doubting the validity of reductio ad impossibile proofs in post-avicennan arabic logic

The rule of reductio ad impossibile (to prove p, assume not-p, and infer a contradiction) is indispensable for aristotle’s deductive system, because it is needed to establish the validity of all but the four purely universal syllogistic moods. in avicenna’s fully modalized syllogistic, the structure of reductio proofs and their place in the deductive system had to be adjusted. avicenna employed, for example, a reductio proof (burhān al-khalf) for the conversion of the absolute universal negative proposition, or to show that an affirmative universal conditional entails a negative universal conditional with the same antecedent and the contradictory consequent. the central importance of the issue for the avicennan theory of science and the completeness of the deductive system on which it was based did not escape the notice of two twelfth century scholars who otherwise are not known for their work as logicians: rashīd al-dīn waṭwaṭ (d. 1182) and ibn ghaylān balkhī (d. ca. 1194). rashīd al-dīn penned a treatise rejecting the validity of reductio proofs which elicited a critical response from ibn ghaylān. the present article discusses and assesses the arguments advanced by the two scholars and traces their afterlife in a number post-avicennan logicians. section i lays out rashīd al-dīn’s three argumentative steps to reject the reductio rule. first, he argues against the claim that the conversion of an absolute universal proposition can be established by reductio. second, he shows that by employing the reductio rule in the avicennan sense, more than one converse can be established for a given proposition. based on the first two claims, he thirdly argues that since the application of the reductio rule to establish the conversion of the absolute universal proposition can be employed to yield three distinct converse propositions, it cannot reliably be used to reduce syllogistic moods to the first figure and thus prove their validity. sections ii-iv present and assess ibn ghaylān’s detailed replies to each of these argumentative steps. ibn ghaylān’s counterarguments seem to have been influential, for the issue ceases to be discussed in this form. however, the remainder of the article argues that it resurfaced in a different guise with afḍal al-dīn al-khūnajī and continued to shift forms in the logical works of athīr al-dīn al-abharī (d. 1265), naṣīr al-dīn al-ṭūsī (d. 1274), and najm al-dīn al-kātibī (d. 1276). section v argues that with khūnajī’s rejection of aristotle’s thesis the focus shifted to the problem of how the reductio rule is to be grounded if an impossible antecedent can imply a proposition and its contradictory, a topic which both abharī and ṭūsi discussed controversially. section vi shows that the problem with reductio informed abharī’s rejection of some of the standardly accepted conversions and closes with a comparison of kātibī’s position on the conversion of conditionals. the conclusion highlights the significance of the logical work by rashīd al-dīn and ibn ghaylān on a central issue not only for the avicennan deductive system, but for the theory of science which is based on it.