What Conditional Probability Must (Almost) Be

Probability theory first reached its modern form in the work of Kolmogorov.1 Along with his famous axioms for unconditional probability, he gave a formula for calculating conditional probabilities. “If P (A) > 0, then the quotient PA(B) = P (AB) P (A) is defined to be the conditional probability of the event B under the condition A.”2 (Throughout this paper, I will use the more standard notation P (B|A) in place of Kolmogorov’s PA(B).) However, since this equation gives no answer for conditional probabilities when the antecedent has probability 0, several philosophers have given different axiomatizations, taking conditional probabilities as basic and defining unconditional probabilities in terms of them.3 In his recent paper [4], Alan Hájek points out that conditional probability is in fact a pre-theoretic notion, and thus can’t be taken to be a purely technical. Thus, each of these proposed sets of axioms is an analysis of the notion, and not a definition, despite Kolmogorov’s use of the word “defined”. Hájek then goes on to argue that Kolmogorov’s analysis is insufficient, and that we must therefore adopt something like Popper’s axioms instead, taking conditional probability to be basic and analyzing unconditional probability in terms of it. However, I will argue that there is no analysis of conditional probability that could be correct while assigning a value to every pair of events, as Popper requires.4 This argument will rely on a sort of “reflection principle”5 stating that if B is the event that exactly one of some pairwise mutually impossible events Eα occurs, then P (A|B) ≥ min{P (A|Eα)}. In addition, I will show that there is a function that satisfies the standard axioms as well as this principle, and that this function was discussed by Kolmogorov already in his foundational work. Any function that could claim to represent conditional probability must almost equal this function. However, this function will take three arguments instead of the standard two for conditional probability. Also, [11] points out that in at least some probability spaces, any such function must violate certain intuitive constraints on a probability function. Thus, conditional probability must often be taken to be defined merely relatively, not absolutely as Hájek