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 Lukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call TRD, 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 TRD 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 TRD.
