Aristotle's Natural Deduction Reconsidered

 

History and Philosophy of Logic, 18 (1997), 1-15

Abstract

John Corcoran's natural deduction system (equivalent to systems by Timothy Smiley and others) for Aristotle's syllogistic is reconsidered. Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogism as arguments and in rejecting Lukasiewicz' treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system. An alternative is presented that is technically more elegant and equally compatible with the texts. The abstract role assigned to Barbara and Celarent by tradition and Lukasiewicz is retained. The two "perfect syllogisms" serve as "basic elements" in the construction of an inductively defined set of valid syllogisms. The proposal departs from Lukasiewicz and follows Corcoran, however, in construing the construction as one in natural deduction. The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions. To compare the theory to Corcoran's, his original is reformulated in current terms and generalized. It is shown to be equivalent to the proposed sequent system, and several variations are discussed. For all systems mentioned a method of Henkin style completeness proofs is given that is more direct and intuitive than Corcoran's original, and that uses Aristotle's notion of "ecthesis" as the basis for the relvant concept of saturation.