Is there any system of modal logic in which the duality between necessity and possibility does not hold? In other words, is there any system of modal logic in which
$\Box p = \neg\Diamond\neg p$ and $\Diamond p = \neg\Box\neg p$
does not hold?
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1Yes: Intuitionistic modal logic – Mauro ALLEGRANZA Apr 02 '17 at 10:04
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1See also Alex Simpson, The Proof Theory and Semantics of Intuitionistic Modal Logic. – Mauro ALLEGRANZA Apr 02 '17 at 10:08