I've only thought of this because superficially they look the same, and seem to be making similar claims. When you prove a statement P=>Q ◻, then is it the same as writing ◻P=>Q in modal logic?

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    No; it is only a way to signify the end of the proof. It seems it is due to Halmos (see here). Mar 7 '16 at 7:34

As Mauro indicates the original intention of Halmos was not ◻ to mean "necessarily". But if you agree that our logic holds in all possible worlds - a necessary assumption when discussing possible worlds at all - then any mathematical(!) proof A => B holds necessarily in any possible world, hence ◻(A=>B).

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