Aristotle’s Proofs of Conversions and Syllogisms

Abstract In Prior Analytics Aristotle proves the conversion principles, using three logical techniques: reductio (indirect derivation), exposition (a kind of existential instantiation: given ‘Some S is a P’, introduce a previously unused name m, and write ‘m is an S’ and ‘m is a P’), and expository syllogism (given ‘m is a P’ and ‘m is an S’ infer ‘Some S is a P’). He assumes four basic forms of syllogism, and uses them together with reductio, exposition, and expository syllogisms to prove the remaining forms. The four basic forms can be proved too, though this was unknown to him or to medieval logicians. Monotonicity properties of the determiners ‘every’, ‘no’, and ‘some’ are explained; Aristotle’s four basic argument forms allow easy proofs of their monotonicity properties. By the 13th century a memorizable verse was developed that encodes Aristotle’s validation of all syllogistic forms from the basic four.