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.
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Hájek, A. What Conditional Probability Could Not Be. Synthese 137, 273–323 (2003). https://doi.org/10.1023/B:SYNT.0000004904.91112.16
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DOI: https://doi.org/10.1023/B:SYNT.0000004904.91112.16