Some epistemological ramifications of the Borel-Kolmogorov paradox

This paper discusses conditional probability P( A | B), or the probability of A given B. When P(B) > 0, the ratio formula determines P( A | B). When P(B) = 0, the ratio formula breaks down. The Borel-Kolmogorov paradox suggests that conditional probabilities in such cases are indeterminate or ill-posed. To analyze the paradox, I explore the relation between probability and intensionality. I argue that the paradox is a Frege case, similar to those that arise in many probabilistic and non-probabilistic contexts. The paradox vividly illustrates how an agent’s way of representing an entity can rationally influence her credal assignments. I deploy my analysis to defend Kolmogorov’s relativistic treatment of conditional probability. §1. The paradox of the sphere Bertrand (1889) introduced several “problems of geometric probability” that continue to spark debate. Especially puzzling is the so-called Borel-Kolmogorov paradox. Imagine that we randomly pick a point  on the surface of the Earth, considered as a perfect sphere. Say that  falls on an arc C comprising half a great circle. Consider a subarc D occupying 1/4 of C. What is the probability that  falls on D, given that it falls on C? In other words, what is the conditional probability P(  falls on D |  falls on C)? Let us notate this conditional probability more compactly as P( D | C). An intuitively compelling argument runs as follows (see Figure 1): We may choose coordinates so that C comprises half the equator (say, the half falling in the Western Hemisphere). Since we picked  randomly, it is equally likely to fall anywhere on the equator. Thus, the conditional probability that  falls on some subarc is proportional to the subarc’s length. So P( D | C) = C of length D of length = 1⁄4.