Existence Judgments and Nikolai Vasiliev's Imaginary Logic

Abstract

The paper attempts to explicate the judgements of N.A. Vasiliev's imaginary logic with the usage of the existence judgments. The existential judgment in traditional logic was understood as a judgement, the predicate of which is the term “exists” and the subject is a certain term or a sequence of terms. According to some predecessors of modern logic, existential judgements have a more fundamental status than attributive judgements, so that the latter can be reduced to the former. As a modern reconstruction of imaginary logic, this paper considers the IL suggested by T.P. Kostyuk and V.I. Markin in the article “Formal Reconstruction of Imaginary Logic by N.A. Vasiliev” (1998). On the base of the analysis of Vasiliev's texts, it is shown that the formal semantics of this system quite well corresponds to the interpretation of judgements of imaginary logic which was kept in mind by the author himself. The way the language is constructed allows to make the formal recording of existential judgements for the case when three extensional characteristics are associated with each general term: not only its extension and anti-extension, but also a contradictory domain of this term. A semantics which uses the same class of models as in IL logic is formulated for this language. The translation of the IL formulas into this language is formulated and it is proved that the truth conditions of judgements in IL-models logic are equivalent to the truth conditions of their translations in the “imaginary” logic of existential judgements (ILΥ logic). The latter means that a given translation embeds IL into ILΥ. An analytic-tableau formalization of the ILΥ logic is carried out. The soundness and completeness theorems are also proved. A decision procedure is formulated to solve the problem of provability for an arbitrary formula of the ILΥ language by the analytic tableau method. This procedure provides an adequate algorithm for checking the validity of any statement (including any syllogism) of imaginary logic.