2 Introducción
4.3 Análisis de covarianza con muestras representativas
In the fourteenth century, a distinct North African tradition of Arabic logic arose. Like their eastern counterparts, North African logicians in the fourteenth and fifteenth centuries took their point of departure from the writings of Avicenna, Fakhr al-Dīn al-Rāzī, Khūnajī, and
4 There does not appear to be a single commentary on Kātibī’s Shamsiyya nor
Urmawī’s Maṭāliʿ by a North African scholar. In contrast, there is a plethora of North African commentaries on Khūnajī’s Jumal and Sanūsī’s Mukhtaṣar.
5 For example, the Ottoman bibliographer Kātib Çelebī (d. 1657) did not know of
Sanūsī’s Mukhtaṣar and falsely noted that Khūnajī’s Jumal was an epitome of a work entitled Nihāyat al-amal by Ibn Marzūq, not realizing that the latter is in fact a com- mentary on the Jumal (see Kashf al-ẓunūn, I, 602). He was obviously not well acquainted with these works.
6 Ibn Khaldūn, al-Taʿrīf bi Ibn Khaldun wa riḥlatihi gharban wa sharqan, ed. by
M. al-Ṭanjī (Cairo: n.p., 1951), 62–64. The entry on this figure in Rescher’s The Development of Arabic Logic, 217, is unreliable and should be corrected on the basis of, for example, the detailed entry on him in Aḥmad Bābā al-Timbuktī, Nayl al-ibtihāj bi-taṭrīz al-Dībāj (Cairo: ʿAbbās b. ʿAbd al-Salām b. Shaqrūn, 1351/1932), 255–264 (margins).
7 This is the text as it appears in the lemma quoted within the commentary on the
work by Khūnajī’s student Ibn Wāṣil. The lemma quoted in later commentaries is slightly different.
Khūnajī’s students and associates, like Ibn Wāṣil and Urmawī. However, they were not systematically apprised of later writings in the East. Furthermore, the handbooks used in North Africa were different from those used in other parts of the Islamic world. Rather than Kātibī’s Shamsiyya and Urmawī’s Maṭāliʿ, the handbooks most used in North Africa in later centuries were Khūnajī’s short epitome al-Jumal and, later, the epitome (Mukhtaṣar) of the North African scholar Muḥammad b. Yūsuf al-Sanūsī (d. 1490).4 Conversely, the writings of North African logicians were almost completely unknown to scholars east of Egypt.5
One of the earliest representatives of the North African tradition of Arabic logic was Muḥammad al-Sharīf al-Tilimsānī (d. 1370), a teacher of the historian Ibn Khaldūn. Tilimsānī is one of the last scholars known to have taught Averroes’ Middle Commentaries on Aristotle.6 However, in his own commentary on Khūnajī’s Jumal he explicitly fol- lowed the tradition of Avicenna, Rāzī, and Khūnajī. In retrospect, Tilimsānī can be seen as marking a turning point after which North African logicians accepted, and participated in, the post-Avicennian tradition of Arabic logic that had come to prevail in other parts of the Islamic world a century earlier.
In Khūnajī’s Jumal, a syllogism is defined as “a statement, composed of statements, that implies another statement” (qawlun muʾallafun min aqwālin yastalzimu qawlan ākhar).7 It is noticeable that the definition makes no mention of the condition that the conclusion should follow from the premise-pair ‘by itself.’ Khūnajī’s student Ibn Wāṣil al-Ḥamawī
8 Ibn Wāṣil, Sharḥ al-Jumal (Manuscript Landberg 103, Beinecke Library, Yale
University), fol. 29b (kaʾannahu fī l-ḥaqīqati ghayru muḥtājin ilayhi fa-inna l-qaḍāyā idhā stalzamat shayʾan bi-wāsiṭati muqaddimatin ajnabiyyatin lam takun bi-dūni l- muqaddimati l-ajnabiyyati mustalzimatan aṣlan).
