Lewis Carroll’s Syllogistic with Negative Terms

Abstract

Lewis Carroll was an author of original syllogistic theory which is different from Traditional syllogistic. Carroll’s system contains term negation, so it made him possible to eliminate $o$-type propositions $SoP$ treating them as a kind of $i$-type propositions $SiP'$. We set out the following axiom schemes for Carroll’s syllogistic:$(MaP\&SaM)\supset SaP$, $SiP_PiS$, $SiP_SaS$, $SaP_SiP$, $SeP_¬SiP$, $ SaP\equiv(SeP'\&SiS)$, $ SaP'\equiv(SeP\&SiS)$, $SiS _ S_iS_$. We prove that this system embeds into the Predicate calculus by the following interpretation (equivalent to Carroll’s understanding) of categorical propositions: $SaP\rightarrow(\forall x(Sx\supset Px)\& \exists xSx)$, $SiP\rightarrow \exists x(Sx\& Px)$,$SeP __x(Sx_¬Px)$