I still agree with what I said originally about the second problem. You could compare it to these two statements:
A) A is true if and only if A and B are either both provable, or both non-provable.
B) B is true if and only if B and A are either both provable, or both non-provable.
It is much more obvious that both of these statements are necessarily true and provable (by symmetry), than it is that “this is true if and only if it is provable” is true or provable. This is why at first I only accepted the second case.
I still agree with what I said originally about the second problem. You could compare it to these two statements:
A) A is true if and only if A and B are either both provable, or both non-provable. B) B is true if and only if B and A are either both provable, or both non-provable.
It is much more obvious that both of these statements are necessarily true and provable (by symmetry), than it is that “this is true if and only if it is provable” is true or provable. This is why at first I only accepted the second case.