9 Tilimsānī, Sharḥ al-Jumal (Manuscript Add. 9617. British Library), fol. 77b
(yushʿiru qawluhu mustalzimun ann lā yakūnu mutawaqqifan fī istilzāmihi li-l-natījati ʿalā muqaddimatin ajnabiyyatin fa-innahu in tawaqqafa ʿalā muqaddimatin ajnabiyya- tin lam yakun al-mustalzimu li-l-natījati l-madhkūra faqaṭ bal al-majmūʿa min al-qawli l-madhkūri maʿa l-muqaddimati l-ajnabiyya).
10 Tilimsānī, Sharḥ al-Jumal, fol. 77b (fa-yakhruju ʿanhu qiyāsu l-musāwāti
ka-qawlinā A musāwin li-B wa B musāwin li-J fa-inna dhālika yastalzimu A musāwin li-J lākin bi-wāsiṭati muqaddimatin ajnabiyyatin wa hiya anna l-ashyāʾa l-musāwiyata li-shayʾin wāḥidin mutasāwiyatun).
(d. 1298), in his commentary on the Jumal, explained that Khūnajī had excised this condition since:
It is as if in reality there is no need for it, for if propositions imply some- thing by the mediation of an extrinsic premise, then they do not imply it at all without this extrinsic premise.8
Tilimsānī reiterated the point in his own commentary on the Jumal. He wrote:
His [i.e. Khūnajī’s] saying ‘implies’ indicates that its implication of the conclusion should not depend on an extrinsic premise. For if it depends on an extrinsic premise then that which implies the conclusion is not what has been [explicitly] mentioned alone but rather the whole consist- ing of the mentioned composite statement and the extrinsic premise.9
Tilimsānī went on to apply this point to the case of the syllogism of equality. He stated that the ‘syllogism of equality’ is excluded from the definition of syllogism by the condition that the premises ‘imply’ the conclusion:
This excludes the syllogism of equality, such as our statement ‘A is equal to B & B is equal to J,’ for this implies that ‘A is equal to J’ by the media- tion of an extrinsic premise, and this is that ‘things that are equal to one thing are equal.’10
In other words, Ibn Wāṣil and Tilimsānī departed from the hitherto reigning assumption that the premise-pair of the syllogism of equality implies the conclusion but does not do so ‘by itself.’ Their position seems to have been that there is no implication (luzūm) that is not implication by the premises alone. It is not the case that elliptical arguments with suppressed premises imply a conclusion but not ‘by themselves.’ They simply do not imply the conclusion at all. As will be
11 Tilimsānī, Sharḥ al-Jumal, fol. 78a–78b.
seen below, a similar opinion was sometimes voiced in the eastern parts of the Islamic world. However, it seems never to have become the generally accepted view of later logicians, for the condition that the conclusion should follow from the premise-pair ‘by itself ’ contin- ued to be expounded in the Arabic logical works written in the eight- eenth and nineteenth centuries.
Tilimsānī initially wrote – as quoted above – that the premises of the syllogism of equality only imply the conclusion with the addition of a further premise to the effect that ‘things that are equal to one thing are equal to one another.’ The formulation is oddly reminiscent of Ibn Zurʿa’s reconstruction of the suppressed premise, and it can- not be precluded that Tilimsānī had access to works in the more Aristotelian tradition of Fārābī and his students. However, he went on to reproduce the detailed derivation of the conclusion of the ‘syllogism of equality’ given by Khūnajī in Kashf al-asrār, in which the additional premise is ‘Everything that is equal to B is equal to what B is equal to (kullu musāwin li-B fa-huwa musāwin limā yusāwīhi B).’11 As in the case of Ṭūsī, what he stated was the missing premise is not the premise that he used in the actual regimentation of the argument.
