Syntactically, adding conditional probability doesn't do anything new, besides serving as short-hand for expressions that look like "P(A∩B)/P(B)".

The issue you're recognizing is that, semantically, conditional probabilities should mean something and behave coherently with the system of probability you've built up to this point.

This is not really a question of probability theory but instead a question of interpretations of probability. As such, it has a very philosophical feel and I'm not sure it's going to have the solid answers you're looking for.