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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic

[issues concerning the varieties of modal logic]

6 ideas
If something is only possible relative to another possibility, the possibility relation is not transitive [Dummett]
     Full Idea: If T is only possible if S obtains, and S is possible but doesn't obtain, then T is only possible in the world where S obtains, but T is not possible in the actual world. It follows that the relation of relative possibility is not transitive.
     From: Michael Dummett (Could There Be Unicorns? [1983], 1)
     A reaction: [compressed]
Relative possibility one way may be impossible coming back, so it isn't symmetrical [Dummett]
     Full Idea: If T is only possible if S obtains, T and S hold in the actual world, and S does not obtain in world v possible relative to the actual world, then the actual is not possible relative to v, since T holds in the actual. Accessibility can't be symmetrical.
     From: Michael Dummett (Could There Be Unicorns? [1983], 1)
With possible worlds, S4 and S5 are sound and complete, but S1-S3 are not even sound [Kripke, by Rossberg]
     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'.
     From: report of Saul A. Kripke (A Completeness Theorem in Modal Logic [1959]) by Marcus Rossberg - First-order Logic, 2nd-order, Completeness §4
Non-S5 can talk of contingent or necessary necessities [Stalnaker]
     Full Idea: One can make sense of necessary versus contingent necessities in a non-S5 modal semantics.
     From: Robert C. Stalnaker (Mere Possibilities [2012], 4.3 n17)
     A reaction: In S5 □φ → □□φ, so all necessities are necessary. Does it make any sense to say 'I suppose this might have been necessarily true'?
Modal logic is multiple systems, shown in the variety of accessibility relations between worlds [Jacquette]
     Full Idea: Modal logic by its very nature is not monolithic, but fragmented into multiple systems of modal qualifications, reflected in the plurality of accessibility relations on modal model structures or logically possible worlds.
     From: Dale Jacquette (Intro to 'Philosophy of Logic' [2002], §3)
     A reaction: He implies the multiplicity is basic, and is only 'reflected' in the relations, but maybe the multiplicity is caused by incompetent logicians who can't decide whether possible worlds really are reflexive or symmetrical or transitive in their relations.
Necessity is provability in S4, and true in all worlds in S5 [Read]
     Full Idea: In S4 necessity is said to be informal 'provability', and in S5 it is said to be 'true in every possible world'.
     From: Stephen Read (Thinking About Logic [1995], Ch.4)
     A reaction: It seems that the S4 version is proof-theoretic, and the S5 version is semantic.