On the Imaginary Logic of N. A. Vasil'év

Abstract

This chapter focuses on the imaginary life of Vasil'ēv and emphasizes that Vasil'ēv can be considered as a forerunner of non-classical logics constructed for the study of inconsistent non-trivial theories. According to some of Vasil'ēv insights three propositional calculi (V1, V2, and V3) are constructed, and they are related to propositional calculi of Costa 1974 and of Routley and Heyer 1976. The chapter aims at developing and studying the systems V1, V2, and V3 and presents some results obtained about the corresponding predicate calculi and their extension. A certain formal system is a formalization of a given interpretation of Vasil'ēv imaginary logic. The chapter also presents a summary of Vasil'ēv's ideas that are taken as motivations for the construction of V1, V2, and V3. Opinions of the Russian logic is interpreted in a way that justifies systems as Vasil'ēv propositional calculi. Vasil'ēv's logical system is composed of two parts—that he calls metalogic and ontological basis of logic. Vasil'ēv also tried to show that his imaginary logic with his law of excluded fourth has a classical interpretation, as is the case with the imaginary geometry of Lobachevsky.