Two arguments against modeling probabilities by size of propositions
To my surprise, there are quite a few people here at ANU who believe that probabilities of various kinds can be modeled in terms of relative size of propositions: something has probability 1 if it is true in all (or 100%) of the relevant worlds, probability 0 if it is true in none (or 0%), and probability 0.5 if it is true in half of the worlds (or 50%). I also find it surprisingly hard to explain why I think that's wrong. Here are two arguments I've come up with so far (apart from obvious worries about making sense of these fractions in infinite and proper-class cases).
First, an argument from relativity. Take objective chance, the kind of thing quantum physics talks about. In the actual world, the chance of a radium atom decaying within 1600 years is roughly 0.5. But this is a contingent fact. In other worlds the chance is 0.9, or 0.1. On the probabilities-by-sizes account, the relative size of the proposition will therefore differ depending on which world is actual. But why would it? In fact, it seems not unreasonable to assume that for any world w that is possible relative to the actual world, there is a corresponding world w' that is possible relative to worlds where the chances are different such that w and w' agree on all relevant matters of particular fact, e.g. on when individual radium atoms decay. After all, our laws do not rule out that all radium atoms decay within a millisecond, or only after a billion years; they only declare this extremely improbable. So any possible decay pattern that is nomologically possible relative to this world is also possible relative to the other worlds. But then it can't be the case that within the worlds accessible from here, 50% of all radium atoms decay within 1600 years whereas within the worlds accessible from elsewhere, only 10% do so.
Likewise for credence. It often happens that you and I assign different credence to some proposition, say, to whether the latest US elections were rigged. According to the probabilities-by-sizes account, this means that such worlds are more common in your belief space than in mine (or vice versa). But again, couldn't we assign different credence to this proposition even though there is no proposition that I rule out and you do not (or vice versa)? If there is no such proposition, there is presumably a 1-1 map between the worlds in your space and the worlds in mine which maps rigged worlds to rigged worlds and non-rigged worlds to non-rigged worlds. And then it's hard to see how the fraction of rigged worlds could be higher in your space than in mine (or vice versa).
Second, an argument from probability distributions. Let's again begin with objective single-case chance. Physics tells us how the probabilities are distributed, and this distribution is almost never uniform: the chance that a radium atom decays within between 1000 and 2000 years is much higher than that it decays between 2000 and 3000 years, and that again is much higher than a decay between 3000 and 4000 years, and so on, without end. So there are lots of possibilities that have an extremely low chance of coming true, whereas there are comparatively few possibilities that have a comparatively high chance of coming true. So chance doesn't correspond to proportion among possible cases. This is particularly obvious if we assume that the total number of nomologically possible worlds is finite. Suppose for example that there are exactly 1 billion times at which an atom can decay. Then physics tells us that a handful of these have a high chance of truth, whereas the overwhelming majority is very improbable.
Again, this carries over to credence. Our credences are also probability distributions, which often assign higher probability to more specific propositions. To take a dummy example, suppose I give credence 0.5 to some very specific proposition p, true at only one world w. In case p is false, I am pretty much undecided between various options: ~p & q & r, ~p & ~q & r, ~p & q & ~r, ~p & ~q & ~r; all get about the same credence of 0.125. But now there must be at least 5 worlds in my belief space, and p isn't true at half of them. But still I give it credence 0.5.
I suppose one could get out of these problems by using indistinguishable worlds and adjusting their numbers in such a way that the probabilities-by-sizes account becomes true: whenever p has probability 0.5, it is true in half the relevant worlds. If only one world comes into question, simply add in sufficiently many perfect duplicates of it. The 1-1 maps considered above then turn out to be 1-many or many-1, where the manys are indistinguishable duplicates. But that's cheating, and we learn nothing at all about probability from the fact that it can be interpreted in this way.
That's good - I've discussed these things with some of the people at ANU and never been totally convinced by their point. I can see why it might be appealing to say that probability corresponds to size of the set of possibilities (whether possible worlds or whatever you've got in the subjective case), but I think your point about uniform distributions is right. There's no reason for things to have uniform distributions (as we see from Bertrand's paradox) and in fact, we know that there can be no uniform distribution either over the natural numbers, or even over the set of all real numbers (though there can be a uniform distribution over any bounded interval of real numbers). The only way to make probabilities always work out as uniform distributions seems to be just to stipulate that this is how you'll measure size. Or else hope that the size works out in such a way that it's possible to have a uniform distribution.
However, maybe there is a way to make sense of some of the symmetry arguments that Alan Hájek likes to make, in a less direct way than what you're considering here.