Aristotle's Second-Order Logic

Abstract: This paper has the following aims. We study the metalogic of the Organon and other works of Aristotle with special attention to the evidence provided by the later work of Galen. We attempt to formalize the metalogic implicit in the Analytics and the Physics which is at the same time, so we argue, the logic implicit in the Topics, and our proposal is that this consists in an extension of second-order logic presented in a variant of natural deduction which includes non-monadic predicates which we call Aristotelian second-order logic (ASOL) Furthermore we argue that Aristotle actually made conscious use of the key quantifier rules as well as propositional rules thus lending further support to the thesis argued by Bobzien\cite{bob1} that ancient logic had sufficient means to deal effectively with multiple generality. The second part of the paper is dedicated to deploying ASOL to formalize the Topics and showing the logical, linguistic and philosophical interest of this often neglected work.