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Synthese

, Volume 137, Issue 3, pp 273–323 | Cite as

What Conditional Probability Could Not Be

  • Alan Hájek
Article

Abstract

Kolmogorov's axiomatization of probability includes the familiarratio formula for conditional probability:

\(({\text{RATIO}}) P(A|B) = \frac{{P(A \cap B)}}{{P(B)}}{\text{ }}(P(B) >0).\)

Call this the ratio analysis of conditional probability. It has become so entrenched that it is often referred to as the definition of conditional probability.I argue that it is not even an adequate analysis of that concept. I prove what I call the Four Horn theorem, concluding that every probability assignment has uncountably many ‘trouble spots’. Trouble spots come in four varieties: assignments of zero togenuine possibilities; assignments of infinitesimals to such possibilities; vague assignments to such possibilities; and no assignment whatsoever to such possibilities. Each sort of trouble spot can create serious problems for the ratio analysis. I marshal manyexamples from scientific and philosophical practice against the ratio analysis. I conclude more positively: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it.

“I'd probably be famous now If I wasn't such a good waitress.”

Jane Siberry, “Waitress”

Keywords

Trouble Spot Conditional Probability Ratio Analysis Probability Assignment Unconditional Probability 
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. Appiah, Anthony: 1985, Assertion and Conditionals, Cambridge University Press, Cambridge.Google Scholar
  2. Carnap, Rudolf: 1952 The Continuum of Inductive Methods, University of Chicago Press.Google Scholar
  3. Carnap, Rudolf: 1963, 'Carnap's Intellectual Autobiography', in Paul Arthur Schilpp (ed.), The Philosophy of Rudolf Carnap, The Library of Living Philosophers, Vol. XI.Google Scholar
  4. Cartwright, Nancy: 1979, 'Causal Laws and Effective Strategies', Noûs 13, 419–437.Google Scholar
  5. Cartwright, Nancy: 1999, The Dappled World, Cambridge University Press, Cambridge.Google Scholar
  6. Cushing, James T. and Ernan McMullin (eds.): 1989, Philosophical Consequences of Quantum Theory-Reflections on Bell's Theorem, University of Notre Dame Press, Indiana, pp. 224–253.Google Scholar
  7. de Finetti, Bruno: 1972, Probability, Induction and Statistics, Wiley, New York.Google Scholar
  8. de Finetti, Bruno: 1980, 'Foresight: Its Logical Laws, Its Subjective Sources', in Henry E. Kyburg, Jr. and Howard E. Smokler (eds.), Studies in Subjective Probability, Robert E. Krieger Publishing Company, pp. 53–118.Google Scholar
  9. de Finetti, Bruno: 1990 (originally published 1974), Theory of Probability, Vol. 1, Wiley Classics Library, John Wiley & Sons.Google Scholar
  10. Earman, John: 1992, Bayes or Bust?, MIT Press.Google Scholar
  11. Edgington, Dorothy: 1995, 'On Conditionals', Mind 104, 235–329.Google Scholar
  12. Edgington, Dorothy: 1996, 'Lowe on Conditional Probability', Mind 105, 617–630.Google Scholar
  13. Edwards, W., Lindman, H., and Savage, L. J.: 1963, 'Bayesian Statistical Inference for Psychological Research', Psychological Review LXX, 193–242.Google Scholar
  14. Eells, Ellery: 1991, Probabilistic Causality, Cambridge University Press, Cambridge.Google Scholar
  15. Etchemendy, John: 1990, The Concept of Logical Consequence, Harvard University Press.Google Scholar
  16. Fetzer, James: 1983, 'Probability and Objectivity in Deterministic and Indeterministic Situations', Synthese 57, 367–386.Google Scholar
  17. Fine, Arthur: 1989, 'Do Correlations Need to be Explained?', in Cushing and McMullin, pp. 175–194.Google Scholar
  18. Fine, Terrence: 1973, Theories of Probability, Academic Press.Google Scholar
  19. Gaifman, Haim:1988, 'A Theory of Higher Order Probabilities', in Brian Skyrms and William L. Harper (eds.), Causation, Chance, and Credence, Kluwer Academic Publishers, pp. 191–219.Google Scholar
  20. Gillies, Donald: 2000, 'Varieties of Propensity', British Journal for the Philosophy of Science 51, 807–835.Google Scholar
