  The Aristotelian syllogistic cannot account for the validity of many
inferences involving relational facts. In this paper, we investigate the
prospects for providing a relational syllogistic. We identify several fragments
based on (a) whether negation is permitted on all nouns, including those in the
subject of a sentence; and (b) whether the subject noun phrase may contain a
relative clause. The logics we present are extensions of the classical
syllogistic, and we pay special attention to the question of whether reductio
ad absurdum is needed. Thus our main goal is to derive results on the existence
(or non-existence) of syllogistic proof systems for relational fragments. We
also determine the computational complexity of all our fragments.
