operations P ⊙ Q =  max {0, P + Q − 1}, the restricted difference, and P ⊕ Q =  min {1, P + Q}, the restricted sum, later named by ”Łukasiewicz t-norm” and ”Łukasiewicz t-conorm”, respectively. The logic Ł$_{\aleph}$ is compared with the fuzzy logic Fuzzy_L with the set of truth-values [0,1]. It appears that Fuzzy_L is a Łukasiewicz-like many-valued logic with a dual pair of modifiers (or fuzzifiers). Otherwise, the only reasonable version of Fuzzy_L is Ł$_{\aleph}$, without the additional connectives corresponding to the ”Łukasiewicz norms” which are not original operations of Łukasiewicz’ logics.