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.)