this is a "deep" thought because everything is either possible or impossible, right? but take the proposition Mp. Is it in general true that either MMp or N~Mp? Not without substantive assumptions. Let Mp be true in W and let W have access only to W' (not to W) at which ~p is true and which has access only to itself. In that case we have Mp true at W, but neither MMp nor N~Mp. So there is a proposition (and corresponding state of affairs) which is such that it is neither possible nor impossible, viz. Mp. If an intuitive case is needed, take moral possibility or permissibility.
I assume "N" = "necessary"?While it needs substantive assumptions to see that MMp or N~Mp, it needs no substantive assumptions, besides excluded middle, to see that MMp or ~MMp. And it seems to me that "~Mq" is a better definition of "impossible" than "N~q". Moreover, the interchange of M with ~N~ doesn't require any assumptions about accessibility, but only the standard claims: Mp at w iff p is true at some w' such that Aww' Np at w iff p is true at all w' such that Aww'.From these the equivalence of Mp with ~N~p follows by FOL with no assumptions whatsoever on A.No?
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