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Single Idea 10760

[filed under theme 4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic ]

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

The 6 ideas with the same theme [issues concerning the varieties of modal logic]:

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]