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IBN SĪNĀ’S APPROACH TO EQUALITY AND UNITY

Published online by Cambridge University Press:  05 August 2014

Shahid Rahman*
Affiliation:
Université Lille3, STL: UMR 8163, Rue du Barreau, BP 60149, 59653 Villeneuve d'Ascq Cedex, France
Johan Georg Granström*
Affiliation:
Tiergartenstrasse 23A, CH-8802 Kilchberg ZH, Switzerland
Zaynab Salloum*
Affiliation:
Université Libanaise, Faculté de Sciences, Section I, Département de Mathématiques, Beyrouth, Liban

Abstract

Aristotle did not develop the quantification of the predicate, but, as shown in a recent paper by Hasnawi, Ibn Sīnā did. In fact, assuming the Aristotelian subject-predicate structure, Ibn Sīnā qualifies those propositions that carry a quantified predicate as deviating (muḥarrafa) propositions. A consequence of Ibn Sīnā’s approach is that the second quantification is absorbed by the predicate term. The clear differentiation between a quantified subject, that settles the domain of quantification, and a predicative part, that builds a proposition over this domain, corresponds structurally to the distinction, made in constructive type theory, between the type of sets and the type of propositions.

Neither did Aristotle combine his logical analysis of quantification with his ontological theory of relations or equality. But Ibn Sīnā makes use of syllogisms that require a logic of equality, and considered cases where quantification combines via equality with singular terms. Moreover these reflections provide the basis for his theory of numbers that is based on the interplay between the One and the Many. If we combine Ibn Sīnā’s metaphysical theory of equality with his work on the quantification of the predicate, a logic of equality comes out naturally. Indeed, the interaction between quantification of the predicate and equality can be applied to Ibn Sīnā’s own examples of syllogisms involving these notions. By using the formal instruments provided Martin-Löf's constructive type theory, the present paper establishes links between Ibn Sīnā’s metaphysics and his logical work: links that have been discussed in relation to other topics by Thom and Street. Ibn Sīnā did not develop a logic of identity, but he did develop the conceptual means to do so.

Résumé

Aristote n'a pas développé une théorie de la quantification du prédicat, mais une étude récente de Hasnawi a montré qu'Ibn Sīnā a consacré à celle-ci une étude rigoureuse. Assumant la structure aristotélicienne sujet-prédicat, Ibn Sīnā qualifie les propositions qui comportent un prédicat quantifié, de propositions déviantes (muḥarrafa). Une conséquence de cette approche avicennienne est que la seconde quantification est absorbée par le prédicat. La distinction claire ainsi opérée entre un sujet quantifié, qui pose le domaine de la quantification, et une partie prédicative, qui construit une proposition sur ce domaine, correspond structurellement à la distinction, faite dans la théorie constructive des types, entre le type des ensembles et le type des propositions.

Aristote n'a pas non plus combiné son analyse logique de la quantification avec sa théorie ontologique des relations ou de l’égalité. Ibn Sīnā, en revanche, a utilisé les syllogismes qui nécessitent une logique de l’égalité et il a examiné des cas où la quantification se combine, via l’égalité, avec des termes singuliers. En outre, ces réflexions sont essentielles pour sa théorie des nombres qui est fondée sur l'interaction entre l’un et le multiple. Lorsqu'on combine la théorie métaphysique de l’égalité telle qu'on la trouve chez Ibn Sīnā avec son travail sur la quantification du prédicat, il en résulte tout naturellement une logique de l’égalité. En effet, l'interaction entre la quantification du prédicat et l’égalité peut être appliquée aux exemples de syllogismes mentionnés par Ibn Sīnā dans lesquels ces notions interviennent. En utilisant les instruments formels fournis par la théorie constructive des types de Martin-Löf, cet article établit des liens entre la métaphysique d'Ibn Sīnā et son travail logique, liens qui ont été discutés en relation avec d'autres sujets par Thom et Street. Ibn Sīnā n'a pas développé une logique de l'identité, mais il a développé les moyens conceptuels de le faire.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2014 

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References

1 Sundholm, G., “Conservative generalized quantifiers”, Synthese, 79 (1989): 112CrossRefGoogle Scholar. Cf. Fernando, T.. “Conservative generalized quantifiers and presupposition”, in Hastings, R., Jackson, B., and Zvolenski, Z. (eds.), Eleventh Semantic and Linguistic Theory Conference (New York, 2001), pp. 172–91Google Scholar.

2 Aristote, Premiers Analytiques, trans. by M. Crubellier (Paris, to appear).

3 Crubellier, M., “Platon, les nombres et Aristote”, in Goff, J.P. Le (ed.), La mémoire des nombres (Cherbourg, Caen, 1997), pp. 81100Google Scholar.

4 Lorenz, K. and Mittelstrass, J., “On rational philosophy of language: the programme in Plato's Cratylus reconsidered”, Mind, 76.301 (1967): 120CrossRefGoogle Scholar.

5 In this paper, we use the sign ≡ for definitional equality: in this case, the definition of S has type ‘set’ and the definition of P(x) has type ‘prop’.

6 S. Rahman and Z. Salloum, “The One, the Many and Ibn Sīnā’s logic of identity”, paper in preparation.

7 Personal communication of Ahmad Hasnawi.

8 Aristotle, Metaphysics, trans. by Tredennick, H., Loeb Classical Library 271 & 287 (Cambridge, MA, 1933 & 1935)Google ScholarPubMed.

9 Hasnawi, A., “Avicenna on the quantication of the predicate”, in Rahman, S., Street, T., and Tahiri, H. (eds.), The Unity of Science in the Arabic Tradition, vol. 11 (Dordrecht, 2008), pp. 295328CrossRefGoogle Scholar.

10 In the 19th century, this debate was revived, with, on one side, W. Hamilton defending quantification of the predicate, and, on the other side, A. De Morgan strongly objecting to it. Cf. Fogelin, R. J.. “Hamilton's quantification of the predicate”, The Philosophical Quarterly, 26 (1976): 217–28CrossRefGoogle Scholar.

11 Hasnawi, “Avicenna on the quantication of the predicate”, p. 304.

12 Ibid., p. 323.

13 Cf., e.g., van Dalen, D., Logic and Structure, 3rd edn (Dordrecht, 1997)Google Scholar.

14 Cf. Maritain, J.: “In every affirmative, the predicate is taken particularly”, in Introduction to Logic, trans. by Choquette, I., 2nd edn (London, 1946), p. 125Google Scholar.

15 Avicenne, Le Livre de science, trans. by Achena, M. and Massé, H. (Paris, 1955), pp. 121–5Google Scholar.

16 Avicenne, La Métaphysique du Shifāʾ, trans. by Anawati, G. C. (Paris, 1978), pp. 160–4Google Scholar. The Schoolmen referred to the distinction between one in nature and one in an aspect as “duplex est unum”, cf. Aquinas De Potentia, q. 3, a. 16, ad 3 (in Quaestiones Disputatae, ed. by Pession, P. M., 10th edn, vol. 2 [Turin/Rome, 1965], pp. 1276Google Scholar).

17 Transl. W. Hodges.

18 This classification of the various senses of identity is derived from Aristotle, Metaph., Bk. 5, Ch. 6.

19 Nota, in the scholastic terminology or Merkmal in the terminology of Frege.

20 Frege, G., Die Grundlagen der Arithmetik (Breslau, 1884)Google Scholar.

21 An. Pr. I 27, 43b20 (Aristotle, Analytica priora, trans. by Tredennick, H., Loeb classical library 325 [Cambridge, MA, 1938], pp. 181531Google Scholar).

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