What conditional probability could not be.

*(English)*Zbl 1047.03003Summary: Kolmogorov’s axiomatization of probability includes the familiar ratio formula for conditional probability:
\[
\text{(RATIO)}\quad P(A \mid B) = \frac{P(A \cap B)}{P(B)}\quad (P(B) > 0).
\]
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.