Skip to main content
SpringerLink
Log in

Conditional Probability in the Light of Qualitative Belief Change

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on the qualitative part of probabilistic inference leads to a very broad class of ‘proto-probability’ functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alchourrón, C., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: partial meet contraction and revision functions. The Journal of Symbolic Logic, 50, 510–530.

    Article  Google Scholar 

  2. Arló Costa, H. (2001). Bayesian epistemology and epistemic conditionals: on the status of export-import laws. The Journal of Philosophy, 98, 555–593.

    Article  Google Scholar 

  3. Arló Costa, H., & Parikh, R. (2005). Conditional probability and defeasible inference. Journal of Philosophical Logic, 34, 97–119.

    Article  Google Scholar 

  4. Balcázar, J. L. (2010). Redundancy, deduction schemes, and minimum-size bases for association rules. Logical Methods in Computer Science, 6, 1–33.

    Article  Google Scholar 

  5. Bohlmann, G. (1909). Die Grundbegriffe der Wahrscheinlichkeitsrechnung in ihrer Anwendung auf die Lebensversicherung, pages 244–278 of Atti del IV Congresso Internazionale dei Matematici, Roma 6–11 Aprile 1908 (Roma: Accademia dei Lincei), vol. III.

  6. Boutilier, C. (1995). On the revision of probabilistic belief states. Notre Dame Journal of Formal Logic, 36, 158–183.

    Article  Google Scholar 

  7. Carnap, R. (1950). Logical foundations of probability. Chicago: Chicago University Press.

    Google Scholar 

  8. Carnap, R. (1952). The continuum of inductive methods. Chicago: Chicago University Press.

    Google Scholar 

  9. Carnap, R. (1971). A basic system of inductive logic. In R. Carnap & R. Jeffrey (Eds.), Studies in inductive logic and probability (pp. 33–165). Berkeley: University of California Press.

    Google Scholar 

  10. de Finetti, B. (1974). Theory of probability. New York: Wiley.

    Google Scholar 

  11. Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Annals of Probability, 3, 89–99.

    Article  Google Scholar 

  12. Dubois, D., & Prade, H. (1998). Possibility theory: Qualitative and quantitative aspects. In D. M. Gabbay & P. Smets (Eds.), Handbook of defeasible reasoning and uncertainty management systems, vol I (pp. 169–226). Dordrecht: Kluwer.

    Google Scholar 

  13. Gärdenfors, P. (1984). The dynamics of belief as a basis for logic. British Journal for the Philosophy of Science, 35, 1–10.

    Article  Google Scholar 

  14. Gärdenfors, P. (1986). The dynamics of belief: contractions and revisions of probability functions. Topoi, 5, 29–37.

    Article  Google Scholar 

  15. Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. Cambridge: MIT Press.

    Google Scholar 

  16. Hájek, A. (2003). What conditional probability could not be. Synthese, 137, 273–323.

    Article  Google Scholar 

  17. Hailperin, T. (1988). The development of probability logic from Leibniz to MacColl. History and Philosophy of Logic, 9, 131–191.

    Article  Google Scholar 

  18. Hailperin, T. (1991). Probability logic in the twentieth century. History and Philosophy of Logic, 12, 71–110.

    Article  Google Scholar 

  19. Halpern, J. (2001). Plausibility measures: A general approach for representing uncertainty. In Proceedings of the 17th Joint Conference on AI (IJCAI 2001). San Francisco: Morgan Kaufmann.

  20. Halpern, J. (to appear). Lexicographic probability, conditional probability, and nonstandard probability. Games and Economic Behaviour. Meanwhile, see http://www.c.s.cornell.edu/home/halpern/papers/lex.pdf.

  21. Harper, W. L. (1975). Rational belief change, Popper functions and counterfactuals. Synthese, 30, 221–262. Reprinted with minor editorial changes in pages 73–115 of W.L. Harper and C.A. Hooker eds Foundations of Probability, Statistical Inference, and Statistical Theories of Science, Vol I (Dordrecht: Reidel).

  22. Harper, W. L. (1976). Rational conceptual change. In PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1976, Vol. II (pp. 462–494). University of Chicago Press.

  23. Hawthorne, J. (1996). On the logic of nonmonotonic conditionals and conditional probabilities. Journal of Philosophical Logic, 25, 185–218.

    Google Scholar 

  24. Hawthorne, J., & Makinson, D. (2007). The quantitative/qualitative watershed for rules of uncertain inference. Studia Logica, 86, 249–299.

    Article  Google Scholar 

  25. Hosiasson-Lindenbaum, J. (1940). On confirmation. The Journal of Symbolic Logic, 4, 133–148.

    Article  Google Scholar 

  26. Koons, R. (2009). Supplement to defeasible reasoning. Stanford Encyclopedia of Philosophy http://plato.stanford.edu. Accessed 05 April 2009.

