[lisper]

> A continuous distribution does not assign probabilities to each proposition but subsets of them.

Nope. It assigns probabilities to individual propositions.

> Most people would not consider 2 a probability.

That's true, but in your example 2 is not a probability but a probability density, and a probability density is not the same as a probability. To get a probability out of a probability density you have to integrate. For each possible interval over which you could integrate there is a corresponding proposition whose probability of being true is exactly the value of the integral.

[cf]

Ok I see where my confusion was. For you propositions are things that can be associated with measurable sets.

[lisper]

No, propositions are simply statements for which is it meaningful to assign a truth value. It might be possible to come up with a proposition that is associated with a non-measurable set, though I can't offhand think of an example. But remember: Bayesian probabilities are models of belief. Priors are peronsal. And so in order to assign a prior to a proposition, the statement of the proposition must have some referent in your personal ontology. You can't hold a belief about the truth value of a statement unless you know (or at least think you know) what that statement means. So unless a person has something in their ontology that corresponds to a non-measurable set (and I suspect most people don't) then that person cannot assign a Bayesian prior to a statement associated with a non-measurable set. But that's a limitation of that particular individual, not a limitation of Bayesian reasoning. For example: you cannot assign a Bayesian prior to the statement, "The frobnostication of any integer is even" because you don't know what a frobnostication is.

(Note that there are all kinds of ways that statements can fail to be propositions. For example, you can't assign a Bayesian prior to the statement: "All even integers are green" despite the fact that you know what all the words mean.)