The idea that “probability” is some preexisting thing that needs to be “interpreted” as something always seemed a little bit backwards to me. Isn’t it more straightforward to say:

Beliefs exist, and obey the Kolmogorov axioms (at least, “correct” beliefs do, as formalized by generalizations of logic (Cox’s theorem), or by possible-world-counting). This is what we refer to as “bayesian probabilities”, and code into AIs when we want to them to represent beliefs.

Measures over imaginary event classes / ensembles also obey the Kolmogorov axioms. “Frequentist probabilities” fall into this category.

Personally I mostly think about #1 because I’m interested in figuring out what I should believe, not about frequencies in arbitrary ensembles. But the fact is that both of these obey the same “probability” axioms, the Kolmogorov axioms. Denying one or the other because “probability” must be “interpreted” as *exclusively either* #1 or #2 is simply wrong (but that’s what frequentists effectively do when they loudly shout that you “can’t” apply probability to beliefs).

Now, sometimes you *do* need to interpret “probability” as something—in the specific case where someone else makes an utterance containing the word “probability” and you want to figure out what they meant. But the answer there is probably that in many cases people don’t even distinguish between #1 and #2, because they’ll only commit to a specific number when there’s a convenient instance of #2 that make #1 easy to calculate. For instance, saying ^{1}⁄_{6} for a roll of a “fair” die.

People often act as though their utterances about probability refer to #1 though. For instance when they misinterpret p-values as the post-data probability of the null hypothesis and go around believing that the effect is real...

Doesn’t it mean the same thing in either case? Either way, I don’t know which way the coin will land or has landed, and I have some odds at which I’ll be willing to make a bet. I don’t see the problem.

(Though my willingness to bet at all will generally go down over time in the “already flipped” case, due to the increasing possibility that whoever is offering the bet somehow looked at the coin in the intervening time.)