Jason (jcreed) wrote,
Jason
jcreed

One more weird fact about Pfenning-Davies modal logic that I never noticed before: box isn't dual to diamond in the following sense: the sequent

~[]~~A |- ~~<>~A

is not provable. Whereas with other connectives, even intuitionistically, if you stuff in enough negations, you get entailments like this. E.g.

~(∀ x . ~~A(x)) -||- ~~(∃ x . ~A(x))

This means that classical S4 box and diamond are generated by Pfenning-Davies box and its DeMorgan dual --- and Pfenning-Davies diamond is just some other crazy thing with its own classical dual, for all I can tell.
Tags: logic, modal
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