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Over the course of the eleventh and twelfth centuries, the influence of Avicenna’s philosophical and logical works spread throughout the Islamic world, and by the thirteenth century they had – in most parts of the Islamic world – effectively replaced the works of Aristotle as the point of departure and reference for philosophers and logicians.25 As will be seen in the following chapters, it was the regimentation of the syllogism of equality by Avicenna and his followers that was the starting-point for later discussions, and the slightly different regimen- tation of Ibn Zurʿa was forgotten.

This spectacular spread of Avicenna’s influence did not occur without resistance. In the remaining part of this section I will present the discussions of syllogisms by some of the more prominent twelfth- century philosophers whose attitude to Avicenna’s influence ranged from skepticism to outright hostility: (1) the Iraqi-Jewish convert to Islam Abū l-Barakāt Ibn Malka al-Baghdādī (d. 1165); (2) the famous

26 _{Abū l-Barakāt al-Baghdādī, al-Muʿtabar fī l-ḥikma (Hyderabad, Deccan: Dāʾirat}

al-Maʿārif al-ʿUthmāniyya, 1357/1938–1939), I, 123 (wa dhālika anna l-qarīnata takūnu min qawlayni humā muqaddimatāni wa fī kulli muqaddimatin ḥaddāni ḥaddin mawḍūʿin wa ḥaddīn maḥmūlin wa yalzamu ʿanhā mā yalzamu li-shirkatin bayna l-muqaddimatayni wa tilka l-shirkatu takūnu fī juzʾin lā maḥālata idh law kānat fī l-kulli la-kānat iḥdāhumā hiya l-ukhrā bi-ʿaynihā wa dhālika l-juzʾu immā an yakūna huwa l-maḥmūla wa immā an yakūna l-mawḍūʿa fī kilayhimā wa immā an yakūna mawḍūʿan fī iḥdāhumā maḥmūlan fī l-ukhrā).

Andalusian Aristotelian Averroes (Ibn Rushd al-Ḥafīd, d. 1198); and (3) the self-styled reviver of anti-Peripatetic Platonism Yaḥyā al- Suhrawardī (d. 1191) who was executed for heresy in Aleppo. None of these scholars appear to have been particularly troubled by the anomaly posed by the ‘syllogism of equality’ or other relational inferences.

(1) Abū l-Barakāt al-Baghdādī’s discussion of syllogism in his survey of logic, physics, and metaphysics entitled al-Muʿtabar fī l-ḥikma does not deal with the argument from equality. He cannot have considered such an argument to be a syllogism as it stands, for he explicitly expounded the principle that in a simple categorical syllogism there must be three and only three terms:

The syllogistic combination (qarīna) is from two statements which are the premises. In every premise there are two extremes: an extreme that is the subject and an extreme that is the predicate. What is implied by these [premises] is implied because of something that is common to both premises. That which is common must be a part [of a premise], for if it were the whole then one [of the premises] would be identical to the other. This part is either the subject or the predicate in both premises, or the subject of one and the predicate of the other.26

Of course, the final statement – that what is common to both premises is either the subject or the predicate – simply does not follow from what precedes it. As will be seen in subsequent chapters, several later logicians agreed that the two premises in a syllogism must have a term in common, but denied that this term must be the predicate or the subject. Rather, they argued that the common term could be part of the subject or predicate of one of the premises. As Avicenna had noted, the two premises ‘A is equal to B’ and ‘B is equal to J’ at least appear to imply the conclusion ‘A is equal to J,’ even though they do not have an entire predicate or subject in common. Abū l-Barakāt was simply asserting standard Aristotelian orthodoxy, without taking into

27 _{Abū l-Barakāt al-Baghdādī, al-Muʿtabar, 173.} 28 _{Abū l-Barakāt al-Baghdādī, al-Muʿtabar, 170.}

29 _{Abū l-Barakāt is here clearly taking the original premises as constituting a second-}

figure syllogism of the form: Every J is B

No non-A is B No J is non-A

consideration apparent counter-examples that had been familiar at least since the time of Alexander of Aphrodisias.

