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benelliott comments on Reflection in Probabilistic Logic - Less Wrong
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In ZF set theory, consider the following three statements.
I) The axiom of choice is false
II) The axiom of choice is true and the continuum hypothesis is false
III) The axiom of choice is true and the continuum hypothesis is true
None of these is provably true or false so they all get assigned probability 0.5 under your scheme. This is a blatant absurdity as they are mutually exclusive so their probabilities cannot possibly sum to more than 1
Ok, it seems that this is covered in the P(phi) = P(phi and psi) + P(phi and not psi) condition. Thanks.