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But then someone comes along and says: How much would you bet on X? And bayes comes back.
By “ill-posed” I meant that it’s wrong to ask about the “probability” of this thing, as if that had a well-defined answer. Instead it would be better to ask, “At what odds would you bet that...?” But then the question isn’t interesting :P
I think the issue with that statement is that it shows a problem in bayesian epistemology: one must give probability=1 or =0 to logically true statements, but the framework also assumes that we are omniscient so that we could be able to answer 0 or 1 on the spot
For example, P(1=1) is well defined, as 1 in the framework. It gets a bit quantum mechanical, doesn't it? with superposed 0 and 1. There is some random quantum epistemology papers that try to formalise that I read years ago, maybe they eventually make sense.
The Kolmogorov axioms just define the type of measure that we mean by "probability". It doesn't say anything about what can/can't be assigned probabilities. Or am I missing something in this discussion?
Ah I misread. If you're saying that the Kolmogorov axioms don't tell you *which* probability function to use, that's definitely true. But neither does Cox's theorem or anything similar, so you're in the same position.
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