Metaphysical Modality and S5
Note to self: I sometimes say that metaphysical modality is of S5 type, when I should rather say only that it satisfies the characteristic axiom of S5, Mp -> LMp.
It isn't clear to me that metaphysical modality obeys all the S5 principles because it isn't even clear that it obeys T. One of the problems is what to say about Lp if p contains names of objects which may exist only contingently. The two most obvious proposals are: a) Lp is true iff p holds at all worlds where the named objects exist (in this sense, Hesperus is necessarily Hesperus, even though Hesperus exists contingently); b) Lp is true if p holds at all worlds (in this sense, Hesperus is not necessarily Hesperus, but it is necessary that if Hesperus exists then Hesperus is Hesperus). Either way violates T. On (a), let "F" express the property of coexisting with Hubert Humphrey; then "L(F(Hesperus) & F(Humphrey) -> F(Hesperus)) -> L(F(Hesperus) & F(Humphrey)) -> LF(Hesperus)" is false, even though it's an axiom of T. On (b), "L(Hesperus = Hesperus -> Hesperus = Hesperus)" is false, even though it's a theorem of T.