Advertisement

The Relationship of the Logic of Big-Stepped Probabilities to Standard Probabilistic Logics

  • Christoph Beierle
  • Gabriele Kern-Isberner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5956)

Abstract

Different forms of semantics have been proposed for conditionals of the form ”Usually, if A then B”, ranging from quantitative probability distributions to qualitative approaches using plausibility orderings or possibility distributions. Atomic-bound systems, also called big-stepped probabilities, allow qualitative reasoning with probabilities, aiming at bridging the gap between qualitative and quantitative argumentation and providing a model for the nonmonotonic reasoning system P. By using Goguen and Burstall’s notion of institutions for the formalization of logical systems, we elaborate precisely which formal connections exist between big-stepped probabilities and standard probabilities, thereby establishing the exact relationships among these logics.

Keywords

conditional logic probabilistic logic big-stepped probability institution institution morphism 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, E.W.: The Logic of Conditionals. D. Reidel, Dordrecht (1975)CrossRefMATHGoogle Scholar
  2. 2.
    Arló Costa, H., Parikh, R.: Conditional probability and defeasible inference. Journal of Philosophical Logic 34, 97–119 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beierle, C., Kern-Isberner, G.: Using institutions for the study of qualitative and quantitative conditional logics. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 161–172. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Beierle, C., Kern-Isberner, G.: Looking at probabilistic conditionals from an institutional point of view. In: Kern-Isberner, G., Rödder, W., Kulmann, F. (eds.) WCII 2002. LNCS (LNAI), vol. 3301, pp. 162–179. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Beierle, C., Kern-Isberner, G.: Formal similarities and differences among qualitative conditional semantics. International Journal of Approximate Reasoning (2009) (to appear)Google Scholar
  6. 6.
    Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic semantics of conditional knowledge bases. Journal of Logic and Computation 9(6), 873–895 (1999)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the ACM 39(1), 95–146 (1992)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Goguen, J.A., Rosu, G.: Institution morphisms. Formal Aspects of Computing 13(3-5), 274–307 (2002)CrossRefMATHGoogle Scholar
  9. 9.
    Hawthorne, J.: On the logic of nonmonotonic conditionals and conditional probabilities. Journal of Philosophical Logic 25(2), 185–218 (1996)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hawthorne, J., Makinson, D.: The quantitative/qualitative watershed for rules of uncertain inference. Studia Logica 86(2), 247–297 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Herrlich, H., Strecker, G.E.: Category theory. Allyn and Bacon, Boston (1973)MATHGoogle Scholar
  12. 12.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44, 167–207 (1990)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kyburg Jr., H.E., Teng, C.-M., Wheeler, G.R.: Conditionals and consequences. J. Applied Logic 5(4), 638–650 (2007)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Lehmann, D., Magidor, M.: What does a conditional knowledge base entail? Artificial Intelligence 55, 1–60 (1992)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973)Google Scholar
  16. 16.
    Nute, D.: Topics in Conditional Logic. D. Reidel Publishing Company, Dordrecht (1980)CrossRefMATHGoogle Scholar
  17. 17.
    Pearl, J.: Probabilistic semantics for nonmonotonic reasoning: A survey. In: Shafer, G., Pearl, J. (eds.) Readings in uncertain reasoning, pp. 699–710. Morgan Kaufmann, San Mateo (1989)Google Scholar
  18. 18.
    Snow, P.: The emergence of ordered belief from initial ignorance. In: Proceedings AAAI 1994, Seattle, WA, pp. 281–286 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christoph Beierle
    • 1
  • Gabriele Kern-Isberner
    • 2
  1. 1.Dept. of Computer ScienceFernUniversität in HagenHagenGermany
  2. 2.Dept. of Computer ScienceTU DortmundDortmundGermany

Personalised recommendations