Advertisement

Studia Logica

, 92:281 | Cite as

Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction

  • M. Abraham
  • Dov M. GabbayEmail author
  • U. Schild
Article

Abstract

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix \({\mathbb {A}}\) with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a i,j =?. The method allows us to decide whether to declare a i,j = 0 or a i,j = 1 or a i,j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.

Keywords

Matrix Abduction Talmudic logic Argumentum A Fortiori Qal-Vachomer argumentation 

References

  1. 1.
    Abraham, M., D. M. Gabbay, and U. Schild, Paper 340 Kal-Vachomer in Hebrew, BDD Journal, Bar Ilan University, 112 pages.Google Scholar
  2. 2.
    Barker S., Boella G., Gabbay D.M., Genovese V.: ‘A Meta-model of Access Control in a Fibred Security Language’. Studia Logica 92, 437–476 (2009)CrossRefGoogle Scholar
  3. 3.
    Barringer, H., D. M. Gabbay, and J. Woods, ‘Temporal dynamics of argumentation networks’, in D. Hutter and W. Stephan (eds.), Mechanising Mathematical Reasoning, LNCS 2605, Springer, 2005, pp. 59–98.Google Scholar
  4. 4.
    Becker, M. Y., J. F. Mackay, and B. Dillaway, ‘Abductive Authorization Credential Gathering. http://research.microsoft.com/pubs/80508/becker2009ieee-policysubmission. pdf
  5. 5.
    Besnard, P., and A. B. Hunter, Elements of Argumentation, MIT Press, 2008, 300 pages.Google Scholar
  6. 6.
    Boella G., Gabbay D.M., Genovese V., van der Torre L.: ‘Fibred Security Language’. Studia Logica 92, 395–436 (2009)CrossRefGoogle Scholar
  7. 7.
    Brams, S. J., D. M. Kilgour, and W. S. Zwicker, ‘The paradox of multiple elections’, Social Choice and Welfare 15: 211–236, 1998.CrossRefGoogle Scholar
  8. 8.
    Caminada, M. W. A., and L. Amgoud, ‘On the evaluation of argumentation formalisms’, Artificial Intelligence 171 (5–6): 286–310, 2007.CrossRefGoogle Scholar
  9. 9.
    Dung, Phan Minh, ‘An argumentation theoretic foundation for logic programming, Journal of Logic Programming 22 (2): 151–171, 1995.Google Scholar
  10. 10.
    Fisch, M., Rational Rabbis: Science and Talmudic Culture, Indiana University Press, Bloomignton, 1997.Google Scholar
  11. 11.
    Gabbay, D. M., Labelled Deductive Systems, OUP, 1996.Google Scholar
  12. 12.
    Gabbay, D. M., and A. Garcez, ‘Logical modes of attack in argumentation networks’. To appear in Studia Logica.Google Scholar
  13. 13.
    Gabbay, D. M., and J. Woods, The Reach of Abduction, Elsevier, 2005.Google Scholar
  14. 14.
    Gabbay, D.M., and J.Woods, ‘Resource origins of non-monotonicity’, Studia Logica 88: 85–112, 2008.CrossRefGoogle Scholar
  15. 15.
    Grabenhorst, T. K., Das Argumentum A Fortiori, Peter Lang, 1990.Google Scholar
  16. 16.
    Hartmann, S., G. Pigozzi, and J. Sprenger, ‘Reliable methods of judgement aggregation’, Draft, 23 February 2009.Google Scholar
  17. 17.
    Hasan, A., Analogical reasoning in Islamic jurisprudence, 1986, Republished Adam Publishers, 2007, 486 pages.Google Scholar
  18. 18.
    Jacobs, L., Studies in Talmudic Logic and Methodology, London, Vallentine-Mitchell, 1061. Republished paperback, 2006.Google Scholar
  19. 19.
    Kamali, M. H., Principles of Islamic Jurisprudence, Islamic Text Society, 3rd Revised Eition, 2002, 550 pages.Google Scholar
  20. 20.
    Kornhauser L., Sager L.: ‘Unpacking the court’. Yale Law Journal 96, 82–117 (1986)CrossRefGoogle Scholar
  21. 21.
    Kornhauser L., Sager L.: ‘The one and the many: adjudication in collegial courts’. California Law Review 81, 1–51 (1993)CrossRefGoogle Scholar
  22. 22.
    Kunst, A., ‘An overlooked type of inference’, Bulletin of the School of Oriental and African Studies, X, part 4, 1942, pp. 976–991.Google Scholar
  23. 23.
    Neusner, J., ‘The making of the mind of Judaism; the formative age’, Brown Judaic Studies, vol. 133, Scholars press, Atlanta, 1987.Google Scholar
  24. 24.
    Pigozzi, G., and L. van der Torre, ‘Premise independence in judgement aggregation’, Dagstuhl Seminar 07531, 2007.Google Scholar
  25. 25.
    Prakken, H., and G. Sartor, ‘Argument based extended logic programming with defeasible priorities’, Journal of Applied Non-classical Logics 7: 25–75, 1997.Google Scholar
  26. 26.
    Schild, U. J., ‘Criminal Sentencing and Intelligent Decision Support’, Artificial Intelligence and Law 6: 2–4, 1998.CrossRefGoogle Scholar
  27. 27.
    Schwarz, A., Der Hermeneutische Syllogismus in der Talmudischen, Ltitteratur, Karlsruhe, 1901.Google Scholar
  28. 28.
    Stebbing, V. L. S., A Modern Introduction to Logic, London, 1945.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Bar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of Computer ScienceKing’s CollegeLondonUK

Personalised recommendations