Logica Universalis

, Volume 12, Issue 1–2, pp 37–54 | Cite as

Two Early Arabic Applications of Model-Theoretic Consequence

Article

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.

Keywords

Logic diagram model-theoretic consequence Arabic logic Barakāt 

Mathematics Subject Classification

Primary 01-02 Secondary 01A30 03C98 00A66 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)Google Scholar
  2. 2.
    Aristotle: Prior Analytics Book I, trans. and commentary by Gisela Striker. Clarendon Press, Oxford (2009)Google Scholar
  3. 3.
    Baron, M.E.: A note on the historical development of logic diagrams: Leibniz, Euler and Venn. Math. Gaz. 53(384), 113–125 (1969)CrossRefMATHGoogle Scholar
  4. 4.
    Bolzano, B.: Grundlegung der Logik (Wissenschaftslehre I/II), ed. Friedrich Kambartel, Meiner, Hamburg (1963)Google Scholar
  5. 5.
    Couturat, L.: Opuscules et Fragments Inédits de Leibniz. Alcan, Paris (1903)Google Scholar
  6. 6.
    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. 1–21Google Scholar
  7. 7.
    De Morgan, A.: On the Syllogism and Other Logical Writings, ed. Peter Heath. Routledge, London, (1966)Google Scholar
  8. 8.
    Frege, G.: Über die Grundlagen der Geometrie. Jahresber. Dtsch. Math. 15, 293–309, 377–403, 423–430 (1906)Google Scholar
  9. 9.
    Gergonne, J.D.: Variétés. Essai de dialectique rationnelle. Ann. Math. Pures Appl. 7, 189–228 (1816/7)Google Scholar
  10. 10.
    Gil, M.: Jews in Islamic Countries in the Middle Ages. Brill, Leiden (2004)Google Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Hodges, W.: Mathematical Background to the Logic of Ibn Sīnā. Perspectives in Mathematical Logic, Association for Symbolic Logic (forthcoming)Google Scholar
  13. 13.
    Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 1. Aristotelian and logical background. Draft online at http://wilfridhodges.co.uk/history26.pdf
  14. 14.
    Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 2. Alexander and Paul (in preparation)Google Scholar
  15. 15.
    Hodges, W.: Nonproductivity proofs from Alexander to Abū al-Barakāt: 3. The Arabic logicians (in preparation)Google Scholar
  16. 16.
    Hodges, W.: Identifying Ibn Sīnā’s hypothetical logic: I. Sentence forms. Draft online at http://wilfridhodges.co.uk/arabic59.pdf
  17. 17.
    Hodges, W.: Identifying Ibn Sīnā’s hypothetical logic: II. Interpretations. Draft online at http://wilfridhodges.co.uk/arabic59a.pdf
  18. 18.
    Ibn Sīnā, Al-qiyās (Syllogism), ed. S. Zayed, Cairo (1964)Google Scholar
  19. 19.
    Naṣīr al-Dīn al-Ṭūsī: Asās al-iqtibās (Persian). Ferdows, Tehran (2010)Google Scholar
  20. 20.
    Paul the Persian: Logica. In: Land, Jan Pieter Nicolaas (ed.) Anecdota Syriaca, vol. 4, pp. 1–30. Brill, Lugdunum Batavorum (Katwijk) (1875)Google Scholar
  21. 21.
    Pavlov, M.M.: Abu’l-Barakat al-Baghdadi’s Metaphysical Philosophy, The Kitab al-Mu\({}^c\)tabar. Routledge, Abingdon (2017)Google Scholar
  22. 22.
    Read, S.: Aristotle and Lukasiewicz on existential import. J. Am. Philos. Assoc. 1(3), 535–544 (2015)CrossRefGoogle Scholar
  23. 23.
    Ross, W.D.: Aristotle’s Prior and Posterior Analytics: A Revised Text with Introduction and Commentary. Clarendon Press, Oxford (1949)Google Scholar
  24. 24.
    Tarski, A.: On the concept of logical consequence. In: Corcoran, J. (ed.) Alfred Tarski, Logic, Semantics, Metamathematics. Hackett, Indianapolis, pp. 409–420 (1983)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Herons Brook, Sticklepath, OkehamptonDevonEngland, UK

Personalised recommendations