Preferential Logics for Reasoning with Graded Uncertainty

  • Ofer Arieli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2711)

Abstract

We introduce a family of preferential logics that are useful for handling information with different levels of uncertainty. The corresponding consequence relations are non-monotonic, paraconsistent, adaptive, and rational. It is also shown that any formalism in this family that is based on a well-founded ordering of the different types of uncertainty, can be embedded in a corresponding four-valued logic with at most three uncertainty levels.

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References

  1. 1.
    Arieli, O.: Reasoning with modularly pointwise preferential relations. In: van den Bosch, A., Weigand, H. (eds.) Proc. BNAIC 2000, pp. 61–68. BNVKI (2000)Google Scholar
  2. 2.
    Arieli, O.: Useful adaptive logics for rational and paraconsistent reasoning. Technical Report CW286, Depatrment of Computer Science, University of Leuven (2001)Google Scholar
  3. 3.
    Arieli, O.: Paraconsistent declarative semantics for extended logic programs. Annals of Mathematics and Artificial Intelligence 36(4), 381–417 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arieli, O., Avron, A.: Reasoning with logical bilattices. Journal of Logic, Language, and Information 5(1), 25–63 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arieli, O., Avron, A.: The logical role of the four-valued bilattice. In: Proc. LICS 1998, pp. 218–226. IEEE Press, Los Alamitos (1998)Google Scholar
  6. 6.
    Arieli, O., Avron, A.: Nonmonotonic and paraconsistent reasoning: From basic entailments to plausible relations. In: Hunter, A., Parsons, S. (eds.) ECSQARU 1999. LNCS (LNAI), vol. 1638, pp. 11–22. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Arieli, O., Denecker, M.: Modeling paraconsistent reasoning by classical logic. In: Eiter, T., Schewe, K.-D. (eds.) FoIKS 2002. LNCS, vol. 2284, pp. 1–14. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Avron, A.: Simple consequence relations. Journal of Information and Computation 92, 105–139 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Avron, A.: Classical Gentzen-type methods in propositional many-valued logics. In: Fitting, M., Orlowska, E. (eds.) Theory and Applications in Multiple-Valued Logics, pp. 113–151. Springer, Heidelberg (2002)Google Scholar
  10. 10.
    Batens, D.: Inconsistency-adaptive logics. In: Orlowska, E. (ed.) Logic at Work, pp. 445–472. Physica Verlag, Heidelberg (1998)Google Scholar
  11. 11.
    Batens, D.: On a partial decision method for dynamic proofs. In: Decker, H., Villadsen, J., Waragai, T. (eds.) Proc. PCL 2002, ICLP 2002 Workshop on Paraconsistent Computational Logic, pp. 91–108 (2002)Google Scholar
  12. 12.
    Batens, D., Mortensen, C., Priest, G., Van Bendegem, J.: Frontiers of Paraconsistent Logic. In: Studies in Logic and Computation Vol. 8. Research Studies Press, Hertfordshire (2000)Google Scholar
  13. 13.
    Belnap, N.D.: A useful four-valued logic. In: Epstein, G., Dunn, J.M. (eds.) Modern Uses of Multiple-Valued Logic, pp. 7–37. Reidel Publishing Company, Dordrechtz (1977)Google Scholar
  14. 14.
    Belnap, N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press (1977)Google Scholar
  15. 15.
    Benferhat, S., Besnard, P. (eds.): ECSQARU 2001. LNCS (LNAI), vol. 2143. Springer, Heidelberg (2001)MATHGoogle Scholar
  16. 16.
    Bialynicki-Birula, A.: Remarks on quasi-boolean algebras. Bull. Acad. Polonaise des Sciences Cl. III V(6), 615–619 (1957)MathSciNetGoogle Scholar
  17. 17.
    Bialynicki-Birula, A., Rasiowa, H.: On the representation of quasi-boolean algebras. Bull. Acad. Polonaise des Sciences Cl. III V(3), 259–261 (1957)MathSciNetGoogle Scholar
  18. 18.
    Carnielli, W., Coniglio, M.E., D’Ottaviano, I.M.L.: Paraconsistency: The logical way to the inconsistent. Lecture Notes in Pure and Applied Mathematics, vol. 228. Marcel Dekker (2002)Google Scholar
  19. 19.
    da-Costa, N.C.A.: On the theory of inconsistent formal systems. Notre Dam Journal of Formal Logic 15, 497–510 (1974)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Damasio, C.M., Pereira, L.M.: A survey on paraconsistent semantics for extended logic programs. In: Gabbay, D.M., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2, pp. 241–320. Kluwer, Dordrecht (1998)Google Scholar
  21. 21.
    Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In: Gabbay, D., Hogger, C., Robinson, J. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, pp. 439–513. Oxford Science Publications (1994)Google Scholar
  22. 22.
    Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  23. 23.
    Hunter, A., Parsons, S. (eds.): ECSQARU 1999. LNCS (LNAI), vol. 1638. Springer, Heidelberg (1999)MATHGoogle Scholar
  24. 24.
    Gabbay, D.M.: Theoretical foundation for non-monotonic reasoning in expert systems. In: Apt, K.P. (ed.) Proc. of the NATO Advanced Study Inst. on Logic and Models of Concurrent Systems, pp. 439–457. Springer, Heidelberg (1985)Google Scholar
  25. 25.
    Kalman, J.A.: Lattices with involution. Trans. of the American Mathematical Society 87, 485–491 (1958)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Kifer, M., Lozinskii, E.L.: A logic for reasoning with inconsistency. Automated Reasoning 9(2), 179–215 (1992)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1–2), 167–207 (1990)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artificial Intelligence 55, 1–60 (1992)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Lukasiewicz, T.: Fixpoint characterizations for many-valued disjunctive logic programs with probabilistic semantics. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 336–350. Springer, Heidelberg (2001)Google Scholar
  30. 30.
    Makinson, D.: General theory of cumulative inference. In: Reinfrank, M., Ginsberg, M.L., de Kleer, J., Sandewall, E. (eds.) Non-Monotonic Reasoning 1988. LNCS (LNAI), vol. 346, pp. 1–18. Springer, Heidelberg (1988)Google Scholar
  31. 31.
    Makinson, D.: General patterns in nonmonotonic reasoning. In: Gabbay, D., Hogger, C., Robinson, J. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming 3, pp. 35–110. Oxford Science Publications (1994)Google Scholar
  32. 32.
    McCarthy, J.: Circumscription – A form of non monotonic reasoning. Artificial Intelligence 13(1–2), 27–39 (1980)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Pearl, J.: Reasoning under uncertainty. Annual Review of Computer Science 4, 37–72 (1989)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Priest, G.: Minimally inconsistent LP. Studia Logica 50, 321–331 (1991)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Schlechta, K.: Unrestricted preferential structures. Journal of Logic and Computation 10(4), 573–581 (2000)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Shoham, Y.: Reasoning about change. MIT Press, Cambridge (1988)Google Scholar
  37. 37.
    Subrahmanian, V.S.: Mechanical proof procedures for many-valued lattice-based logic programming. Journal of Non-Classical Logic 7, 7–41 (1990)MathSciNetGoogle Scholar
  38. 38.
    Tarski, A.: Introduction to logic. Oxford University Press, Oxford (1941)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ofer Arieli
    • 1
  1. 1.Department of Computer ScienceThe Academic College of Tel-AvivIsrael

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