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lukeprog comments on Logical Uncertainty as Probability - Less Wrong Discussion
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I don't know much about logical uncertainty, but have ya'll seen Gaifman's paper on the subject?
Gaifman (2004), Reasoning with limited resources and assigning probabilities to arithmetical statements.
From taking a quick look at page 8 above (numbered 104 in the document), if gRR used X instead of OLC(X), it would result in something like the above article. The point is in having consistent probability distribution over held beliefs, not over some infinite closure of beliefs.