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Patel-Schneider Paradox ...

From: Jeremy Carroll <jjc@hplb.hpl.hp.com>
Date: Fri, 18 Jan 2002 12:23:29 -0000
To: <www-webont-wg@w3.org>

I was looking at my notes on Peter's paradox:


_:1 rdf:type owl:Restriction .
_:1 owl:onProperty rdf:type .
_:1 owl:maxCardinalityQ "0" .
_:1 owl:hasClassQ _:2 .
_:2 owl:oneOf _:3 .
_:3 owl:first _:1 .
_:3 owl:rest owl:nil .
_:1 rdf:type _: 1 .


Aside: Isn't the last triple not meant to be there. The question is whether
the last triple is entailed by the others.

In N3:
   :_g1 a owl:Restriction;
      owl:onProperty rdf:type .
      owl:maxCardinalityQ "0";
      owl:hasClassQ  [
         owl:oneOf  [
            owl:first :_g1;
            owl:rest owl:nil


In RDF/XML (naming the restruction):

<Restriction xmlns="...owl..."
    <onProperty rdf:resource="rdf:type" />
       <oneOf rdf:parseType="owl:collection">
            <rdf:Description rdf:about="#PatelSchneider"/>

In English:

The PatelSchneider class is formed as those things that have at most 0
rdf:type arcs leading to the PatelSchneider class.
The PatelSchneider class is those things that are not of type PatelSchneider


But .....

Russell's paradox (in English)

The Russell set is the set of all things that do not belong to themselves.

This is a degree of self-reference that the DAML+OIL language is not
demonstrated to permit. (i.e. if it does permit it this example does not
show it).

Now as I see it, while Russell's paradox is about finite or infinite
descent, the PatelScheider paradox is about class membership as a first
class relationship.

If Peter had encoded Russell's paradox in OWL, then we could try and address
it by formulating a well-founded theory, or by a more rigorous following of
a set theory with anti-foundation, or by a flat theory (Peter's proposal).
But a flat theory in which rdf:type is still a restrictable property still
suffers the paradox - hence the depth of class embedding appears to be

In terms of Peter's triples, the above paragraph relates principally to the
last. We can remove that triple and consider a triple showing any other
member of the questionable class and still have a problem.



_:4 rdf:type _:1 .

So, I do see the Patel-Schneider paradox as an attack on the meta-model  but
it is an attack on the class rdfs:Property not rdfs:Class, and it certainly
isn't an attack on classes as members of classes.

I think I am suggesting trying to attack the paradox by:
 - denying
    rdf:type rdf:type rdfs:Property .
 - considering moving to a well-founded system for rdfs:Class rather than
the current system articulated by Pat's antifoundationalisam.

I am not convinced of the necessity of the second step, and I am a long way
from convinced of the necessity of a flat model.

I note that if rdf:type is not a property then the problem of relationship
typing may have a fairly different theoretical flavour.

Received on Friday, 18 January 2002 07:24:03 UTC

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