Abstract This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and Łukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties of the natural deduction counterpart – that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. Acknowledgements We thank John Corcoran and Jonathan Zvesper for their kind help in polishing the prose of previous versions of this study. Notes 1There are, of course, many other systems of the syllogistic. However, due to the characteristics of the logical systems, all of them are to be classified either as natural deduction or axiomatic systems. 2Martin's point is different from ours. He is not so much interested in comparing Łukasiewicz's theory with Corcoran's, but in showing that the perfect syllogisms Barbara and Celarent can be treated as elements of the object language in a natural deduction theory and not as rules thereof. Different theories presented by Martin can be compared in terms of their extensional equivalence, but he does not do that in his study. We present in what follows some reasons why we do not take over this task. 3This is the semantics for the language of the syllogistic that Martin works with (Cf. Martin 1979 p. 5). We will show in §6 below a precise formulation of the problem that we are now pointing out. 4It is not hard to see that the present definition provides categorical propositions with their traditional semantics, namely, SaP: ‘Every S is P’, SeP: ‘No S is P’, SiP: ‘Some S is P’, SoP: ‘Some S is not P’. 5Rules (I), (II), and (V) are known, according to the tradition, as the conversion rules among categorical propositions. Rules (III), (IV), (VI), and (VII) are known as the perfect syllogisms. 6The proof of this claim follows straightforwardly once we get to grips with the proof of theorem 5. 7Find the proof of this fact below in section 4. 8As is done in, for instance, Caicedo 1989. 9In making this move, however, the propositions a and o are no longer contradictory according to the proposed first-order translation. The same goes for propositions e and i. Therefore, it is not trivial to keep the contradictory relations between the categorical propositions and at the same time to meet the non-empty set requirement. 10It has been recently brought to our attention by John Corcoran that an axiomatization fully writen down in first-order logic already existed in (Mates 1965§11.2). However, the set of axioms presented therein is bigger than ours. It is also interesting that he uses model theory to get at some results, though the section in question does not exploit these tools as we do in this study—see also Andrade and Becerra 2007. 11The consequence of leaving out the equality symbol is that the atomic sentences are only of the form xAy or xEy. 12One of the outcomes of the identification of the set V of terms of RD with the set of variables of the first-order language is that this g : V → M turns out to be an assignment function. 13It is not hard to prove that the same result holds for axiom L 2. 14This class of interpretations is the one used in Martin 1997. 15What we are proving here is an adequacy lemma: every consistent set of propositions has a true interpretation. The completeness theorem can be derived from this fact in the standard way.