In a later section of his work, Abū l-Barakāt gave some clues as to how he would have dealt with the anomaly of the argument from equals. He noted with admirable forthrightness that arguments in ordinary language hardly ever have explicit syllogistic form, and that even Aristotle’s prose hardly ever contained such explicit syllogisms.27 Often, a premise was left unmentioned because it is evident or because one wishes to mask a fallacy. Sometimes, an argument appears entirely convincing even though it does not have the form of a recognizable syllogism. One example given by Abū l-Barakāt is by now familiar:

The part of the substance, its removal necessitates the removal of the substance

The removal of what is not a substance does not necessitate the removal of the substance

The part of the substance is a substance

There was, Abū l-Barakāt wrote, a number of ways of regimenting such an argument into syllogistic form.28 _{One could, for example, take} care to give the conclusion that actually follows from the premises: ‘The part of the substance, its removal is not the removal of what is not a substance,’ and then seek to derive the desired conclusion from this intermediate conclusion.29 _{Alternatively, one could replace the} second premise with an equipollent proposition, thus obtaining a straightforward syllogism in the first figure:

The part of the substance, its removal necessitates the removal of the substance

Everything whose removal necessitates the removal of the sub- stance is a substance

The part of the substance is a substance

It is thus clear that Abū l-Barakāt subscribed to the standard Aris- totelian procedure for regimenting arguments that appear not to have

30 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, ed. M. Qāsim, C.E. Butterworth, and A. Harīdī}

(Cairo: al-Hayʾa al-miṣriyya al-ʿāmma li-l-kitāb, 1983), 65.

31 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 65.} 32 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 65.} 33 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 65.} 34 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 190–192.}

a familiar syllogistic form: either treat them like enthymemes that call for the addition of a further premise, or replace one or more of the original premises with propositions that are ‘equipollent’ to them. (2) Averroes’s discussion of syllogisms in his Middle Commentary on Aristotle’s Prior Analytics likewise did not engage with the issue of the syllogism of equality. He defined a syllogism as “a statement that, if things more than one are posited in it, then something else different from it is implied with necessity from these posited things by them- selves and not accidentally” (qawlun idhā wuḍiʿat fīhi ashyāʾu akhtharu min wāḥidin lazima min al-iḍṭirāri ʿan tilka l-ashyāʾi l-mawḍūʿati bi- dhātihā lā bi-l-ʿaraḍi shayʾun mā ākharu ghayruhā).30 _{He held that the} stipulation that the conclusion must follow from the premises ‘by themselves’ is to ensure that the argument is not elliptical, i.e. “that it does not lack something that would make it complete (tāmm).”31 _Like Avicenna in al-Shifāʾ, Averroes held this stipulation to be distinct from the two stipulations that the conclusion must follow necessarily and that it must not follow accidentally. However, his understanding of these other two stipulations was different. Avicenna understood the stipulation that the conclusion must not follow accidentally to rule out cases in which the conclusion only follows when a premise is replaced by an equipollent proposition. Averroes understood it to rule out cases of productivity due to matter, such as “the production from two affirm- ative propositions in the second figure if the predicates are co-extensive with the subjects in predication.”32 _{Avicenna understood the stipula-} tion that the conclusion must follow necessarily to rule out cases of material rather than formal productivity. Averroes understood it to rule out cases like analogy or induction in which the conclusion fol- lows from the premises but not necessarily.33

Averroes attempted to show by abstract considerations that a syllo- gism thus defined must consist of two premises with three terms in total, one of which is common to both premises.34 _{A conclusion will,} he wrote, either assert or deny that a predicate belongs to a subject. One could not show that the predicate belongs or does not belong to

the subject by means of a single premise, for this single premise must have either one term or two terms in common with the desired con- clusion. It cannot have both terms in common, for then one would be proving a proposition by means of itself. It also cannot have one term only in common with the conclusion, for there is no syllogistic argu- ment from the premise ‘A is B’ or ‘A is not B’ to the conclusion ‘A is C’ or ‘A is not C.’ The premise obviously cannot have no term in common with the conclusion, for there is no syllogistic argument from a premise of the form ‘D is J’ to the conclusion ‘B is A.’ In an argu- ment that meets the aforementioned definition of a syllogism, there must therefore be two premises. These two premises each have two extremes. It cannot be the case that the premises have no extreme in common, nor can they have two extremes in common, so they must share one and only one term. It also must be the case that both extremes of the desired conclusion appear in the pair of premises, though not in the same premise. Thus, each of the two premises must have one term in common with the desired conclusion. In this way, it can be shown – or so Averroes claimed – that a syllogism that seeks to establish that a predicate belongs or does not belong to a subject must have the form of a standard Aristotelian syllogism with three terms.

