Abstract
Abstract We define the classical syllogistic, together with its extensions by either noun-level negation or transitive verbs, showing that the satisfiability problem for all three fragments is NLogSpace-complete. We show, however, that the satisfiability problem for the extension with both these features is ExpTime-complete. We go on to establish that the Aristotelian syllogisms---together with some auxiliary rules---constitute a sound and complete proof system for the classical syllogistic and its extension by noun-level negation. We exhibit a set of syllogism-like rules that is sound and refutation-complete for the extension by transitive verbs, and show that there is no such system which is sound and complete for this fragment. Finally, we show that, for the extension by both noun-level negation and transitive verbs, there is no system of syllogism-like rules that is even sound and refutation-complete.
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