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>

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 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, 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? > >Jeremy I believe it is necessary that the inspected ontology entails some "x rdf:type owl:Nothing" statement to be really inconsistent. Cheers, Michael -- 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 Haid-und-Neu-Str. 10-14, D-76131 Karlsruhe Tel.: +49-721-9654-0, Fax: +49-721-9654-959 Stiftung des bürgerlichen Rechts Az: 14-0563.1 Regierungspräsidium Karlsruhe Vorstand: Rüdiger Dillmann, Michael Flor, Jivka Ovtcharova, Rudi Studer Vorsitzender des Kuratoriums: Ministerialdirigent Günther LeßnerkrausReceived on Tuesday, 23 October 2007 11:49:54 GMT

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