From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>

Date: Tue, 23 Oct 2007 08:51:08 -0400 (EDT)

Message-Id: <20071023.085108.147787497.pfps@research.bell-labs.com>

To: schneid@fzi.de

Cc: jjc@hpl.hp.com, public-owl-dev@w3.org, phayes@ihmc.us

Date: Tue, 23 Oct 2007 08:51:08 -0400 (EDT)

Message-Id: <20071023.085108.147787497.pfps@research.bell-labs.com>

To: schneid@fzi.de

Cc: jjc@hpl.hp.com, public-owl-dev@w3.org, phayes@ihmc.us

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

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