The problem is update procedure. 

When you are conditioning on empirical fact, you are imaging set of logically consistent worlds where this empirical fact is true and ask yourself about frequency of other empirical facts inside this set.

But it is very hard to define update procedure for fact "P=NP", because one of the worlds here is logically inconsistent, which implies all other possible facts which makes notion of "frequency of other facts inside this set" kinda undefined.

I can reason as follows: There is 0.5 chance that it is Heads. Let P represent the actual, unknown, state of the outcome of the toss (Heads or Tails); and let Q represent the other state. If Q, then anything follows. For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000, and I should be willing to pay that much to take the bet.

This doesn't go through, what you have are two separate propositions "H -> (T -> [insert absurdity here]" and "T -> (H -> [insert absurdity here]" ^[1], and actually deriving a contradiction from the consequent requires proving which antecedent obtains, which you can't do since neither is a theorem. 

The distinction then with logical uncertainty is supposedly that you do already have a proof of the analogue of H or T, so you can derive the consequent that either H or T derives a contradiction 

1. ^{^}

   You don't really have these either, unless you can prove NOT(H AND T) i.e. can you definitively rule out a coin landing both heads and tails? But that's kinda pedantic

Yes, both of these credences should obey the axioms of a probability space.

This sort of thing is applied in cryptography with the concept of "probable primes", which are numbers (typically with many thousands of decimal digits) that pass a number of randomized tests. The exact nature of the tests isn't particularly important, but the idea is that for every composite number, most (at least 3/4) of the numbers less than it are "witnesses" such that when you apply a particular procedure using that number, the composite number fails the test but primes have no such failures.

So the idea is that you pick many random numbers, and each pass gives you more confidence that the number is actually prime. The probability of any composite number passing (say) 50 such tests is no more than 4^-50, and for most composite numbers it is very much less than that.

No such randomized test is known for parity of the googolth digit of pi, but we also don't know that there isn't one. If there was one, it would make sense to update credence using the results of such tests using probability axioms.

With empirical uncertainty, it's easier to abstract updating from reasoning. You can reason without restrictions, and avoid the need to update on new observations, because you are not making new observations. You can decide to make new observations at the time of your own choosing, and then again freely reason about how to update on them.

With logical uncertainty, reasoning simultaneously updates you on all kinds of logical claims that you didn't necessarily set out to observe at this time, so the two processes are hard to disentangle. It would be nice to have better conceptual tools for describing what it means to have a certain state of logical uncertainty, and how it should be updated. But that doesn't quite promise to solve the problem of reasoning always getting entangled with unintended logical updating.

In theories of Bayesianism, the axioms of probability theory are conventionally assumed to say that all logical truths have probability one, and that the probability of a disjunction of logically inconsistent statements is the sum of their probabilities. Corresponding to the second and third Kolmogorov axiom.

If one then e.g. regards the Peano axioms as certain, then all theorems of Peano arithmetic must also be certain, because those are just logical consequences. And all statements which can be disproved in Peano arithmetic then must have probability zero. So the above version of the Kolmogorov axioms is assuming we are logically omniscient. So this form of Bayesianism doesn't allow us to assign anything like 0.5 probability to the googolth digit of pi being odd: We must assign 1 if it's odd, or 0 if it's even.

I think the simple solution is to not talk about logical tautologies and contradictions when expressing the Kolmogorov axioms for a theory of subjective Bayesianism. Instead talk about what we actually know a priori, not about tautologies which we merely could know a priori (if we were logically omniscient). Then the second Kolmogorov axiom says that statements we actually know a priori have to be assigned probability 1, and disjunctions of statements actually known a priori to be mutually exclusive have to be assigned the sum of their probabilities.

Then we are allowed to assign probabilities less than 1 to statements where we don't actually know that they are tautologies, e.g. 0.5 to "the googolth digit of pi is odd" even if this happens to be, unbeknownst to us, a theorem of Peano arithmetic.

I agree that there should not be a fundamental difference. Actually, I think that when an A.I. is reasoning about improving its reasoning ability some difficulties arise that are tricky to work out with probability theory, but similar to themes that have been explored in logic / recursion theory. But that only implies we haven't worked out the versions of the logical results on reflectivity for uncertain reasoning, not that logical uncertainty is in general qualitatively different than probability. In the example you gave I think it is perfectly reasonable to use probabilities, because we have the tools to do this. 

See also my comment on a recent interesting post from Jessica Taylor: https://www.lesswrong.com/posts/Bi4yt7onyttmroRZb/executable-philosophy-as-a-failed-totalizing-meta-worldview?commentId=JYYqqpppE7sFfm9xs

If Q, then anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, Q implies that I will win $1 billion.

I'm not sure even this is the case.

Maybe there's a more sophisticsted version of this argument, but at this level, we only know the implication Q=>$1M is true, not that $1M is true. If Q is false, the implication being true says nothing about $1M.

But more generally, I agree there's no meaningful difference. I'm in the de Finetti school of probability in that I think it only and always expresses our personal lack of knowledge of facts.

If you make a lot of educated guesses about the googolth digit of pi based on chains of reasoning that are actually possible for humans, around 50% of them will get its parity right.

(Of course that's reversed, in a way, since it isn't stating that the digit is uncertain.)

Contrary to what too many want to believe, probability theory does not define what "the probability" is. It only defines these (simplified) rules that the values must adhere to:

1. Every probability is greater than, or equal to, zero.
2. The probability of the union of two distinct outcomes A and B is Pr(A)+Pr(B).
3. The probability of the universal event (all possible outcomes) is 1.

Let A="googolth digit of pi is odd", and B="googolth digit of pi is even." These required properties only guarantee that Pr(A)+Pr(B)=1, and that each is a non-zero number. We only "intuitively" say that Pr(A)=Pr(B)=0.5 because we have no reason to state otherwise. That is, we can't assert that Pr(A)>Pr(B) or Pr(A)<Pr(B), so we can only assume that Pr(A)=Pr(B). But given a reason, we can change this.

The point is that there are no "right" or "wrong" statements in probability. Only statements where the probabilities adhere to these requirements. We can never say what a "probability is," but we can rule out some sets of probabilities that violate these rules.