Completeness Theorems for Syllogistic Fragments

Traditional syllogisms involve sentences of the following simple forms: All X are Y , Some X are Y , No X are Y ; similar sentences with proper names as subjects, and identities between names. These sentences come with the natural semantics using subsets of a given universe, and so it is natural to ask about complete proof systems. Logical systems are important in this area due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in logic from Aristotle onwards. We present complete systems for the entire syllogistic fragment and many sub-fragments. These begin with the fragment of All sentences, for which we obtain one of the easiest completeness theorems in logic. The last system extends syllogistic reasoning with the classical boolean operations and cardinality comparisons.