  21. Gilboa, Itzhak: 1994, 'Can Free Choice Be Known?', in C. Bicchieri, R. Jeffrey and B. Skyrms (eds.), The Logic of Strategy, Oxford University Press.Google Scholar
  22. Goldstein, M.: 1983, 'The Prevision of a Prevision', Journal of the American Statistical Association 78, 817–819.Google Scholar
  23. Hacking, Ian: 1965, The Logic of Statistical Inference, Cambridge University Press, Cambridge.Google Scholar
  24. Hájek, Alan: 1997, '“Mises Redux”-Redux: Fifteen Arguments Against Finite Frequentism', Erkenntnis 45, 209–227.Google Scholar
  25. Howard, Don: 1989, 'Holism, Separability and the Metaphysical Implications of the Bell Experiments', in Cushing and McMullin.Google Scholar
  26. Howson, Colin and Peter Urbach: 1993, Scientific Reasoning: The Bayesian Approach, Open Court, 2nd edition.Google Scholar
  27. Jackson, Frank: 1987, Conditionals, Blackwell.Google Scholar
  28. Jarrett, Jon P.: 1989, 'Bell's Theorem: A Guide to the Implications', in Cushing and McMullin, pp. 60–79.Google Scholar
  29. Jeffrey, Richard: 1983a, 'Bayesianism with a Human Face', in John Earman (ed.), Testing Scientific Theories, Minnesota Studies in the Philosophy of Science, vol. X, pp. 133–156.Google Scholar
  30. Jeffrey, Richard: 1983b, The Logic of Decision, 2nd ed., University of Chicago Press.Google Scholar
  31. Jeffreys, Harold: 1961, Theory of Probability, 3rd ed., Oxford, Clarendon Press.Google Scholar
  32. Johnson, W. E.: 1921, Logic, Cambridge University Press.Google Scholar
  33. Keynes, John Maynard: 1921, A Treatise on Probability, Macmillan and co.Google Scholar
  34. Kemeny, J.: 1955, 'Fair Bets and Inductive Probabilities', Journal of Symbolic Logic 20, 263–273.Google Scholar
  35. Kolmogorov, A. N.: 1933, Grundbegriffe der Wahrscheinlichkeitrechnung, Ergebnisse Der Mathematik; translated as Foundations of Probability, Chelsea Publishing Company, 1950.Google Scholar
  36. Kyburg, Henry: 1988, 'Powers', in William L. Harper and Brian Skyrms (eds.), Causation in Decision, Belief Change, and Statistics, Kluwer Academic Publishers, pp. 71–82.Google Scholar
  37. Levi, Isaac: 1974, 'On Indeterminate Probabilities', The Journal of Philosophy 71, 391–418.Google Scholar
  38. Levi, Isaac: 1978, 'Coherence, Regularity and Conditional Probability', Theory and Decision 9, 1–15.Google Scholar
  39. Levi, Isaac: 1980, The Enterprise of Knowledge, The MIT Press.Google Scholar
  40. Levi, Isaac: 1997, The Covenant of Reason: Rationality and the Commitments of Thought, Cambridge University Press.Google Scholar
  41. Lewis, David: 1973, Counterfactuals, Harvard University Press, Cambridge, Massachusetts.Google Scholar
  42. Lewis, David: 1980, 'A Subjectivist's Guide to Objective Chance', in Studies in Inductive Logic and Probability, Vol. II., University of California Press, pp. 263–293; reprinted in Lewis 1986b.Google Scholar
  43. Lewis, David: 1986a, 'Probabilities of Conditionals and Conditional Probabilities II', Philosophical Review 95, 581–589.Google Scholar
  44. Lewis, David: 1986b, Philosophical Papers Volume II, Oxford University Press.Google Scholar
  45. Lewis, David: 1986c, On the Plurality of Worlds, Blackwell.Google Scholar
  46. Lewis, David: 1994a, 'Reduction of Mind', in Samuel Guttenplan (ed.), A Companion to the Philosophy of Mind, Blackwell, pp. 412–431.Google Scholar
  47. Lewis, David: 1994b, 'Humean Supervenience Debugged', Mind 103, 473–490.Google Scholar
  48. Lindstrøm, Tom: 1988, 'An Invitation to Nonstandard Analysis', in Nigel Cutland (ed.), Nonstandard Analysis and its Applications, pp. 1–105.Google Scholar
  49. Mellor, D. H.: 1971, The Matter of Chance, Cambridge University Press, Cambridge.Google Scholar
  50. Mellor, D. H.: 1993, 'How to Believe a Conditional', Journal of Philosophy 90, 233–248.Google Scholar
  51. Pearl, Judea: 1988, Probabilistic Reasoning in Intelligent Systems, Morgan-Kaufman.Google Scholar
  52. Pitowski, Itamar: 1989, Quantum Probability-Quantum Logic, Springer-Verlag.Google Scholar
  53. Popper, Karl: 1959a, 'The Propensity Interpretation of Probability', British Journal of the Philosophy of Science 10, 25–42.Google Scholar
