In an exposition about the modal logic system K4, after introducing the box "necessity" quantifier [itex]\Box[/itex] (where [itex]\Box[/itex]P is essentially that the Gödel number of P is provable), then introduces the "strong box" quantifier (I don't know how to put an s inside a box in LaTex) as:(adsbygoogle = window.adsbygoogle || []).push({});

A = A [itex]\wedge[/itex] [itex]\Box[/itex]A.

But since K4 is sound, shouldn't [itex]\Box[/itex]A imply A? In that case, I do not see the difference between box and strong box.

Thanks in advance for clearing this up. (The exposition that I am following is in Smorynski's "Self-Reference and Modal Logic")

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# Modal logic K4: strong box quantifier

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