Nested Indirect Proofs in Aristotle’s Deductive Logic

Aristotle’s Analytica Priora contains three systems of syllogistic proof, what are known as ‘figures’ 1–3. Aristotle states that all syllogistic inferences reduce to the first one but does not carry that through. It turns out that this reduction hides up to three nested instances of indirect proof. Syllogistic derivations are given as two-dimensional derivation trees—the only addition of substance to Aristotle’s proofs—with the result that nested indirect proofs can be resolved into a single such step, in perfect analogy to steps of normalization in natural deduction.