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.
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.
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.
ReplyDeleteI assume "N" = "necessary"?
ReplyDeleteWhile 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?