Обобщенная позитивная силлогистика.

Abstract

The paper concerns the problem of the representation of all possible extensional relations between two general terms by means of positive syllogistic - syllogistic without negative terms. I introduce new syllogistic constants u and q: the statement of the form $S\textbf{u}P$ means «Everything is either $S$ or $P$», the statement of the form $S\textbf{q}P$ means «Something is neither $S$ nor $P%». I offer two syllogistic systems based on propositional calculus in the language with constants \textbf{a, i, e, o, u, q}. The first system is the generalization of the positive fragment of Brentano-Leibnitz fundamental syllogistic. I demonstrate that it is embedded into the predicate calculus under the following translation *: $(S\textbf{a}P)^*=\forall x(S_x\supset P_s,(S\textbf{i}P)^*=\exists x (S_x\& P_x),(S\textbf{e}P)^*=\forall(S_x\supset\neg P_x),(S\textbf{o}P)^*=\exists(S_x \&\neg P_x),(S\textbf{u}P)^*=\forall(S_x\vee P_x), (S\textbf{q}P)^*=\exists(\neg S_x\&\neg P_x),(\neg A)^*=\neg A^*, (A\bigtriangledown B)^*=F^*\bigtriangledown B^*$, where $\bigtriangledown$ is any binary connective. The second system is the generalization of Lukasiewicz' syllogistic which is a formalization of the traditional one. I prove that generalized traditional syllogistic is embedded into the predicate calculus under the translation $\Theta(A)=(\exists S_1x\&\exists x\neg S_1x\&...\&\exists xS_nx\&\exists\neg S_nx)\supset A^*$, where $S_1,...,S_n$ is a list of all general terms in $A$.