~~~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.