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Re: [ontolog-forum] Logic, Datalog and SQL

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Bill Andersen <andersen@xxxxxxxxxxxxxxxxx>
Date: Sun, 11 Feb 2007 21:57:44 -0500
Message-id: <EE478CDC-F93F-4C7E-A44A-0873B86C3B6B@xxxxxxxxxxxxxxxxx>
Hi John,    (01)

See comments below.    (02)

On Feb 11, 2007, at 18:57 , John F. Sowa wrote:    (03)

> Following is a revised version:
>     A formal ontology consists of a theory T stated in
>     some version of logic and a nonempty vocabulary V
>     of types and relations.  The names in V are divided
>     in three disjoint subsets:
>     1. Names defined elsewhere, which are used in one
>        or more axioms of T.
>     2. Names defined in T, which may be used in other
>        theories.
>     3. Names that are never used in any other theories.
>     To be considered an ontology, the set of names in
>     subset #2 must be nonempty; i.e., the theory T
>     must define one or more types or relations, whose
>     names may be used in statements other than T.    (04)

While I like the direction this discussion is going, I think there  
are technical problems with it.    (05)

One of them has to do with the intended force of your last sentence.   
Let's call your "names in subset #2 being non-empty" condition O.   
The issue has to do with whether condition O is both necessary and  
sufficient, just sufficient, or just necessary for ontologyhood of  
some theory T.    (06)

Second, there is the notion of definability being invoked.  That is,  
of course, always a notion that is relative to a particular logical  
system being considered.  By any reading of your sentence where  
condition O is necessary, no first-order theory T can contain the  
term 'Integer' with the intended meaning we have for it and be  
considered an ontology.  I would think that to be a bad thing.    (07)

I would consider a theory meaningful that tells me, for example, that    (08)

        (x) Integer(x) => ~Dog(x)    (09)

But this sort of theory, it seems, doesn't make the cut, at least not  
when expressed in FOL (and any weaker language).  That spells all  
kinds of trouble for any proposed ontology that attempts, for example  
to make sense of any countably infinite classes of objects and into  
those classes fall all kinds of things other than integers -- big  
swaths of abstract objects of all kinds (like computer programs), for  
starters.    (010)

        .bill    (011)

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