[F]or all of the power and intricacies of [George] Boole’s system, he somehow managed to overlook indirect reasoning, or reductio ad absurdum, one of the most important and productive forms of inference. Indirect reasoning, also called reasoning by contradiction, is one of the most common forms of reasoning found in Euclid’s Elements and in general humanistic and scientific reasoning. . . . In Aristotle’s system there are two parallel types of deductions, direct and indirect. For direct deduction, after assuming the premises and identifying the ultimate conclusion to be reached, the reasoner derives intermediate conclusions one-after-the-other by epistemically immediate inferences until the ultimate conclusion is achieved. In Boole’s system the deductions are all direct. Readers should confirm this astounding point for themselves. . . .
For indirect reasoning, as demonstrated repeatedly in Aristotle’s work, a conclusion is deduced from given premises by showing that from the given premises augmented by the denial of the conclusion a contradiction can be derived. After the premises have been set and the conclusion to be reached has been identified, the next step is to assume for purposes of reasoning the denial of the conclusion. Then intermediate conclusions are entered step-by-step until a contradiction has been reached, which means that the last intermediate conclusion in the chain of intermediate conclusions is the contradictory opposite of one of the previously deduced conclusions or of one of the assumptions. In other words, to show indirectly that a conclusion follows from given premises, one shows that the denial of the conclusion contradicts the premises, i.e. that if the premises were true it would be logically impossible for the conclusion not to be true. Indirect reasoning, though commonplace in mathematical analysis, is absent from Boole’s system. . . . The avoidance of indirect reasoning sometimes requires painfully circuitous and unnatural paths of reasoning. The fact that Boole omitted even the mention of indirect deduction, or reductio reasoning, is astounding. It is not that he disapproved of it on some puristic principle; he did not even mention it. How can this glaring omission be explained?
(John Corcoran, "Aristotle's Prior Analytics and Boole's Laws of Thought," History and Philosophy of Logic 24 [December 2003]: 261-88, at 280 [parenthetical references omitted])
Keith Burgess-Jackson, J.D., Ph.D. 