[FOM] A minor issue in modal logic

Richard Heck rgheck at brown.edu
Mon Jul 5 21:23:06 EDT 2010

On 07/05/2010 02:36 PM, Michael Lee Finney wrote:
>   I would agree that Np =>  p does not hold in all modal logics, such as
>   denotic or provability logics. However, it does appears to be the
>   defining minimal characteristic for logics of necessity. It is the
>   basis of the classical system T. There are many other modal logics in
>   which this doesn't hold, but they aren't generally regarded as logics
>   of necessity.
>   It appears to me that you confuse the general idea of modal logics
>   with the more specific idea of necessity logics. So if Np =>  p is not
>   validated in all necessity logics, what properties do you see as
>   defining the minimal properties of necessity -- as opposed to the
>   many other modal logics? You could have Np =>  Pp as a weaker
>   condition (the basis of the classical system D), but that is usually
>   considered to be basis of denotic logic (at least when Np =>  p is not
>   also present).
I don't have a general view on this topic, and I'm not sure why I need 
to have a view about what characterizes "necessity logics". I'm not even 
sure there are "necessity logics", i.e., that there is any natural kind 
so characterized. I simply meant to leave it open that someone might 
want to deny:
(*)    Np |- p
I don't think that's insane. One can of course stipulate that one isn't 
dealing with a notion of necessity unless (*) holds, but I'm not sure 
what the argument for that would be. Maybe there is one; maybe not.

In any event, as I've said elsewhere, the issue concerned:
     NAp |- Np
or more weakly:
     NAp |- p
and the remark about (*) was by the way.


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