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Single Idea 11004

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

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.

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]