Re: Some basic questions about OWL-Full

I thought that this would be hard.

From: "Michael Schneider" <schneid@fzi.de>
Subject: RE: Some basic questions about OWL-Full
Date: Tue, 23 Oct 2007 13:49:38 +0200

> Hey, Jeremy, this was *my* homework! ;-)
> 
> Jeremy Carroll wrote:
> >Peter F. Patel-Schneider wrote:
> >> 
> >> For homework:  Is EquivalentProperties(owl:sameAs owl:differentFrom)
> >>     	       itself inconsisten?
> 
> I think yes!
> 
> After looking at chapter 5 of the OWL semantics document
> 
>   http://www.w3.org/TR/owl-semantics/rdfs.html
> 
> I would say the following:
> 
> *IF* the class R_I of all resources is non-empty,

and it is, from RDF, restated at the beginning of 5.2

> then a resource x exists with x IN R_I,
> and for this x I can conclude the following:
> 
>   ==> x = x
> 
> Because: This is always true.
> 
>   ==> x owl:sameAs x
> 
> Because: According to the fourth table in chapter 5.2, if x=y, then <x,y> IN
> EXT(S(owl:sameAs)), i.e. the tuple <x,y> is an instance of the extension of
> the 'owl:sameAs' property. 
> 
>   ==> x owl:differentFrom x
> 
> Because: Our axiom above is "EquivalentProperties(owl:sameAs
> owl:differentFrom)", and this means according to the fourth table in 5.2
> that the extensions of the properties 'owl:sameAs' and 'owl:differentFrom'
> are the same. And because we had <x,x> in EXT(S(owl:sameAs)) before, we then
> also have <x,x> in EXT(S(owl:differentFrom)).
> 
>   ==> x =!= x

I think that you're done already.  

> Because: Again fourth table of 5.2: if <x,x> is an instance of the extension
> of the 'owl:differentFrom' property, then x =!= x.  
> 
>   ==> x IN {y| y =!= y}
> 
> Because: This is just a reformulation of "x=!=x" (I hope this is allowed,
> because I do not find a backing for this in chapter 5).
> 
>   ==> x rdf:type owl:Nothing
> 
> Because: According to table 1 of 5.2, the extension of owl:Nothing is the
> empty set. And the above set "{y|y=!=y}" is just a fancy way to write the
> emtpy set (again: I hope this is allowed).
> 
> So I get an inconsistency from the above equivalence axiom,
> *IF* R_I is non-empty.
> But in section 5.1 it is stated that
> 
>     "R_I is the domain of discourse or universe, 
>  -> i.e., a nonempty set
>     that contains the denotations of URI references 
>     and literals in V."  
> 
> So R_I *IS* actually non-empty, hence the above equivalence of 'owl:sameAs'
> and 'owl:differentFrom' introduces an inconsistency.
> 
> Was this ok? This has been the very first time I worked with this chapter 5,
> so I am still pretty uncertain about its correct usage.
> 
> >I'm afraid I'm several years' late on my (easier) homework of:
> >    Is (*empty*) itself inconsistent?
> >
> >Jeremy
> 
> I believe it is necessary that the inspected ontology entails some "x
> rdf:type owl:Nothing" statement to be really inconsistent.
> 
> Cheers,
> Michael

peter

Received on Tuesday, 23 October 2007 13:01:50 UTC