  27. Koopman, B. O. (1940). The axioms and algebra of intuitive probability. The Annals of Mathematics, 41, 269–292.

    Article  Google Scholar 

  28. Koopman, B. O. (1940). The bases of probability. Bulletin of the American Mathematical Society, 46, 763–774.

    Article  Google Scholar 

  29. Leblanc, H. (1989). Popper’s formal contributions to probability theory. In M. A. Notturno (Ed.), Perspectives on psychologism (pp. 341–367). Leiden: E.J. Brill.

  30. Leblanc, H., & Roeper, P. (1989). On relativizing Kolmogorov’s absolute probability functions. Notre Dame Journal of Formal Logic, 30, 485–512.

    Article  Google Scholar 

  31. Lindström, S., & Rabinowicz, W. (1989). On probabilistic representation of non-probabilistic belief revision. Journal of Philosophical Logic, 18, 69–101.

    Article  Google Scholar 

  32. Makinson, D. (1997). Screened revision. Theoria, 63, 14–23.

    Article  Google Scholar 

  33. Makinson, D. (2005). Bridges from classical to nonmonotonic logic. London: College Publications.

    Google Scholar 

  34. Makinson, D. (2007). Completeness theorems, representation theorems: What’s the difference? In Rønnow-Rasmussen et al. (Ed.), Hommage à Wlodek: Philosophical Papers dedicated to Wlodek Rabinowicz, www.fil.lu.se/hommageawlodek.

  35. Makinson, D. (to appear). Logical questions behind the lottery and preface paradoxes: lossy rules for uncertain inference. Synthese, issue commemorating Henry Kyburg.

  36. Mazurkiewicz, S. (1932). Zur Axiomatik der Wahrscheinlichkeitsrechnung. Comptes Rendus des Séance de la Société des Sciences et des Lettres de Varsovie (Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego), 25, 1–4.

    Google Scholar 

  37. McGee, V. (1994). Learning the impossible. In E. Eells & B. Skyrms (Eds.), Probability and conditionals: Belief revision and rational decision (pp. 179–199). New York: Cambridge University Press.

    Google Scholar 

  38. Paris, J., & Simmonds, R. (2009). O is not enough. Review of Symbolic Logic, 2, 298–309.

    Article  Google Scholar 

  39. Peppas, P. (2007). Belief revision. In F. van Harmelen, V. Lifschitz, & B. Porter (Eds.), Handbook of knowledge representation, chapter 8. Amsterdam: Elsevier.

    Google Scholar 

  40. Popper, K. (1959). The logic of scientific discovery (2nd ed.). New York: Basic Books.

    Google Scholar 

  41. Rényi, A. (1955). On a new axiomatic theory of probability. Acta Mathematica Academiae Scientiae Hungaricae, 6, 268–335.

    Google Scholar 

  42. Renyi, A. (1970). Foundations of probability theory. San Francisco: Holden-Day.

    Google Scholar 

  43. Rényi, A. (1970). Probability theory. Amsterdam: North-Holland.

    Google Scholar 

  44. Roeper, P., & Leblanc, H. (1999). Probability theory and probability logic. Toronto: University of Toronto Press.

    Google Scholar 

  45. Simmonds, R. (2010). On Horn closure conditions for probabilistic consequence relations. (PhD thesis, University of Manchester).

  46. Stalnaker, R. C. (1970). Probability and conditionals. Philosophy of Science, 37, 64–80, reprinted in pages 107–128 of W.L. Harper et al eds Ifs (Dordrecht: Reidel, 1981).

  47. Spohn, W. (1986). The representation of Popper measures. Topoi, 5, 69–74.

    Article  Google Scholar 

  48. Spohn, W. (2009). A survey of ranking theory. In F. Huber et al. (Eds.), Degrees of belief (pp. 185–228). New York: Springer.

    Chapter  Google Scholar 

  49. van Fraassen, B. (1976). Representation of conditional probabilities. Journal of Philosophical Logic, 5, 417–430.

    Google Scholar 

  50. van Fraassen, B. (1995). Fine-grained opinion, probability, and the logic of full belief. Journal of Philosophical Logic, 24, 349–377.

    Article  Google Scholar 

  51. Weydert, E. (1994). General belief measures. Uncertainty in Artificial Intelligence, 94, 575–582 (San Francisco: Morgan Kaufmann).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Logic and Scientific Method, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    David Makinson

Authors
  1. David Makinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Makinson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Makinson, D. Conditional Probability in the Light of Qualitative Belief Change. J Philos Logic 40, 121–153 (2011). https://doi.org/10.1007/s10992-011-9176-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-011-9176-4

Keywords

Search

Navigation