Re: subclass loops, proper subclasses and so on

From: pat hayes (
Date: 08/30/01

>From: pat hayes <>
>Subject: Re: subclass loops, proper subclasses and so on
>Date: Wed, 29 Aug 2001 15:25:39 -0700
> > > > 2. Suppose Subclass is the subclass relation, Psubclass the
> > > > proper-subclass relation, and R is any binary irreflexive relation
> > > > between classes; then (R intersect Subclass) = (R intersect
> > > > PSubclass). From which is follows that if some formal relation symbol
> > > > on classes is loop-free, then if it can be interpreted as meaning
> > > > subclass then it can also be interpreted as meaning proper subclass,
> > > > without changing the truthvalues of any expressions in the language.
> > > > So, what justification can there be for asserting that
> > > > rdfs:subClassOf does *not* mean proper subclass?
> > >
> > >I don't see how this can be true.
> > >
> > >Suppose R is ``has a different ID than'',
> >
> > That is not an irreflexive relation on classes, since the same class
> > may have two distinct IDs (ie two IDs may denote the same class.)
>Ahh.  This depends on what you mean by ``class''.  If by ``class'' you
>indeed mean ``the extension in all models'' then fine.  If, however, a
>class is a node (in an RDF graph) or a name (in a description logic KB),
>then it makes sense to distinguish between the class and its extension.
>I believe that your model theory for RDF makes this distinction in the
>model theory itself.   A class is a resource.  The extension of two classes
>may be the same in all models.  (There is, of course, few or no ways to
>enforce this in RDF, but DAML+OIL can easily state that two classes have
>the same extension in all models.)

Yes, Ive been thinking about this. Indeed, the MT 'extensions' trick 
does allow one to insist that classes (and properties) are always 
uniquely named, since two distinct classes may have identical 
extensions. In fact one could think of classes as being pairs of the 
form <class extension, class ID>, and reconstruct the MT on that 
basis, and it would produce the same truthvalues. So, I concede your 
point. If I am going to sell a nonextensional model theory, I guess 
that I have to stop relying on extensional intuitions.



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