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reification, negation & paradox in daml

From: Daniel Mahler <mahler@cyc.com>
Date: Tue, 16 Jan 2001 17:00:39 -0600
Message-ID: <14948.53911.89156.869524@mcallister.cyc.com>
To: www-rdf-logic@w3.org

I believe the reification approach to negation leads
directly to Tarski's paradox.
In hindsight this is not be a surprise since
<http://www.w3.org/DesignIssues/Toolbox#truth>
is a truth predicate.
Normally, one would require more logical machinery
then graond atoms and conjuction to do real dammage.
However, the peculiarities of the graph model
seem to make it very easy to construct a Goedel sentence,
since we can explicitly construct cycles in the representations of
reified statements, thus creating representations
of fixed points of parametrized statements.

<rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-schema-ns#"
          xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#"
          xmlns:daml="http://www.daml.org/2000/12/daml+oil#"
	  xmlns:toolbox="http://www.w3.org/DesignIssues/Toolbox#"
          >

<rdf:Description rdf:ID="goedel">
		 <rdf:type resource="rdf:Statement"/>
                 <rdf:subject resource="goedel">
                 <rdf:predicate resource="toolbox:truth"/>
                 <rdf:subject rdf:value="0"/>
<rdf:Description>

<rdf:RDF>

If we then wanted to query the model
about the rdf:truth of "goedel",
it can be neither 1 nor 0.
We could just say it is 0,
since there is no rdf:truth statement
actually asserted about "goedel".
However, that would be a very strong form
of the closed world assumption
and it would render reificataion devoid
of any logical content.

Since we are using the truth predicate to define
negation, rather then attempting to describe
an existing operator like Tarski,
it seems we are forced to abandon classical
logic to avoid the paradox.

This problem is not limited to negation,
but will also apply to using reification to simulate
second order predicates and modalities.
The general scope of this problem
was discussed by Montague.
There is also a very detailed discusion of the issues in
Raymond Turner's "Truth and Modality for Knowledge Representation"
book. The upshot is that only fairly weak operators
can be handled consistently using predicates over reified sentences.

Daniel Mahler
Cycorp Inc
Received on Tuesday, 16 January 2001 18:00:51 GMT

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