Averroes’ argument on this point – which is derived from Aristotle (Pr An. A25) – is more developed than that of Abū l-Barakāt al- Baghdādī, but is also beset by a number of difficulties. One difficulty that is particularly relevant in the present context is that an apparent counter-example is the ‘syllogism of equality.’ It would seem – at least at first sight – that one could show that ‘A is equal to J’ by adducing the premises ‘A is equal to B’ and ‘B is equal to J,’ and these premises do not have an entire extreme in common. Averroes must have held such an argument not to be formally productive, and hence not to be a gen- uine counter-example. The problem is that – as has been seen above – it is far from clear how the ‘syllogism of equality’ can be supplemented with additional premises so as to become formally productive from an Aristotelian perspective. Furthermore, one could present other relational arguments that cannot plausibly be dismissed as elliptical. It is for example possible to argue – as some later logicians writing in Arabic did – that one may want to prove a proposition such as ‘Zayd is the brother of a writer’ by adducing the premises ‘Zayd is the brother of ʿAmr’ and ‘ʿAmr is a writer.’ It is difficult to see why seeking to establish such a conclusion by means of such premises should be

35 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 227.}

dismissed as illegitimate. And the argument would seem to have as good a claim as any to formal productivity. Yet Averroes cannot con- cede this, for the argument does not consist of two premises with three terms in total, one of which is shared. Such examples highlight the familiar limitation of the orthodox Aristotelian assumption that the only multi-premised arguments that imply a conclusion ‘by them- selves’ are those syllogisms presented and discussed by Aristotle.

Averroes too devoted a section of his Middle Commentary to show- ing how ordinary arguments can be captured in syllogistic form. This endeavor would allow us, he wrote, to show by induction that any argument that meets the aforementioned definition of a syllogism can be formulated as an Aristotelian syllogism with two premises and three terms.35 _{Averroes wrote that an argument often only has a single} premise, or contains statements that are not part of the proof but have been added for clarification or to mask the structure of the argument for dialectical purposes. One should therefore seek to uncover super- fluous premises and add missing premises to the argument. Consider, for example, the following argument: ‘The parts of the substance are substances, for the removal of what is not a substance does not neces- sitate the removal of the substance, and the removal of parts of the substance necessitates the removal of the substance.’ For the desired conclusion to follow from the premises ‘by themselves,’ one must replace the first premise with the premise ‘Whatever necessitates the removal of the substance is a substance.’ This premise is then con- joined as a major to the second premise, thus forming a first-figure syllogism:

The part of the substance, its removal necessitates the removal of the substance

Everything whose removal necessitates the removal of the sub- stance is a substance

The part of the substance is a substance

Another example given by Averroes is the following argument: ‘If a human exists then a living thing exists, and if a living thing exists then a substance exists, therefore if a human exists then a substance exists.’ The argument would seem to be a straightforward example of what is known in the western logical tradition as a ‘purely hypothetical

36 _{See N. Shehaby, The Propositional Logic of Avicenna (Dordrecht, Reidel, 1973).} 37 _{Abū l-Barakāt al-Baghdādī, al-Muʿtabar, I, 155.}

38 _{See for example his treatise al-Qawl fī l-qiyās al-ḥamlī wa l-sharṭī wa naqd al-qiyās}

al-iqtirānī l-sharṭī ʿinda Ibn Sīnā, in Rasāʿil falsafiyya: Maqālāt fī l-manṭiq wa l-ʿilm al-ṭabīʿī li-Ibn Rushd, ed. J. ʿAlawī (Casablanca: Dār al-Nashr al-Maghribiyya, 1983), 187–207.

39 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 229–230.}

40 _{Ibn Rushd, Talkhīṣ kitāb al-qiyās, 229 (laysa kullu mā yalzamu ʿan shayʾin bi-}

l-iḍṭirāri fa-huwa lāzimun luzūman qiyāsiyyan bal mā lazima bi-iḍṭirārin ʿan muqaddi- matayni nisbatu iḥdāhumā ilā l-ukhrā nisbatu l-kulli ilā l-juzʾi fa-huwa qiyās).

syllogism’ and Avicennian logicians called ‘combinatorial conditional syllogisms’ (al-qiyāsāt al-sharṭiyya al-iqtirāniyya), with the form:

If P then Q If Q then R If P then R

The recognition of such ‘combinatorial hypothetical syllogisms’ was one of the hallmarks of the Avicennian tradition of Arabic logic. Avicenna had devoted a considerable part of his book on ‘Syllogism’ in al-Shifāʾ to investigating the logic of such arguments, and later logicians working in his wake expanded his treatment.36 _However, more orthodox Aristotelian scholars tended to be skeptical of this new-fangled interest. Abū l-Barakāt al-Baghdādī conceded the exist- ence of combinatorial hypothetical syllogisms, but added that they were of little importance.37 _{Averroes went even further, presenting} arguments to the effect that postulating such syllogisms is wrong- headed – arguments that were widely ignored by the later tradition and that are not relevant to the present study.38 _{In the example just} cited, he simply stated that the argument lacks the premises ‘Every human is a living thing’ and ‘Every living thing is a substance.’39 _{He in} effect seems to have held that both premises in such an argument would have to be reformulated as paradigmatic Aristotelian categori- cal premises before they could be said to meet the definition of a syllogism.

