Формальная реконструкция традиционной сингулярной негативной силлогистики.


V.I. Markin


I set out the formal reconstruction o f traditional singular negative syllogistic by means of modern logic. I introduce the formal language o f syllogistic with singular and negative terms adequate for the problem solution. The traditional logic had several dominant assumptions concerning the usage of singular and negative terms within the propositions. The main idea was to minimize the syntactical difference between singular and general terms. According to this idea, singular terms can take predicate as well as subject position. Singular propositions are treated as a kind o f general or particular propositions. Negative terms could be constructed not only from general but also from singular terms.

In this language I formulate a deductive system \textbf{TS} - the extension of Lukasiewicz ’ syllogistic.

The translation of syllogistic formulas is done into the language of first order predicate calculus with equality. Let $\varSigma$ be a function assigning for each syllogistic term a predicate o f this language: $\varSigma(\nu)=(x=\nu),\varSigma(S)=Sx,\varSigma(\sim\alpha)=\neg\varSigma(\alpha)$, where $\nu$ is a singular term, $S$ is a primitive general term, $\alpha$ is any syllogistic term. Let $^*$ be standard («fundamental») translation of \textbf{TS} syllogistic formulas into the language of the first order predicate calculus with equality: $(\alpha \textbf{a}\beta)^*=\forall x(\varSigma(\alpha)\supset\varSigma(\beta)),(\alpha \textbf{e}\beta)^*=\forall x(\varSigma(\alpha)\supset\neg\varSigma(\beta)),(\alpha \textbf{i}\beta)^*=\exists x(\varSigma(\alpha)\&\varSigma(\beta)),(\alpha \textbf{o}\beta)^*=\exists x(\Sigma(\alpha)\&\neg\varSigma(\beta)),(\neg \textbf{A})^*=\neg \textbf{A}^*, (\textbf{A}\bigtriangledown \textbf{B})^*=\textbf{A}^*\bigtriangledown \textbf{B}^*$. Finally I define the translation $\Theta:\Theta(\textbf{A})=(\exists xS_1x\&\exists x\neg S_1x\&...\&\exists xS_nx\&\exists\neg S_nx)\supset \textbf{A}^*$, where $S_1,...,S_n$ is a list of all primitive general terms in \textbf{A}.

I prove that for each syllogistic formula \textbf{A}, \textbf{A} is a theorem of \textbf{TS} iff its translation $\Theta(\textbf{A})$ is a theorem of the first order predicate calculus with equality completed with the axiom $\exists x\exists y\neg(x=y)$.