Aristotle’s Natural Deduction System

Abstract

Here and elsewhere we shalI not obtain the best insight into things untiI we actually see them growing from the beginning. AristotIe In the present article we attempt to show that Aristotie's syllogistic is an underlying logie which includes a natural deductive system and that it is not an axiomatic theoryas had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic and we examine both the mathematical properties of the model and the relation of the model to the system of logic envisaged in certain scattered parts of Prior and Posterior Ana/ytjes. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses. In the :fint place, his theory of deduction is logicaIly sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) In the second place, Aristotle's logic presupposes no other logical concepts, not even those of propositionallogic. In the third place, the Aristotelian system is seen to be complete in the sense that every valid argument expres-sible in his system admits of a deduction within his deductive system; i.e., every semantically valid argument is deducible. There are six sections in this article. The :fint section includes method-ological remarks, a preliminary survey of the present interpretation and a discussion of the differences between our interpretation and that of Lukasiewicz. The next three sections develop the three parts of the mathematical model. The fifth section deals with general properties ofthe model and its relation to the Aristotelian system. The final section contains conclusions.