Syllogistic Theory as Logic of Anti-extensions of Terms

Abstract

For two syllogistics (fundamental and traditional) we define nonstandard translations into the predicate calculus. These translations make it possible to treat syllogistic theories as logics of anti-extensions of the subjects and the predicates of categorical statements. In compliance with the first translation, $SaP$ means that anti-extension of $S$ is included in anti-extension of $P$, $SeP$ means that anti-extensions of $S$ and $P$ don’t contain common elements, $SiP$ means that the intersection of anti-extensions of $S$ and $P$ is nonempty,$SoP$ means that anti-extension of $S$ is not included in anti-extension of $P$. For arbitrary syllogistic formula $A$, the second translation includes additionally the precondition that anti-extensions of all the terms in $A$ are nonempty. It is proved that these two syllogistics are embedded into the classical predicate calculus under the given translations.