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JoshuaZ comments on Reflection in Probabilistic Logic - Less Wrong
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Not really my field of expertise, thus I may be missing something, but can't you just set P = 1 for all classically provably true statements, P = 0 for all provably false statements and P = 0.5 to the rest, essentially obtaining a version of three-valued logic?
If that's correct, does your approach prove anything fundamentally different than what has been proved about known three-valued logics such as Kleene's or Łukasiewicz's?
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.
That won't work because some of the unprovable claims are, through Gödel-coding, asserting that their own probabilities are different from 0.5.