Excluded Middle and Graduation

Abstract

Excluded middle is one of the fundamental laws of the thought according to Aristotle and Boolean algebra. The valued interpretation of this law in the classical (two-valued) case is: an object has or does not have the analyzed property (for example, in the propositional logic: a proposition is either truth or untruth). It is known that the excluded middle is not valid in the frame of the conventional fuzzy logic in a wider sense (fuzzy sets theory, fuzzy logic in narrow sense, fuzzy relations) and, as a consequence, the conventional approaches are not in the Boolean frame (They are not Boolean consistent generalization of the classical case). In this paper, the algebraic explanation of the excluded middle is presented. To each property uniquely corresponds its complementary property (For example: in logic the complement of truth is untruth). Complementary property is determined by (a) excluded middle: it contains everything which analyzed property doesn’t contain (there is nothing between); and by (b) non-contradiction: there is nothing common with the analyzed property. Excluded middle and non-contradiction are the fundamental algebraic principles (value indifferent: valid in all valued realizations)!KeywordsExcluded middleClassical logicConventional fuzzy logicConsistent real-valued realization of Boolean algebra