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ISSUE-119 (russell paradox): OWL 2 Full may become inconsistent due to self restrictions

From: OWL Working Group Issue Tracker <sysbot+tracker@w3.org>
Date: Sun, 20 Apr 2008 22:58:08 +0000 (GMT)
To: public-owl-wg@w3.org
Message-Id: <20080420225808.4B9DA5F74D@stu.w3.org>


ISSUE-119 (russell paradox): OWL 2 Full may become inconsistent due to self restrictions

http://www.w3.org/2007/OWL/tracker/issues/

Raised by: Michael Schneider
On product: 

If the OWL 2 Full semantics for self restrictions is specified analogue to the semantics of restrictions in OWL 1 Full, then OWL 2 Full will become unsatisfiable.

RDF compatible semantics treat rdf:type as a normal property. The introduction of self restrictions into OWL 2 Full allows to define classes which have the famous Russell set "{x| x notin x}" as their class extension.

  _:c rdf:type owl:Class .
  _:c owl:complementOf _:s .
  _:s rdf:type owl2:SelfRestriction .
  _:s owl:onProperty rdf:type .

For every restriction and every boolean constructor in OWL 1 Full there are (unconstrained) comprehension principles. An analogue comprehension principle for self restrictions would have the form (informally stated):

  "IF 
    we have property p
  THEN 
    there exists a class S 
    which is a self restriction on p"

Since

  rdf:type rdf:type rdf:Property

is true in every RDF ontology, the above comprehension principle would provide the existence of a Russell class in every OWL 2 Full ontology. This would lead to inconsistency of OWL 2 Full, which means that the empty RDF graph would not have any model when interpreted under OWL 2 Full semantics.

There are different possible ways to avoid this problem by weakening the semantics accordingly. It has to be decided, which solution to choose.
Received on Sunday, 20 April 2008 22:58:43 GMT

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