Tilimsānī then added his own somewhat simplified regimentation, which involves a suppressed premise that is a conditional rather than categorical proposition:
Always: If B is equal to J, then what is equal to B is equal to J This premise is the conditional premise of a so-called ‘reduplicative’ (istithnāʾī) syllogism – a hypothetical syllogism in modus ponens – whose other premise is the second premise of the original syllogism of equality:
Always: If B is equal to J, then what is equal to B is equal to J B is equal to J
What is equal to B is equal to J
This intermediate conclusion is then conjoined to the first premise of the syllogism of equality to produce the desired conclusion:
A is equal to B
What is equal to B is equal to J A is equal to J
12 On Ibn Marzūq and ʿUqbānī, see Timbuktī, Nayl al-ibtihāj, 293–299 and 125–126
respectively.
13 Ibn Marzūq, Nihāyat al-amal bi-sharḥ kitāb al-Jumal (Manuscript Derenbourg
640, Escorial, Madrid), fol. 3b (qawluhum fī qiyāsi l-musāwāti l-mutaqaddami: innamā antaja A musāwin li-J liʾanna musāwī l-musāwī musāwin kalāmun ghayru muḥaqqaqin wa innamā yatimmu qiyāsan bi-an naqūla idhā ʿulima anna B musāwin li-J ṣadaqa kullu musāwin li-B musāwin li-musāwī J wa kullu musāwin li-musāwī J musāwin li-J ammā l-ūlā fa-jaliyyatun baʿda l-ʿilmi bi-musāwāti B li-J wa l-thāniyatu ajlā).
The later North African scholar Ibn Marzūq al-Ḥafīd (d. 1439) wrote a lengthy commentary on Khūnajī’s Jumal that he intended to be a synthesis of earlier commentaries, specifically those of Tilimsānī and another North African scholar, Saʿīd al-ʿUqbānī (d. 1408).12 Ibn Marzūq reiterated Tilimsānī’s discussion of the syllogism of equality, but he also engaged with the views of ʿUqbānī. The latter is quoted by Ibn Marzūq as having written:
Their saying of the syllogism of equality – that it produces ‘A is equal to J’ because ‘what is equal to what is equal is equal’ – is not a verified statement. It only becomes a syllogism if we say: If we know that ‘B is equal to J’ then it is true that ‘Everything that is equal to B is equal to what is equal to J’ and that ‘Everything that is equal to what is equal to J is equal to J.’ As for the first proposition, it is manifest once we know the equality of B to J. As for the second proposition, it is even more manifest.13
This, according to ʿUqbānī, allowed for the regimentation of the syllo- gism of equality into two syllogisms, the first of which is:
A is equal to B
Everything that is equal to B is equal to what is equal to J A is equal to what is equal to J
This intermediate conclusion is then conjoined to the second of the evident propositions posited by ʿUqbānī:
A is equal to what is equal to J
Everything that is equal to what is equal to J is equal to J A is equal to J
The analysis is the one proposed by Ṭūsī in his commentary on Avicenna’s Ishārāt. ʿUqbānī went on to write that this regimentation is an effort to reduce the syllogism of equality to an argument in which there is a standard middle term. Otherwise, he wrote, there would be little point to the exercise, for the following of the conclusion from
14 Ibn Marzūq, Nihāyat al-amal, fol. 3b (qawluhu inna istilzāma hādhā aẓharu min
istilzāmi l-shakli l-rābiʿi musallamun lākin laysa l-qiyāsu ʿibāratan ʿammā kāna istilzāmuhu li-ghayrihi ẓāhiran bal huwa ʿibāratun ʿan tarkībin khāṣṣin laysa hādhā minhu).
