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Ebthgidr comments on Stupid Questions December 2014 - Less Wrong
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Breaking my no-comment commitment because I think I might know what you were thinking that I didn't realise that you were thinking (won't comment after this though): if you start with (provable(provable(P)) or provable(not(provable(P)))), then you can get your desired result, and indeed, provable(provable(P) or not(provable(P))). However, provable(Q or not(Q)) does not imply provable(Q) or provable(not(Q)), since there are undecideable questions in PA.
Ohhh, thanks. That explains it. I feel like there should exist things for which provable(not(p)), but I can't think of any offhand, so that'll do for now.