Re: Patel-Schneider Paradox ...

From: "Jeremy Carroll" <jjc@hplb.hpl.hp.com>
Subject: Patel-Schneider Paradox ...
Date: Fri, 18 Jan 2002 12:23:29 -0000

> 
> 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.

No.  This is not the question at all.  If you put some of the triples on
the left-hand side of the entailment you destroy most of the power of the
example.  The question is whether the above collection of triples is
entailed by an empty collection of triples.

[....]

> 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).

Agreed.  This example is not the Russell paradox.  I'm not sure that the
Russell paradox can created using the DAML+OIL constructors on top of
RDFS.  (There are, however, other DLs close to DAML+OIL in which this would
be the case.)

> 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.

Well, the Russell paradox does depend on having class membership as a first
class relationship as well.  After all if you can't use ``member of'' then
you can't say

	the set of all sets that are not members of themselves.

> 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.

Not really.  If you require a layering, then you are required to determine
a layering position for every class you want to construct.  The attempt to
create a class above does not have a well-defined layering position and
thus fails to create a class.

[...]

> 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.

Well, rdfs:Class can only exist in a non-layered theory, so an attack on
non-layered theories is, in some sense, an attack on rdfs:Class.  Also,
there is no real set-theoretic problem in creating the class of all
properties.  A problem can arise, however, if the class membership
relationship is a property and can be use in the construction of classes.

[By the way, Russell's parardox is also not really an attack on classes as
member of classes.]
> 
> I think I am suggesting trying to attack the paradox by:

>  - denying
>     rdf:type rdf:type rdfs:Property .

I don't know how this would change things.  If you mean that rdf:type is no
longer something that can be mentioned in the theory, then this would go a
long way to getting around the problem.  However, you could not then give
rdf:type a domain or a range or, probably, even use it in

	John rdf:type Person.

>  - 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.

Well, in some sense, no change is necessary.  RDFS does not appear to have
any problems in an of itself.  True extensions to RDFS can be made and also
do not appear to have any problem.  The problems only occur when a
same-syntax extension is mandated.  

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

I'm not sure what you are getting at here.

> Jeremy

peter

Received on Friday, 18 January 2002 09:17:53 UTC