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Full Idea
Kripke gave a possible worlds semantics to a whole range of modal logics, and S4 and S5 turned out to be both sound and complete with this semantics. Hence more systems could be designed. S1-S3 failed in soundness, leading to 'impossible worlds'.
Gist of Idea
With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound
Source
report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Marcus Rossberg - First-order Logic, 2nd-order, Completeness §4
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] |