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Anatoly_Vorobey comments on Logical Uncertainty as Probability - Less Wrong Discussion
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Comments (22)
You're sidestepping the whole point of cousin_it, which is that your mathematical knowledge is certainly enough to determine whether the millionth digit of pi is odd or even. One of these two statements is a trivial consequence of the Peano axioms and some well-known representation of pi as an infinite series. It's just that its being a trivial consequence is witnessed by a very long (and also trivial) proof which you're not aware of, and you don't know which of the two statements is backed by such a long proof.
I don't think I'm sidestepping the issue. The point of cousin_it's comment, as I understood it, was that assigning probabilities to "logically uncertain" statements results in inconsistencies. What I tried to show is that for probabilistic assignments to be consistent, it is only necessary to be logically omniscient at propositional calculus, not at full-power PA. And this is an important difference, because propositional calculus is decidable.
Ah, well, if you're only closing under propositional tautologies, then you're not doing anything interesting. OLC(X) is for practical purposes the same as X (not just because it's decidable, as you say, but more importantly because it's so weak). So your suggestions boils down to assigning P=1 to axioms, P=0 to their negations, and trying to figure out non-trivial probabilities for everything else by constraining on propositional consistency. But propositional consistency is merely a very thin veneer over X.
Because propositional inference isn't going to be able to break down quantifiers and "peer" inside their clauses, any quantified statement that's not already related to X [in the sense of participating in X either as a member or as a clause of a Boolean member] is opaque to X. So if I write A = (exists Y)(Y=2) and B = (exists Z)(Z=2 and Z=3), you'll be forced to deduce P(A)=P(B) under your axioms, or pre-commit to include in your axioms A, not(B) and everything else in a smoothly growing (complexity-wise) list of arithmetical truths I can come up with. That doesn't seem very useful.
That was my goal - to come up with a minimum necessary for consistency, but still sufficient to prove the 1/2 probability for digits of PI :) If you wish to make OLC stronger, you're free to do so, as long as it remains decidable. For example, you can define OLC(X) to be {everything provable from X by at most 10 steps of PA-power reasoning, followed by propositional calculus closure}.
In your scheme you have P=1/2 for anything nontrivial and its negation that's not already in X. It just so happens that this looks reasonable in case of the oddity of a digit of pi, but that's merely a coincidence (e.g. take A="a millionth digit of pi is 3" rather than "...odd").
No, a statement and its negation are distinguishable, unless indeed you maliciously hide them under quantifiers and throw away the intermediate proof steps.