Позитивная силлогистика $C3^{+}$ с константой исчерпываемости

Abstract

We set out the syllogistic system $\mathbf{OC3}^{+}$ with standard constants $\mathbf{a}$, $\mathbf{e}$, $\mathbf{i}$, $\mathbf{о}$ and new constants $\mathbf{u}$ and $\mathbf{q}$: the statement of the form $S \mathbf{u} P$ means “Everything is either $S$ or $P$”; the statement of the form $S \mathbf{q} P$ means “Something is neither S nor P”. We demonstrate that $\mathbf{OC3}^{+}$ is embedded into the predicate calculus under the following translation $\chi$: $\chi(S\mathbf{a}P) = \forall x (Sx \supset Px) \& \exists xSx \& \exists x \neg Px$, $\chi(S\mathbf{e}P) = \forall x (Sx \supset \neg Px) \& \exists xSx \& \exists xPx$, $\chi(S\mathbf{i}P) = \exists x (Sx \& Px) \vee \neg \exists xSx \vee \neg \exists xPx$, $\chi(S\mathbf{o}P) = \exists x (Sx \& \neg Px) \vee \neg \exists xSx \vee \neg \exists x \neg Px$, $\chi$ : $\chi(S\mathbf{u}P) = \forall x (\neg Sx \supset Px) \& \exists x \neg Sx \& \exists x \neg Px$, $\chi(S\mathbf{q}P) = \exists x (\neg Sx \& \neg Px) \vee \neg \exists x \eg Sx \vee \neg \exists x \neg Px$, $\chi(\neg\mathbf{A}) = \neg \chi(\mathbf{A})$, $\chi (\mathbf{A}\nabla\mathbf{B}) = \chi(\mathbf{A}) \nabla \chi(\mathbf{B})$, where $\nabla$ is any binary connective.