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Stuart_Armstrong comments on Reflection in Probabilistic Logic - Less Wrong
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Ok, for [a < P('G') < b] I see why you'd use a schema (because it's an object level axiom, not a meta-language axiom).
But this still seems possibly problematic. We know that adding axioms like [-1 < P('G') < 1] --> [P('-1 < P('G') < 1') = 1], would break the system. But I don't see a reason to suppose that the other reflection axioms don't break the system. It might or it might not, but I'm not sure; was there a proof along the lines of "this system is inconsistent if and only if the initial system is" or something?
I'm not sure whether I'm misunderstanding your point, but the paper proves that there is a coherent probability distribution P(.) that assigns probability 1 to both T and to the collection of reflection axioms [a < P('G') < b] for P(.); this implies that there is a probability distribution over complete theories assigning probability 1 to (T + the reflection axioms for P). But if (T + the reflection axioms for P) were inconsistent, then there would be no complete theory extending it, so this would be the empty event and would have to be assigned probability 0 by any coherent probability distribution. It follows that (T + the reflection axioms for P) is consistent. (NB: by "the reflection axioms for P(.)", I only mean the appropriate instances of [a < P('G') < b], not anything that quantifies over a, b or G inside the object language.)