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RE: Some basic questions about OWL-Full

From: Michael Schneider <schneid@fzi.de>
Date: Tue, 23 Oct 2007 13:49:38 +0200
Message-ID: <0EF30CAA69519C4CB91D01481AEA06A04A87B7@judith.fzi.de>
To: "Jeremy Carroll" <jjc@hpl.hp.com>
Cc: <public-owl-dev@w3.org>, "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>, "Pat Hayes" <phayes@ihmc.us>

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


I would say the following:

*IF* the class R_I of all resources is non-empty,
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

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?

I believe it is necessary that the inspected ontology entails some "x
rdf:type owl:Nothing" statement to be really inconsistent.


Dipl.-Inform. Michael Schneider
FZI Forschungszentrum Informatik Karlsruhe
Abtl. Information Process Engineering (IPE)
Tel  : +49-721-9654-726
Fax  : +49-721-9654-727
Email: Michael.Schneider@fzi.de
Web  : http://www.fzi.de/ipe/eng/mitarbeiter.php?id=555

FZI Forschungszentrum Informatik an der Universität Karlsruhe
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Received on Tuesday, 23 October 2007 11:49:54 UTC

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