, Volume 76, Issue 3, pp 299–318 | Cite as

Two Problems of Direct Inference

  • Paul D. Thorn
Original Article


The article begins by describing two longstanding problems associated with direct inference. One problem concerns the role of uninformative frequency statements in inferring probabilities by direct inference. A second problem concerns the role of frequency statements with gerrymandered reference classes. I show that past approaches to the problem associated with uninformative frequency statements yield the wrong conclusions in some cases. I propose a modification of Kyburg’s approach to the problem that yields the right conclusions. Past theories of direct inference have postponed treatment of the problem associated with gerrymandered reference classes by appealing to an unexplicated notion of projectability. I address the lacuna in past theories by introducing criteria for being a relevant statistic. The prescription that only relevant statistics play a role in direct inference corresponds to the sort of projectability constraints envisioned by past theories.


Frequency Statement Frequency Information Target Class Reference Class Direct Inference 
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.



This work was supported by the LogiCCC EUROCORES program of the ESF and DFG. For helpful comments on earlier presentations of this paper, I thank audiences at the University of Arizona, the University of Düsseldorf, and the Third Formal Epistemology Festival at the University of Toronto. I also thank Terry Horgan, Shaughan Lavine, Gerhard Schurz, and especially John Pollock and two anonymous referees for Erkenntnis for helpful comments on earlier drafts of the paper.


  1. Bacchus, F. (1990). Representing and reasoning with probabilistic knowledge. Cambridge, Massachusetts: MIT Press.Google Scholar
  2. Bacchus, F., Grove, A., Halpern, J., & Koller, D. (1996). From statistical knowledge bases to degrees of belief. Artificial Intelligence, 87(1–2), 75–143.CrossRefGoogle Scholar
  3. Colyvan, M., & Regan, H. (2001). Is it a crime to belong to a reference class? Journal of Political Philosophy, 9(2), 168–181.CrossRefGoogle Scholar
  4. Colyvan, M., Regan, H., & Ferson, S. (2007). Legal decisions and the reference class problem. International Journal of Evidence and Proof, 11(4), 274–286.CrossRefGoogle Scholar
  5. Fetzer, J. (1977). Reichenbach, reference classes, and single case ‘probabilities’. Synthese, 34, 185–217.CrossRefGoogle Scholar
  6. Fitelson, B., Hájek, A., & Hall, N. (2005). Probability. In S. Sarkar & J. Pfeifer (Eds.), Philosophy of science: An encyclopedia. Oxford: Routledge.Google Scholar
  7. Geffner, H., & Pearl, J. (1992). Conditional entailment: bridging two approaches to default reasoning. Artificial Intelligence, 53(2–3), 209–244.CrossRefGoogle Scholar
  8. Goodman, N. (1955). Fact, fiction, and forecast. Cambridge: Harvard University Press.Google Scholar
  9. Hájek, A. (2007). The reference class problem is your problem too. Synthese, 156(3), 563–585.CrossRefGoogle Scholar
  10. Halpern, J. (2003). Reasoning about uncertainty. Cambridge: Harvard University Press.Google Scholar
  11. Hempel, C. (1968). Lawlikeness and maximal specificity in probabilistic explanation. Philosophy of Science, 35(2), 116–133.CrossRefGoogle Scholar
  12. Horty, J., Thomason, R., & Touretzky, D. (1990). A sceptical theory of inheritance in nonmonotonic semantic networks. Artificial Intelligence, 42(2–3), 311–348.CrossRefGoogle Scholar
  13. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1–2), 167–207.CrossRefGoogle Scholar
  14. Kyburg, H. (1961). Probability and the logic of rational belief. Middleton, Connecticut: Wesleyan University Press.Google Scholar
  15. Kyburg, H. (1974). The logical foundations of statistical inference. Dordrecht-Holland: Reidel Publishing Company.CrossRefGoogle Scholar
  16. Kyburg, H., & Teng, C. (2001). Uncertain inference. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. Levi, I. (1982). Direct inference. Journal of Philosophy, 74, 5–29.CrossRefGoogle Scholar
  18. Pollock, J. (1990). Nomic probability and the foundations of induction. Oxford: Oxford University Press.Google Scholar
  19. Pollock, J. (2007). The Y-function. In G. Wheeler & B. Harper (Eds.), Probability and evidence: Essays in honour of Henry E. Kyburg Jr. London: College Publications.Google Scholar
  20. Pollock, J. (unpublished). Probable probabilities. PhilSci Archive, at:
  21. Pust, J. (2011). Sleeping beauty and direct inference. Analysis.Google Scholar
  22. Reichenbach, H. (1949). A theory of probability. Berkeley: Berkeley University Press.Google Scholar
  23. Rhee, R. (2007). Probability, policy and the problem of the reference class. International Journal of Evidence and Proof, 11(4), 286–292.CrossRefGoogle Scholar
  24. Salmon, W. (1971). Statistical explanation. In W. Salmon (Ed.), Statistical explanation and statistical relevance. Pittsburgh: University of Pittsburgh Press.Google Scholar
  25. Salmon, W. (1977). Objectively homogeneous reference classes. Synthese, 36, 399–414.CrossRefGoogle Scholar
  26. Salmon, W. (1984). Scientific explanation and the causal structure of the world. Princeton: Princeton University Press.Google Scholar
  27. Stone, M. (1987). Kyburg, Levi, and Petersen. Philosophy of Science, 54(2), 244–255.CrossRefGoogle Scholar
  28. Thorn, P. (2007). Three problems of direct inference. Dissertation, University of Arizona.Google Scholar
  29. Thorn, P. (forthcoming). Undercutting defeat via reference properties of differing arity: a reply to Pust. Analysis.
  30. Venn, J. (1866). The logic of chance. New York: Chelsea Publishing Company.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Philosophisches InstitutUniversity of DüsseldorfDüsseldorfGermany

Personalised recommendations