The book deals with the formal analysis of Aristotles non-modal syllogistic; with the inter-relations between various syllogistic systems, their sub-systems and extensions; and with the fundamental question of the nature of the syllogism. It aims to effect a synthesis of recent work (both logical and philological) on the non-modal sections of the Analytics, within the framework of a new formal system which combines features of Lukasiewicz's "axiomatic" approach with features of the ,natural deduction' of Corcoran and Smiley. This system is located in relation to other system categorical syllogistic that have been developed from Aristotle's time to our own Also considered are systems with negative or singular terms, and systems which axiomatic structure to the rejected formulae. It is shown that Aristotle's syllogistic can be based on a system of singular syllogisms with the Aristotelian rules of ecthesis; and a new explanation is ventured for Aristotle's failure to complete a logic of singular terms. A minimal "auxiliary theory" is identified for those Systems which base syllogistic on propositional logic. The question of the nature of the syllogism is approached first syntactically: the traditional "rules of the syllogism" are generalised and applied to the elucidation of Aristotle's syntactic metatheory in Prior Analytica Book B. (A side-result here is the formulation of a system of prosleptic syllogisms within which the results of the Kneales and Lejewski are derivable.) A property first defined by Smiley is then used to interpret some semantic metatheorems formulated by Aristotle. Finally, this semantic account of the syllogism is grounded on a theory of fallacies which has important links, via the notion of petitio principii, with epistemic logic. The resulting account of the syllogism is thus to some extent epistemological, but in no way psychologistic. Of interest to: Logicians, mathematicians, philosophers, historians of these disciplines, scholars in classical and medieval studies