To the Problem of Characterization of Logic of the Vasiliev Type: on Tabularity $I_{\langle x,y \rangle}$ ($x,y\in\{0,1,2,\dots\}$ and $x < y$). Part I

Abstract

In this article, continuing the work carried out in [1], the problem of tabularity of the $I$-logics of the Vasiliev type (propositional logic is called tabular if it has a finite characteristic matrix). The main result obtained in this article: for any non-negative integers $x$ and $y$, the first of which is less than the second, the logic $I_{\langle \mathrm{x}, \mathrm{y}\rangle}$is tabular (the class of all such logics is an infinite subclass of the class of all $I$-logics of the Vasiliev type). The proposed study is based on the use of the results obtained in [1], and on the use of the authors’ “cortege semantics”. To achieve the above main result, we show how on arbitrary nonnegative integer numbers$m$ and $n$, satisfying the inequality $m< n$, is constructed logic matrix $\mathfrak{M}(m, n)$, which is the finite characteristic matrix of logic $I_{\langle m,n\rangle}$. Since the carrier of the logical matrix $\mathfrak{M}(m, n)$is some set of 0-1-corteges, the semantics based on this logical matrix is naturally called the cortege semantics. Important note: the article is published in two parts. Before you the first part of the article, the second part of the article is planned to be published in the second issue of “Logical Investigations” for 2017. DOI: 10.21146/2074-1472-2017-23-1-57-82