What Conditional Probability Could Not Be
Hajek (Alan)
Source: Synthese, Volume 137, Number 3, December 2003, pp. 273-323(51)
Paper - Abstract

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Author’s Abstract

  1. Kolmogorov's axiomatization of probability includes the familiar ratio formula for conditional probability:
      (RATIO): P(A | B) = P(A & B) / P(B) …. 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.
  2. 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'.
  3. Trouble spots come in four varieties:
    1. assignments of zero to genuine possibilities;
    2. assignments of infinitesimals to such possibilities;
    3. vague assignments to such possibilities; and
    4. no assignment whatsoever to such possibilities.
    Each sort of trouble spot can create serious problems for the ratio analysis.
  4. I marshal many examples 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”

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