  54. Popper, Karl: 1959b, The Logic of Scientific Discovery, Basic Books; reprint edition 1992, Routledge.Google Scholar
  55. Price, Huw: 1986, 'Conditional Credence', Mind 95, 18–36.Google Scholar
  56. Ramsey, F. P.: 1926, 'Truth and Probability', in R. B. Braithwaite (ed.), Foundations of Mathematics and other Essays, Routledge & Kegan Paul, 1931, pp. 156–198; reprinted in Henry E. Kyburg, Jr. and Howard E. Smokler (eds.), Studies in Subjective Probability, 2nd ed., Robert E. Krieger Publishing Company, 1980, pp. 23–52; reprinted in D. H.Google Scholar
  57. Mellor (ed.), Philosophical Papers, Cambridge University Press, Cambridge, 1990, pp. 52–96.Google Scholar
  58. Reichenbach, Hans: 1949, The Theory of Probability, University of California Press, Berkeley.Google Scholar
  59. Renyi, Alfred: 1970, Foundations of Probability, Holden-Day, Inc.Google Scholar
  60. Robinson, Abraham: 1966, Non-Standard Analysis, North Holland Publishing Company, Amsterdam.Google Scholar
  61. Roeper, P. and H. Leblanc: 1999, Probability Theory and Probability Logic, University of Toronto Press.Google Scholar
  62. Rosenkrantz, R. D.: 1981, Foundations and Applications of Inductive Probability, Ridgeview Publishing Company.Google Scholar
  63. Salmon, Wesley: 1980, 'Probabilistic Causality', Pacific Philosophical Quarterly 61, 50–74.Google Scholar
  64. Savage, L. J.: 1954, The Foundations of Statistics, John Wiley.Google Scholar
  65. Schervish, M., T. Seidenfeld, and J. Kadane: 1984, 'The Extent of Non-Conglomerability of Finitely Additive Probability', Z.War. 66, 205–226.Google Scholar
  66. Seidenfeld, T.: 2001, 'Remarks on the Theory of Conditional Probability: Some Issues of Finite Versus Countable Additivity', in V. F. Hendricks et al. (eds.), Probability Theory, Kluwer Academic Publishers, pp. 167–178.Google Scholar
  67. Seidenfeld, Teddy, Mark J. Schervish, and Joseph B. Kadane: 2001, 'Improper Regular Conditional Distributions', The Annals of Probability 29(4), 1612–1624.Google Scholar
  68. Shimony, Abner: 1970, 'Scientific Inference', in Robert Colodny (ed.), The Nature and Function of Scientific Theories, University of Pittsburgh Press.Google Scholar
  69. Skyrms, Brian: 1980, Causal Necessity, Yale University Press.Google Scholar
  70. Skyrms, Brian: 1995, 'Strict Coherence, Sigma Coherence and the Metaphysics of Quantity', Philosophical Studies 77, 39–56.Google Scholar
  71. Skyrms, Brian: 2000, Choice and Chance, Wadsworth, Inc, 4th ed.Google Scholar
  72. Spohn,Wolfgang: 1977, 'Where Luce and Krantz Do Really Generalize Savage's Decision Model', Erkenntnis 11, 113–134.Google Scholar
  73. Spohn, Wolfgang: 1986, 'The Representation of Popper Measures', Topoi 5, 69–74.Google Scholar
  74. Stalnaker, Robert: 1970, 'Probabilities and Conditionals', Philosophy of Science 37, 64–80.Google Scholar
  75. Suppes, Patrick: 1970, A Probabilistic Theory of Causality, North Holland Publishing Company, Amsterdam.Google Scholar
  76. Tarski, Alfred: 1956, 'On the Concept of Logical Consequence', in Logic, Semantics, Metamathematics, Oxford: Clarendon Press; Hackett Publishing Company, Indianapolis, Indiana, 1983. (Trans. by J. H. Woodger, 2nd. ed. edited and introduced by John Corcoran.)Google Scholar
  77. van Fraassen, Bas: 1984, 'Belief and the Will', Journal of Philosophy 81, 235–256.Google Scholar
  78. van Fraassen, Bas: 1989, Laws and Symmetry, Clarendon Press, Oxford.Google Scholar
  79. van Fraassen, Bas: 1990, 'Figures in a Probability Landscape', in J.M. Dunn and A. Gupta (eds.), Truth or Consequences, Kluwer, pp. 345–356.Google Scholar
  80. van Fraassen, Bas: 1991, Quantum Mechanics-An Empiricist View, Clarendon Press, Oxford.Google Scholar
  81. van Fraassen, Bas: 1995a, 'Belief and the Problem of Ulysses and the Sirens', Philosophical Studies 77, 7–37.Google Scholar
  82. van Fraassen, Bas: 1995b, 'Fine-Grained Opinion, Conditional Probability, and the Logic of Belief', Journal of Philosophical Logic 24, 349–377.Google Scholar

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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Alan Hájek

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