# [FOM] A minor issue in modal logic

Richard Heck rgheck at brown.edu
Thu Jul 8 09:25:53 EDT 2010

On 07/05/2010 09:18 PM, Richard Heck wrote:
> On 07/05/2010 12:53 AM, Michael Carroll wrote:
>> Richard Heck wrote:
>>
>>> Of course, one might sensibly argue that any reasonable notion of
>>> necessity must validate "N\phi -->  \phi" and so argue that any modal
>>> logic modelling any reasonable notion of necessity must have only
>>> models
>>> in which the accessibility relation is reflexive. But that is a
>>> substantive---i.e., not purely logical---claim.
>> I'm sorry but I fail to see how that differs from :
>>
>> "One might sensibly argue that any reasonable notion of conjunction must
>> validate "((A&  B) ->  (B&  A))", so that any logic modeling any
>> reasonable
>> notion of conjunction must have only models in which conjunction is
>> commutative. But that is a substantive -- i.e., not purely logical --
>> claim."
>>
> Since there are logics in which commutation fails for conjunction, I
> suppose I'm willing to agree, to some extent. I think something like:
>     A & B |- A
> is probably non-negotiable, however.
>
Further to this remark, Josh Dever has informed me there are logics in
which this form of conjunction elimination fails, in particular, certain
sorts of dynamic logics. Of course, there are other things going on in
such cases and one might think the failure in some sense misleading, but
this does point out that one needs to be very, very careful indeed about
what one thinks must be "non-negotiable"!!

Richard