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It seems to me that the notions of modal logic are all shaped primarily by one modal, 'can/must' (konnen/mussen). Has anyone looked at all deeply at how this convention compares with the other common modal verbs 'would/might' (wollen/mochten) and 'should/may' (sollen/durfen).

This is, to my mind, the very least useful set of modal verbs, because from a physicalist perspective they just mean things that can be said in complicated non-modal expressions.

It seems to me that the other motivations actually call for a totally different set of deductive conventions, but when someone says 'modal logic', they are talking about a set of conventions that work best for 'necessity' and 'possibility'.

Am I just not seeing how the usual pattern generalizes to the other two pairs?

(I give the German modals for clarity because the English ones drift so much, especially 'should' and 'may' which get used as less certain forms of 'must' and 'might' for no good reason. Also, sorry for not typing umlauts.)

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    Modal logic is as old as formal logic : see Aristotle logic. The "twin" concepts of necessity/possibility stem from a metaphysical perspective : see Time and Necessity: The Sea-Battle.
    – Mauro ALLEGRANZA
    May 9, 2015 at 16:12
  • There are many modal logics: physical possibility, technical possibility, epistemic, logical... There is also a temporal modal logic and also a duty modal logic.
    – Quentin Ruyant
    May 9, 2015 at 18:42
  • So, I think I am hearing a vote for 'We got there first'. It is not that this form is simpler or more meaningful, just that Aristotle did it?
    – user9166
    May 9, 2015 at 20:55

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I regret your not typing umlauts, but I will use them here. (German is my mother tongue.)

The modal logic is well split into deontic and normative modal logic. The difference is very subtle but can be subsummarised mostly as of natural, physical possibility and obligations versus those by convention as in ethics. This is exactly the difference with the German words "dürfen"/"können" and "sollen"/"müssen" (first words of these pairs are normative, second deontic.)

Just like any modal logic, deontic and normative modal logic systems are hierarchally split into systems of strength (S4, S5, K, etc), and they are noted in the Kripke-Semantics. With the variation of the system, you can generate different kinds of logic which capture different habits of using these notions.

Your headline also mentions wishes. Wishes are something completely different, as they are a state of mind that accompany a statement. If you have, for example, the statement "It is raining", then you can have several state of minds along with it: You can wish that it is raining; fear, that it is raining; etc. This has nothing to do with modalities.

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  • 'Deontic' literally means 'concerning duty', which pretty much means what you are contrasting with it as 'normative'. So none of this makes any sense to me. Also the dual operators have to have the relationship that "diamond p = not box not p". And I am pretty sure durfen and konnen are both 'diamond' operators conveying freedom and sollen and mussen are both 'box' operators conveying restriction, in that sense. So I think we are not talking about the same sort of thing at all.
    – user9166
    May 9, 2015 at 20:36
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    I can follow your opinion that the term 'deontic' is not quite expressive for what it is used. On the other hand: Yes, both 'können' and 'dürfen' have the relationships 'diamond p=not box not p', and that makes perfect sense. How does 'freedom' play a role here? Yes, indeed, freedom and restriction are represented as those two operators, not matter if in ethics, physics, or whatever.
    – rexkogitans
    May 13, 2015 at 11:41

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