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Re: Layering LX (or FOL) on RDF

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Mon, 26 Aug 2002 18:04:09 -0400 (EDT)
Message-Id: <20020826.180409.77755860.pfps@research.bell-labs.com>
To: sandro@w3.org
Cc: www-rdf-interest@w3.org

From: Sandro Hawke <sandro@w3.org>
Subject: Layering LX (or FOL) on RDF (was Re: A Rough Guide to Notation3)
Date: Mon, 26 Aug 2002 17:12:19 -0400

> > > > The problem is not their (potential) existence.  It is their existence
> > > > everywhere.  The problem is that if you allow self-referential
> > > > sentences and also need to have sentences exists everywhere, removing just
> > > > the problematic ones is problematic. 
> > > 
> > > I don't quite follow that, sorry.
> > 
> > The problem with some self-referential sentences, such as the self negating
> > one, is that they have no models, not even models that assign them a truth
> > value of false.  
> 
> I'm proposing that the logical formula
> 
>     % there exists a triple whose subject is itself
>     exists t subjTerm predTerm objTerm pred obj (
>       rdf(t, lx_subjectTerm, subjTerm) &
>       rdf(subjTerm, lx_denotation, t) &
>       rdf(t, lx_predicateTerm, predTerm) &
>       rdf(predTerm, lx_denotation, pred) &
>       rdf(t, lx_objectTerm, objTerm) &
>       rdf(objTerm, lx_denotation, obj)       
>     )

This does not look like a logical formula to me.  What is it supposed to mean?

> has the same meaning as any other formula which contradicts the
> layering axioms or contradicts itself, such as
> 
>      % there exists some triple which is both true and false
>      exist a b c (
>        rdf(a,b,c) & -rdf(a,b,c)
>      )

Again, what logical formula is this supposed to be?

> You sound concerned that the triple (self,x,y) doesn't have a truth
> value, but in my proposal the triples one would have to use to
> construct such a triple essentially contradict each other, so one
> can't even phrase the problematic triple to notice that you have no
> truth value for it.  This seems very like FOL, which disallows
> self-referencial terms, although it manages such a restriction in a
> purely-syntactic manner.

Suppose that I want a theory of entailment in this LX.  I need to be able
to do things like
	p  entails  p v q
To do that in RDF, I need comprehension principles, which essentially
require all formulae to exist, including the false ones, because
	p  entails  p v ( q & -q )
and for   p v ( q & -q )  to exist,  q & -q also needs to exist.
So, every formula needs to exist, and has to be given a truth value.

So far, so good.  But now how can the self-referential formulae be handled?
Some of the self-referential formulae are easy, such as x : x (the formula
that references itself positively) but others are impossible, such as x :
-x

So the problem is not that such formulae are false, but that not matter
what truth value they are given, no model can be constructed for them.

If you have disallowed self-referential formulae, then you don't have this
problem.

[...]

> My question is a fairly narrow technical one, not a political one.
> Can one precisely define a way to reach out from the RDF sublanguage
> of FOL to full FOL by means of a pre-arranged vocabulary and
> associated semantics?  
> 
> If you think the answer is "No", then please point out the weak link
> in my demonstration [1].  (If it's in the handling of self-reference
> in the axioms, then please say so (no need to try to find a flaw in
> the axioms), and let me polish that area first.  It'll be tricky, and
> I'd like to avoid the work if it's not necessary.)

The answer, of course, is yes, as long as?
1/ You are allowed to extended the semantics of RDF.
2/ You forbid self-referential statements.

> I seriously appreciate your time in this.
> 
>    -- sandro
> 
> [1] http://www.w3.org/2002/08/LX/RDF/layering
Received on Monday, 26 August 2002 18:04:24 GMT

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