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A Symbolic Approach to Syllogistic Reasoning

  • Mohamed Yasser Khayata
  • Daniel Pacholczyk
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 89)

Abstract

In this paper we present a new approach to a symbolic treatment of quantified statements having the following form “Q A’s are B’s”, knowing that A and B are labels denoting sets, and Q is a linguistic quantifier interpreted as a proportion evaluated in a qualitative way. Our model can be viewed as a symbolic generalization of statistical conditional probability notions as well as a symbolic generalization of the classical probabilistic operators. Our approach is founded on a symbolic finite M-valued logic in which the graduation scale of M symbolic quantifiers is translated in terms of truth degrees of a particular predicate. Then, we present symbolic syllogisms allowing us to deal with quantified statements.

Keywords

Statistical Probability Inference Rule Truth Degree Default Reasoning Syllogistic Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. .
    E. Adams. The logic conditionals. D. Reidel, Dordercht, Netherlands, 1975.Google Scholar
  2. 2.
    S. Amarger, D. Dubois, H. Prade. Imprecise Quantifiers And Conditional Probabilities. In ECSQARU’91, R. Krause and al (eds.), 33–37, 1991.Google Scholar
  3. 3.
    H. Akdag, M. De Glas, D. Pacholczyk. A Qualitative Theory of Uncertainty. Fundamenta Informatica, 17 (4): 333–362, 1992.MATHGoogle Scholar
  4. 4.
    F. Bacchus. Representation and Reasoning with Probabilistic Knowledge. MIT Press, Cambridge, MA, 1990.Google Scholar
  5. 5.
    F. Bacchus, A. J. Grove, J. Y. Halpern, D. Koller. From statistical knowledge hases to degrees of belief. Artificial intelligence, 87: 75–143, 1997.MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Cheeseman. An inquiry into computer understanding. Computational Intelligence, 4 (1): 58–66, 1988.CrossRefGoogle Scholar
  7. 7.
    D. Dubois, H. Prade. On fuzzy syllogisms. Computational Intelligence, 4 (2): 171–179, 1988.CrossRefGoogle Scholar
  8. 8.
    D. Dubois, H. Prade, L. Godo, R. Mantaras. A symbolic approach to reasoning with linguistic quantifiers. In UAI, Stanford, 74–82, 1992.Google Scholar
  9. 9.
    M. Jaeger. Default Reasoning about Probabilities. Ph. D. Thesis, Univ. Of Saarbruchen, 1995.Google Scholar
  10. 10.
    M.Y. Khayata. D. Pacholczyk. A Symbolic Approach to Linguistic Quantifiers. Proc. of IPMU’2000, Madrid, Spain, Vol 3, p 1720–1727.Google Scholar
  11. 11.
    M.Y. Khayata, D. Pacholczyk. A Symbolic Approach to Linguistic Quantification. In First FAST IEEE Student Conference,101–106, Lahore, 1998.Google Scholar
  12. 12.
    H.E. Kyburg. The reference class. Philosophy of Science, 50 (3): 374–397, 1983.MathSciNetCrossRefGoogle Scholar
  13. 13.
    N.J. Nilsson. Probabilistic logic. Artificial Intelligence, 28(1): 71–88, 1986.Google Scholar
  14. 14.
    G. Paass. Probabilistic logic. In Non Standard Logic for Automated Reasoning, P. Smet et al. (eds.), Academic Press, 213–251, 1988.Google Scholar
  15. 15.
    D. Pacholczyk. Contribution au Traitement Logico-symbolique de la Connaissance. Thèse d’Etat. Université de PARIS 6, 1992.Google Scholar
  16. 16.
    D. Pacholczyk. A Logico-symbolic Probability Theory for the Management of Uncertainty. CC-AI, 11(4): 417–484, 1994.Google Scholar
  17. 17.
    J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Revised second printing, Morgan Kaufmann, San Francisco, 1991.Google Scholar
  18. 18.
    J. L. Pollock. Nomic Probabilities and the Foundations of Induction. Oxford University Press, 1990.Google Scholar
  19. 19.
    A.L. Ralescu, B. Bouchon-Meunier, D.A. Ralescu. Combining Fuzzy Quantifiers. Rapport interne de LAFORIA, 1996.Google Scholar
  20. 20.
    L. Sombé. Reasoning under incomplete information in artificial intelligence, John Wiley, 1990.Google Scholar
  21. 21.
    Y. Xiang, M.P. Beddoes, D. Poole. Can Uncertainty Management be realized in a finite totally ordered Probability Algebra. Uncertainty in Artificial Intelligence 5: 41–57, 1990.Google Scholar
  22. 22.
    R.R. Yager. Reasoning with quantified statements. Part I, Kybernetes, 14, 233–240, 1985.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R.R. Yager. Reasoning with quantified statements. Part II, Kybernetes, 14, 111–120, 1986.MathSciNetCrossRefGoogle Scholar
  24. 24.
    L.A. Zadeh. Fuzzy sets. Information and Control, 8: 338–353, 1965.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    L.A. Zadeh. Syllogistic reasoning in Fuzzy logic and its application to usuality and reasoning with dispositions. IEEE Transactions on Systems, Man and Cybernetics, 15 (6): 754–763, 1985.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Mohamed Yasser Khayata
    • 1
  • Daniel Pacholczyk
    • 1
  1. 1.LERIA, U.F.R SciencesAngers Cedex 01France

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