Frequent commenter Kenny Easwaran (who also has a weblog, Antimeta, devoted to philosophy of math) has written several interesting essays on the interpretation of conditional probability:

- What Conditional Probability Must (Almost) Be
- Two Analyses of Conditional Probability in Terms of Unconditional

The question is practically and philosophically interesting in the case that the event you are conditioning on occurs with probability zero.

Thanks for the publicity!

Anyway, that first paper is a development of the ideas in the first half of the second one, so there’s a good deal of redundancy if you look at both.