What Conditional Probability Could Not Be |
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Hajek (Alan) |

Source: Synthese, Volume 137, Number 3, December 2003, pp. 273-323(51) |

Paper - Abstract |

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__Author’s Abstract__

- 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

*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*to genuine possibilities; - assignments of
*infinitesimals*to such possibilities; *vague*assignments to such possibilities; and*no*assignment whatsoever to such possibilities.

- assignments of
- 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.

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