Suppose we have some situation where either event A or event B occurred, but not both, and the two events are on par: our epistemic situation is symmetric between them. Surely:
- One should not assign a different probability to A than to B.
- One should assign the same probability to A as to B.
But while (1) is very plausible (notwithstanding subjective Bayesianism), (2) does not follow from (1), and likewise Indifference does not follow. For (1) is compatible with not assigning any probability to either A or B. And sometimes that is just the right thing to do. For instance, in this post, A and D are on par, but the argument of the post shows that no probability can be assigned to either.
In fact, we can generalize (1):
- One should treat A probabilistically on par with B.
Nonetheless, there is a puzzle. It is very intuitive that sometimes Indifference is correct. Sometimes, we correctly go from the fact that A and B are on par to the claim that they have the same probability. Given (1) (or (3)), to make that move, we need the auxiliary premise that at least one of A and B has a probability.
So the puzzle now is: Under what circumstances do we know of an event that it has a probability? (Cf. this post.)