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Full Idea
In S4 necessity is said to be informal 'provability', and in S5 it is said to be 'true in every possible world'.
Gist of Idea
Necessity is provability in S4, and true in all worlds in S5
Source
Stephen Read (Thinking About Logic [1995], Ch.4)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.118
A Reaction
It seems that the S4 version is proof-theoretic, and the S5 version is semantic.
16952 | If something is only possible relative to another possibility, the possibility relation is not transitive [Dummett] |
16953 | Relative possibility one way may be impossible coming back, so it isn't symmetrical [Dummett] |
10760 | With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound [Kripke, by Rossberg] |
16468 | Non-S5 can talk of contingent or necessary necessities [Stalnaker] |
9456 | Modal logic is multiple systems, shown in the variety of accessibility relations between worlds [Jacquette] |
11004 | Necessity is provability in S4, and true in all worlds in S5 [Read] |