Aristotle's Second-Order Logic and Natural Deduction

This paper has two goals. The first goal is to show how an extension of second-order logic is a natural framework to formalize portions of Aristotle's \emph{Topics} and to bring to the foreground the logical, linguistic and philosophical interest of this work, showing in particular that we are in the presence of a richly intensional and modal conception of logic. Aristotelian logic and its related traditions in antiquity are often held to have been equivalent to monadic predicate logic and as such inadequate to formalize mathematics as well as scientific and philosophical discourse in general. The second goal of this paper is to argue that on the contrary the logical theories of Aristotle (which we argue correspond to a variant of natural deduction) and ancient authors such as Galen and Boethius were in fact quite sufficient to account for the logically complex expressions and reasoning involving multiple generality fundamental to the aforementioned disciplines.