What Conditional Probability Could Not Be

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. 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. Each sort of trouble spot can create serious problems for the ratio analysis. 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” Over three hundred years ago, Fermat and Pascal began the enterprise of systematizing our pre-mathematical, pre-philosophical concept of probability. Orthodoxy has it that this task was completed in 1933, when Kolmogorov provided his axiomatization. In the process, he systematized another pre-theoretical concept: conditional probability. He identified conditional probability with a certain ratio of unconditional probabilities, according to the formula: (RATIO) P(A | B) = P(A ∩ B) P (B) (P (B) > 0). I will call this the ratio analysis of conditional probability.1 It has become so entrenched that we seem to have forgotten that it is an analysis. My main purposes in this paper are to cast doubt on its adequacy, and to argue Synthese 137: 273–323, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.