> 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.
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.