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Full Idea
The five systems of propositional modal logic contain successively stronger conceptions of necessity. In S4 'it is poss that it is poss that p' implies 'it is poss that p'. In S5, 'it is poss that it is nec that p' implies 'it is nec that p'.
Gist of Idea
S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p'
Source
Alex Orenstein (W.V. Quine [2002], Ch.7)
Book Ref
Orenstein,Alex: 'W.V. Quine' [Princeton 2002], p.151
A Reaction
C.I. Lewis originated this stuff. Any serious student of modality is probably going to have to pick a system. E.g. Nathan Salmon says that the correct modal logic is even weaker than S4.
| 9403 | There are three different deductions for actual terms, necessary terms and possible terms [Aristotle] |
| 9470 | Modal logic is not an extensional language [Parsons,C] |
| 16456 | For modality Lewis rejected boxes and diamonds, preferring worlds, and an index for the actual one [Lewis, by Stalnaker] |
| 14670 | Metaphysical (alethic) modal logic concerns simple necessity and possibility (not physical, epistemic..) [Salmon,N] |
| 7689 | The modal logic of C.I.Lewis was only interpreted by Kripke and Hintikka in the 1960s [Jacquette] |
| 8859 | The main modal logics disagree over three key formulae [Yablo] |
| 9404 | Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn] |
| 8480 | S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' [Orenstein] |
| 7787 | Possible worlds logics use true-in-a-world rather than true [Girle] |
| 7788 | Modal logic has four basic modal negation equivalences [Girle] |
| 7796 | Modal logics were studied in terms of axioms, but now possible worlds semantics is added [Girle] |