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Re: [ontolog-forum] Ontology and methodology

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: andersen@xxxxxxxxxxxxxxxxx
Date: Mon, 19 Mar 2007 12:09:22 -0400 (EDT)
Message-id: <56662.>
>>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.    (01)

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):    (02)

(forall (A B)
    (and (class A) (class B))
      (subclass A B)
        (Nec (forall (x) (if (A x) (B x))))
        (Nec (exists (x) (and (A x) (not (B x)))))
     ))))    (03)

and, for all classes A    (04)

(if (class A) (Pos (exists (x) (A x))))    (05)

>From these it follows, for example, that    (06)

(forall (A B)
  (if (subclass A B)
      (not (subclass B A))))    (07)

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