15 Ibn Marzūq, Nihāyat al-amal, fol. 3a (wa ḥtaraza bihi ʿan qiyāsi l-musāwāti
naḥwa A musāwin li-B wa B musāwin li-J fa-innahu yalzamu ʿan hādhā l-qawli A musāwin li-J lākin lā li-mujarradi taʿaqquli l-qawli l-muʾallafi min qaḍiyyatayni bal
the original premises of the syllogism of equality is ‘more obvious’ (aẓhar) than the following of the conclusion from a fourth-figure syl- logism. Indeed, it is not less obvious than the following of a conclusion from a syllogism of the first figure. Ibn Marzūq took ʿUqbānī to be casting doubt on the necessity of the regimentation, and he replied as follows:
We concede his statement that the implication by this [the original syl- logism of equality] is more obvious than the implication by the fourth figure. However, ‘syllogism’ is not an expression that is true of what obviously implies something else, but of what has a specific formation that this [the syllogism of equality] does not have.14
Unfortunately, Ibn Marzūq then went on to add: “and pursuing this [point] would take too long.” It seems that he was giving an unusually straightforward expression of the view that syllogistic implication is not a matter of whether or not we evidently know a conclusion on the basis of certain premises, but a matter of what follows from the form of the premises. However, his remarks are too brief to allow us to be confident of this. If indeed he was making a distinction between psy- chological facts and formal logic in this passage, then he did not stick to the distinction elsewhere in his commentary. In his copy of Khūnajī’s Jumal, the condition that the premise-pair must imply the conclusion ‘by itself ’ is mentioned. Regarding this condition Ibn Marzūq wrote:
He meant to exclude by this the syllogism of equality, such as ‘A is equal to B & B is equal to J,’ for it is implied by this statement that ‘A is equal to J’ – but not from the mere cognizing of the statement consisting of two propositions, but rather after knowing that ‘things that are equal to one thing are themselves equal,’ or that ‘what is equal to what is equal to something is equal to that something’ – these two formulations have the same meaning. This is unlike the case of you saying: ‘Every A is equal to B & Everything that is equal to B is equal to J.’ For this [statement] is such that the mere cognizing of it implies that ‘A is equal to J’ for A is one of the things that the major premise judges to be equal to J.15
baʿda l-ʿilmi bi-anna l-ashyāʾa l-musāwiyata li-shayʾin wāḥidin hiya nafsuhā mutasāwiyatun aw bi-anna musāwī l-musāwī li-l-shayʾi musāwin li-dhālika l-shayʾi fa-l-ʿibāratāni maʿnan wāḥidun wa hādhā bi-khilāfi mā idhā qultu kullu A musāwin li-B wa kullu musāwin li-B musāwin li-J fa-inna hādhā mujarradu taʿaqqilihu yalzamu ʿanhu kullu A musāwin li-J li-anna A min jumlati l-afrādi l-latī ḥukima ʿalayhā fī l-kubrā bi-musāwāti J).
16 On him, see Timbuktī, Nayl al-ibtihāj, 325–330.
17 For example, they were printed in Cairo in 1322/1904 with the gloss of the
Rector of the Azhar College Ibrāhīm al-Bājūrī (d. 1860).
18 Sanūsī, Sharḥ al-Mukhtaṣar fī l-manṭiq (Cairo: n.p., 1875), 88.
Ibn Marzūq in this passage obviously conflated the question of whether a conclusion follows from premises ‘by themselves’ with the question of whether knowledge of these premises is sufficient to know the conclusion. He thus assumed that in the standard syllogism of equality the cognizing of the premises is not sufficient to cognize the conclusion, whereas in the following argument the cognizing of the premises is sufficient to cognize the conclusion:
A is equal to B
What is equal to B is equal to J A is equal to J
As was noted in the previous chapter in connection with the discus- sion of Quṭb al-Dīn al-Rāzī, it is doubtful whether this is anything more than ‘armchair’ psychology.
The writings of Ibn Marzūq’s second-generation student Muḥammad b. Yūsuf al-Sanūsī (d. 1490) exerted a profound influence on later centuries.16 His Ashʿarī creedal works were widely studied in much of the Arabic-speaking Sunnī world until modern times. His epitome of logic, known as Mukhtaṣar al-Sanūsī, and his own com- mentary on it, were also widely studied throughout Muslim Africa until the beginning of the twentieth century.17 Sanūsī had little new to say about the problem of the ‘syllogism of equality.’ However, the his- torical influence of his work is such that a brief account of his discus- sion may be in order.
