A point of confusion: is it enough to prove that A→□B? What about A→B? I’m not sure I can say this well, but here goes:
We might not be able to prove A→B in the theory, or even that ¬□⊥ (which would mean “there are no proofs of inconsistency”). But if we believe our theory is sound, we believe that it can’t show that a copy of itself proves something false.
So A→□B tells us that if A is true, the theory would show that a copy of itself proves B is true. And this is enough to convince us that we can’t simultaneously have A true and B false.
A point of confusion: is it enough to prove that A→□B? What about A→B? I’m not sure I can say this well, but here goes:
We might not be able to prove A→B in the theory, or even that ¬□⊥ (which would mean “there are no proofs of inconsistency”). But if we believe our theory is sound, we believe that it can’t show that a copy of itself proves something false.
So A→□B tells us that if A is true, the theory would show that a copy of itself proves B is true. And this is enough to convince us that we can’t simultaneously have A true and B false.