Interpretation of Łukasiewicz’s Ł3 system with the hidden variable

Abstract

It is generally believed that the law of excluded middle and the law of (non-)contradiction do not apply to Jan Łukasiewicz’s Ł 3 , and many attempts have been made to resolve the issue. Based on the “Interpretation with Hidden Variables” within the framework of the STRF (Special Theory of Relativity of Function), this paper, while fully accepting the syntax and maintaining the truth-value semantics of Ł 3 , demonstrates that the third value in Ł 3 1/2, is actually a combination of “true” and “not true”. Furthermore, this combination can exhibit different arrangements, meaning that there are distinct truth-value distributions corresponding to a single degree of truth. Consequently, it becomes evident that the law of excluded middle and the law of (non-)contradiction remain valid as theorems within Ł 3 . Thus, Ł 3 is simply a classical propositional logic system that has yet to be fully and correctly understood.