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lmm comments on Skepticism about Probability - Less Wrong
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Belief in the axioms of probability theory is justified by the fact that someone with inconsistent beliefs can be Dutch-booked.
If you're willing to put money on your beliefs (i.e. bet on them), then you ought to believe in the axioms in the first place, otherwise your opponent will always be able to come up with a combination of bets that will cause you to lose money.
This fact was proved by Bruno de Finetti in 1930-ties. See e.g. AI: A Modern Approach for an easily approachable technical discussion.
I think De Finetti's justification is fine as far as it goes, but it doesn't go quite as far as people think it does. Here's a couple dialogues to illustrate my point.
Dialogue 1
A: I have secretly flipped a fair coin and looked at the result. What's your probability that the coin came up heads?
B: I guess it's 50%.
A: Great! Will you accept a bet against me that the coin came up heads, at 1:1 odds?
B: Hmm, no, that doesn't seem fair because you already know the outcome of the coinflip and chose the bet accordingly.
A: So rational agents shouldn't necessarily accept either side of a bet according to their stated beliefs?
B: I suppose so.
Dialogue 2
A: I believe the sky is green with probability 90% and also blue with probability 90%.
B: Great! I can Dutch book you now. Here's a bet I want to make with you.
A: No, I don't wanna accept that bet. The theory doesn't force me to, as we learned in Dialogue 1.
In Dialogue 1 I adjust my probability estimate as the bet is offered, no?
That's a reasonable thing to do, but can you obtain something like De Finetti's justification of probability via Dutch books that way?
The obvious steelman of dialogue participant A would keep the coin hidden but ready to inspect, so that A can offer bets having credible ignorance of the outcomes and B isn't justified in updating on A offering the bet.