Abstract
We trace two logical ideas further back than they have previously been traced. One is the idea of using diagrams to prove that certain logical premises do—or don’t—have certain logical consequences. This idea is usually credited to Venn, and before him Euler, and before him Leibniz. We find the idea correctly and vigorously used by Abū al-Barakāt in 12th century Baghdad. The second is the idea that in formal logic, P logically entails Q if and only if every model of P is a model of Q. This idea is usually credited to Tarski, and before him Bolzano. But again we find Abū al-Barakāt already exploiting the idea for logical calculations. Abū al-Barakāt’s work follows on from related but inchoate research of Ibn Sīnā in eleventh century Persia. We briefly trace the notion of model-theoretical consequence back through Paul the Persian (sixth century) and in some form back to Aristotle himself.
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al-Barakāt, A.: Kitāb al-mu\({}^c\)tabar fī al-ḥikmat al-ilāhiyya. Jam\({}^c\)iyyat Dā’irat al-Ma\({}^c\)ārif al-\({}^c\)Uthmāniyya, Hyderabad (1938/9)
Aristotle: Prior Analytics Book I, trans. and commentary by Gisela Striker. Clarendon Press, Oxford (2009)
Baron, M.E.: A note on the historical development of logic diagrams: Leibniz, Euler and Venn. Math. Gaz. 53(384), 113–125 (1969)
Bolzano, B.: Grundlegung der Logik (Wissenschaftslehre I/II), ed. Friedrich Kambartel, Meiner, Hamburg (1963)
Couturat, L.: Opuscules et Fragments Inédits de Leibniz. Alcan, Paris (1903)
De Morgan, A.: On the syllogism: I, On the structure of the syllogism. Trans. Camb. Philos. Soc. 8, 379–408 (1846); reprinted in [7] pp. 1–21
De Morgan, A.: On the Syllogism and Other Logical Writings, ed. Peter Heath. Routledge, London, (1966)
Frege, G.: Über die Grundlagen der Geometrie. Jahresber. Dtsch. Math. 15, 293–309, 377–403, 423–430 (1906)
Gergonne, J.D.: Variétés. Essai de dialectique rationnelle. Ann. Math. Pures Appl. 7, 189–228 (1816/7)
Gil, M.: Jews in Islamic Countries in the Middle Ages. Brill, Leiden (2004)
Gutas, D.: Paul the Persian on the classification of the parts of Aristotle’s philosophy: a milestone between Alexandria and Baghdad. In: Gutas, D. (ed.) Greek Philosophers in the Arabic Tradition, pp. 231–267. Ashgate, Aldershot (2000)
Hodges, W.: Mathematical Background to the Logic of Ibn Sīnā. Perspectives in Mathematical Logic, Association for Symbolic Logic (forthcoming)
Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 1. Aristotelian and logical background. Draft online at http://wilfridhodges.co.uk/history26.pdf
Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 2. Alexander and Paul (in preparation)
Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 3. The Arabic logicians (in preparation)
Hodges, W.: Identifying Ibn Sīnā’s hypothetical logic: I. Sentence forms. Draft online at http://wilfridhodges.co.uk/arabic59.pdf
Hodges, W.: Identifying Ibn Sīnā’s hypothetical logic: II. Interpretations. Draft online at http://wilfridhodges.co.uk/arabic59a.pdf
Ibn Sīnā, Al-qiyās (Syllogism), ed. S. Zayed, Cairo (1964)
Naṣīr al-Dīn al-Ṭūsī: Asās al-iqtibās (Persian). Ferdows, Tehran (2010)
Paul the Persian: Logica. In: Land, Jan Pieter Nicolaas (ed.) Anecdota Syriaca, vol. 4, pp. 1–30. Brill, Lugdunum Batavorum (Katwijk) (1875)
Pavlov, M.M.: Abu’l-Barakat al-Baghdadi’s Metaphysical Philosophy, The Kitab al-Mu\({}^c\)tabar. Routledge, Abingdon (2017)
Read, S.: Aristotle and Lukasiewicz on existential import. J. Am. Philos. Assoc. 1(3), 535–544 (2015)
Ross, W.D.: Aristotle’s Prior and Posterior Analytics: A Revised Text with Introduction and Commentary. Clarendon Press, Oxford (1949)
Tarski, A.: On the concept of logical consequence. In: Corcoran, J. (ed.) Alfred Tarski, Logic, Semantics, Metamathematics. Hackett, Indianapolis, pp. 409–420 (1983)
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I thank Amirouche Moktefi, Lukas Muehlethaler and Robert Wisnovsky for helpful comments and information.
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Hodges, W. Two Early Arabic Applications of Model-Theoretic Consequence. Log. Univers. 12, 37–54 (2018). https://doi.org/10.1007/s11787-018-0187-6
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DOI: https://doi.org/10.1007/s11787-018-0187-6