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Re: [ontolog-forum] Ontology and methodology

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Mon, 19 Mar 2007 11:57:12 -0500
Message-id: <0C564BBC-CD38-4583-8AB7-79C03B7C8753@xxxxxxxx>
On Mar 19, 2007, at 11:09 AM, andersen@xxxxxxxxxxxxxxxxx wrote:
>>> Subtype has a very clear and simple definition:  X < Y
>>> means that every instance of X is an instance of Y.
>> I would prefer to say that it implies this, but subtype is actually
>> somewhat stronger. Not every occassion of one category being included
>> in another need be viewed as a subtyping. BTW, this 'intensional'
>> perspective gives more efficient reasoning, as well as being more
>> natural.
> There are other versions as well.  One stronger version involves  
> modality,
> and corresponds to one view of classes as universals.  Take  
> (subclass A B)
> to read "A is a subclass of B" and (class A) to read "A is a class").
> Then we have (please excuse my extended CLIF - I introduce Nec as the
> necessity operator, and Pos for possibility, assuming S5):
> (forall (A B)
>   (if
>     (and (class A) (class B))
>     (iff
>       (subclass A B)
>       (and
>         (Nec (forall (x) (if (A x) (B x))))
>         (Nec (exists (x) (and (A x) (not (B x)))))
>      ))))    (01)

You've got A and B turned around in that last line, Bill (and given  
that you are generally averse to asserting contradictions, this is  
obviously a hasty typo rather than a thinko); I take it what you  
intended was:    (02)

         (Nec (exists (x) (and (B x) (not (A x)))))    (03)

But that's too strong!  Surely it's possible there are no dogs, but  
Collie is a subclass of Dog all the same.  This might suggest  
replacing the last line instead with:    (04)

         (Nec (if (exists (x) (B x))
                  (exists (x) (and (B x) (not (A x)))))    (05)

But that too entails that Collie is not a subclass of Dog, as there's  
surely a logically possible world where Lassie and her kinfolk are  
the only canines. :-)  I think that, for A to be a genuine subclass  
of B, you only want it to be *possible* that there be
Bs that aren't As:    (06)

         (Poss (exists (x) (and (B x) (not (A x)))))    (07)

-chris    (08)

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