Sanūsī defined a syllogism as “a statement composed of two assents which, if conceded, imply by themselves another assent” (qawlun muʾallafun min taṣdīqayni matā sullimā lazima li-dhātayhimā taṣdīqun ākharu).18 The fact that Sanūsī spoke of ‘assents’ rather than proposi- tions may at first sight seem to indicate a ‘psychologistic’ conflation of logic with epistemology and psychology. However, in his commentary
19 Sanūsī, Sharḥ al-Mukhtaṣar, 89 (an yakūna l-luzūmu li-dhāti taʾlīfi l-taṣḍīqayni
ay lā yakūnu bi-wāsiṭati muqaddimatin ajnabiyyatin ay ghayri lāzimatin li-iḥdā l- muqaddimatayni luzūman ḍarūriyyan).
20 Sanūsī, Sharḥ al-Mukhtaṣar, 89 (lam yuntij hādhā l-taʾlīfu fi qiyāsi l-musāwāti
bi-dhātihi bal bi-wāsiṭati muqaddimatin ajnabiyyatin wa hiya qawlunā kullu musāwin li-B fa-huwa musāwin li-kulli mā yusāwīhi l-B).
Sanūsī made it clear that by ‘assent’ he meant ‘proposition’ (qaḍiyya). Unlike Ibn Marzūq, he made no appeal to the ‘cognition’ of the con- clusion supposedly following from the ‘cognition’ of the premises. Rather, he wrote that the stipulation that the conclusion follow from the premises ‘by themselves’ meant that:
The implication should be by the composition of the two assents itself. In other words, it should not be by the mediation of an extrinsic premise, i.e. a premise that is not implied by one of the premises in a necessary manner.19
This condition, Sanūsī wrote, ruled out the ‘syllogism of equality,’ for its implication of the conclusion is not by means of the very arrange- ment of the composite statement. If it were by means of the arrange- ment itself, then such an argument would be productive “by virtue of its form always (bi-ḥasabi ṣūratihi dāʾiman) and this is not so.” A coun- terexample, Sanūsī wrote, is the following argument which has true premises and a false conclusion:
The human is distinct from the horse (al-insānu mubāyinun li-l-faras)
The horse is distinct from the speaker (al-farasu mubāyinun li-l-nāṭiq)
The human is distinct from the speaker (al-insānu mubāyinun li-l-nāṭiq)
Sanūsī did not invoke the novel semantic analysis of the form of the syllogism of equality proposed by Quṭb al-Dīn al-Rāzī. Nevertheless, he clearly considered the ‘syllogism of equality’ to be an instance of a more general form of argument that is formally invalid:
This composition in syllogisms of equality does not produce a conclu- sion by itself, but by the mediation of an extrinsic premise, and this is our statement ‘Everything that is equal to B is equal to everything which B is equal to.’20
Sanūsī thus accepted the position of Khūnajī and Urmawī concerning the identity of the missing premise in the ‘syllogism of equality,’ and
21 Jarbī, Sharḥ Īsāghūjī (Tunis: al-Maṭbaʿa al-Rasmiyya, 1309/1891), 22.
22 Ṭāṣköprüzāde, Miftāḥ al-saʿāda, I, 299; Rescher, The Development of Arabic Logic,
220. Rescher’s estimated date of birth for this scholar (1340) must be amended to allow for the fact that Ibn Mubārak was a teacher of Jurjānī who was born in 1340.
23 This was printed in Kazan in 1902 with the glosses of Jurjānī (d. 1413) and Mīrzā
Jān Bāghnavī (d. 1586).
he reproduced their intricate analysis of how to derive the conclusion from the original premises with this missing premise.
Sanūsī appears to have been the last major figure in the North African logical tradition in the period from 1350 to 1600. There are apparently only two sixteenth-century works on logic by North African scholars that continued to be studied in later times. One is the commentary by the Tunisian scholar Sulaymān al-Jarbī (fl. 1507) on the elementary introduction to logic entitled Īsāghūjī by Athīr al-Dīn al-Abharī (d. 1265). Jarbī studied in Cairo with a Persian-born scholar