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Re: circular paradox gizmo

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Thu, 02 May 2002 19:49:10 -0400
To: jos.deroo.jd@belgium.agfa.com
Cc: www-webont-wg@w3.org
Message-Id: <20020502194910B.pfps@research.bell-labs.com>
From: "Jos De_Roo" <jos.deroo.jd@belgium.agfa.com>
Subject: circular paradox gizmo
Date: Fri, 3 May 2002 00:42:32 +0200

> [from the 2002-05-02 WebOnt WG telecon minutes]

[...]

> Peter, you seem to go like
> either
>   every formula has to exist in every model
>   and one can write a paradoxical formula in OWL
>   so there exists a paradox in every model
>   and therefore the system is fundamentally broken
>   --
>   That sounds like a self fulfilling prophecy.

Well, if the formulae do not exist in all models, then there are some
interesting consequences.  You *could* go this way but then you end up with
things *like*

       John belongs to the intersection of student and employee

does not entail that

	John belongs to the intersection of employee and student.

The problem does not really arise from OWL, but from the RDF desire to have
classes (or formulae) be elements of the domain of discourse and to also
allow for circular structures.  This, along with the ability to express a
contradiction and the desire to have useful entailment, leads to paradoxes.

So, if you want classes or formulae to denote and you allow for
circular structures and you want reasonable expressive power and you want
what I consider to be reasonable inference, then you can easily have a
problem.  I don't consider this to be a self-fulfilling prophecy.

Look, if someone else can come up with a coherent theory then be my guest.
However, I do reserve the right to point out the consequences of your
theory, both positive and negative.  This is really all that I have done
with respect to OWL layered on top of RDF, pointed out some consequences of
what happens when you try to build what I consider to be a reasonable
theory for OWL that fits into the (rather severe) constraints that were
imposed on OWL.  

>   I find one of the most important characteristics
>   of a model that it captures the *correspondence*
>   between the-name-of-the-thing that we have in the
>   assertional language and the-named-thing expressed
>   in the object language (using e.g. URI decoupling)
>   I just can't find such correpondence for a paradox
>   so how could they exist in every model ???

I don't understand this.  Where do names come in here?  If by names you
include things like ``the set that consists of all sets that are not
elements of themselves'', then I've been trying to define the
correspondence.  If by names you mean atomic names, then to devise an
expressive formalism you need much more in your model to give meaning to
the other entities in the domain of discourse, if you by into the RDF
vision, or you need to have syntactic constructs that do not correspond to
entities in the domain of discourse, if you don't buy into the RDF vision.

> or
>   one wants to prove a paradox by assertion of its
>   negation and finding a contradiction
>   which is succeeding here, so the paradox is proved
>   of course it is, you have *asserted* an inconsistency
>   and one should not do that

I haven't asserted anything.  I've just pointed out a flaw in the
definitions of OWL interpretations.

>   --
>   I would make the claim that one cannot prove nor
>   disprove a paradox, we just have incompleteness
>   (no evidence/correspondence)

Generally, paradoxes are paradoxes because you can (or should be able to)
both prove and disprove them (at least in one way of thinking about
paradoxes).

> --
> Jos

Peter F. Patel-Schneider
Received on Thursday, 2 May 2002 19:49:20 GMT

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