But propositional consistency is merely a very thin veneer over X.
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.
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.