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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>
Message-ID: <CEECKEAMDAJDDEDGJNBEKEBACAAA.jjc@hpl.hp.com>

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..."
             rdf:ID="PatelSchneider">
    <onProperty rdf:resource="rdf:type" />
    <maxCardinalityQ>0</maxCardinalityQ>
    <hasClassQ>
       <oneOf rdf:parseType="owl:collection">
            <rdf:Description rdf:about="#PatelSchneider"/>
       </oneOf>
     </hasClassQ>
</Restriction>


In English:


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

=========


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
spurious.

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.

e.g.

adding

_: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.

Jeremy
Received on Friday, 18 January 2002 07:24:03 GMT

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