In general, Averroes wrote, it is a mistake to assume that every instance of a conclusion following from premises ‘with necessity’ is an instance of a syllogism.

It is not the case that whatever is implied by something else with neces- sity is implied syllogistically. Rather, what is implied with necessity from two premises, one of which is related to the other as a whole to a part, is a syllogism.40

41 _{J. Barnes, “Logical Form and Logical Matter,” in Logica, mente e persona, ed.}

A. Alberti (Florence: Olschki, 1990), 55–57.

42 _{Suhrawardī, Manṭiq al-Talwīḥāt, ed. ʿA.A. Fayyāḍ (Tehran: Intishārāt-i Dānishgāh-i}

Tehrān, 1955), 46.

The principle that not all cases of necessary implication are cases of syllogistic implication had already been recognized by Aristotle and by the Greek commentators.41 _{It is not, despite appearances, an} acknowledgement of the existence of formally productive arguments that are not syllogisms. Such an understanding would make nonsense of much of what Aristotle, his Greek commentators, and Averroes had to say elsewhere. Rather, Averroes was simply expressing the view that there are arguments in which the conclusion follows from the premises with necessity but are nevertheless not syllogisms because they do not meet other conditions stipulated in the definition of a syllogism – namely, the condition that the conclusion follow from the premises ‘by themselves’ and not ‘accidentally.’ His previous discussion of the defi- nition of syllogism clearly shows that he considered both (i) enthyme- mes with true suppressed premises and (ii) materially productive but formally sterile arguments to imply conclusions necessarily. Conceding the existence of such arguments does not in any way amount to a departure from the orthodox Aristotelian position that any multi- premise argument that implies another categorical proposition ‘by itself ’ and not ‘accidentally’ is a categorical syllogism with three and only three terms.

Averroes thus recognized the familiar techniques for regimenting arguments in ‘books and speech’ into standard Aristotelian syllogisms: treating an argument as enthymematic, or reformulating one or more premises. There can be little doubt that he would have used much the same techniques to solve the anomaly of the ‘syllogism of equality’ as those used by Ibn Zurʿa or Avicenna and his followers.

(3) One might expect to find a radically non-Aristotelian concep- tion of syllogistic implication in the writings of the anti-Peripatetic Platonist Suhrawardī. However, his view of the topic appears to have differed little, if at all, from the standard Aristotelian and Avicennian view. In his al-Talwīḥāt, he followed word for word the definition of a syllogism presented in Avicenna’s ʿUyūn al-ḥikma: “a statement com- posed of propositions which, if they are conceded, implies by itself another statement.”42 _{The condition that the conclusion follow from}

43 _{Suhrawardī, Manṭiq al-Talwīḥāt, 87.}

44 _{Shahrazūrī, Sharḥ Ḥikmat al-ishrāq, ed. H. Ziai (Tehran: Institute for Humanities}

and Cultural Studies, 2001), 138; Ibn Kammūna, Sharḥ al-Talwīḥāt, ed. by Najafqoli Habibi (Tehran: Miras-i Maktub, 2009), I, 328.

45 _{The point is made by Mullā Ṣ̣adrā (d. 1635) in his gloss on Quṭb al-Dīn al-Shīrāzī’s}

commentary on Ḥikmat al-ishrāq, see Mullā Ṣ̣adrā, Taʿlīqāt ʿala Sharḥ Hikmat al-ishrāq (Lithograph: 1315/1897–1898; reprint, Qom: Intisharat Baydar, n.d.), 140 (margins).

46 _{Suhrawardī, Ḥikmat al-ishrāq, ed. and trans. H. Ziai and J. Walbridge (Provo,}

Utah: Brigham Young University Press, 1999), 21.

the pair of premises ‘by itself ’ was, Suhrawardī noted, to exclude two cases: (i) arguments in formally sterile moods in which the conclu- sion happens to be true by virtue of the matter of the premises, and (ii) arguments that require the addition of another premise in order to be productive of the desired conclusion. In a later section of the work, Suhrawardī mentioned that one type of fallacy comes about when the middle term is not repeated in its entirety in the two premises (li-ʿadami naqli l-awsaṭi bi-kulliyatihi).43 _{His two earliest commenta-} tors, Shams al-Dīn al-Shahrazūrī (fl. 1288) and the Jewish philosopher Ibn Kammūna (fl. 1268), both mentioned as an example the following