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RE: Expressiveness of RDF as Rule Conclusion Language (was Re: W hat is an RDF Query? )

From: Wagner, G.R. <G.R.Wagner@tm.tue.nl>
Date: Fri, 12 Oct 2001 09:32:26 +0200
Message-ID: <511BB18E82E9D11188230008C724064602D9DE15@tmex1.tm.tue.nl>
To: "'Pat Hayes'" <phayes@ai.uwf.edu>
Cc: www-rdf-rules@w3.org
> >  > I always liked the example that was given in the KIF 
> documentation
> >>  (KIF 3.0 Ref. Manual,
> >>  http://logic.stanford.edu/kif/Hypertext/node37.html):
> >>
> >>  ... On the other hand, in some cases, replacing <<= by <= would be
> >>  semantically unacceptable. For instance, the rules
> >>
> >>    (<<= (status-known ?x) (citizen ?x))
> >>    (<<= (status-known ?x) (not (citizen ?x)))
> >>
> >>  allow us to infer (status-known Joe) only if one of the sentences
> >>
> >>    (citizen Joe),  (not (citizen Joe))
> >>
> >>  can be inferred. Replacing the rules by implications would make
> >>  (status-known ?x) identically true."
> >
> >But according to classical (2-valued) logic, there is no difference
> >here between these rules and the corresponding implications.
> 
> Why not? The 2-valued status of the logic says nothing about how to 
> interpret *rules*. 

Notice that we can associate a model-theoretic semantics with rules 
in a natural way: an interpretation I satisfies a rule (is a model
of it) if it satisfies its consequent whenever it satisfies its 
antecedcent. 

> >The sentence (status-known Joe) could also be inferred from the
> >two rules alone,
> 
> From the two implications, but not from the rules. In fact, strictly 
> speaking, nothing can be inferred *from* a rule, only *by* a rule.

Yes, we can infer from a rule set: using the above definition of a
model of a rule, we can define that a rule set R entails a sentence F 
if all models of R satisfy F (in logic programming we say that 
R entails F if all stable models of R satisfy F).

> >since every classical (i.e. total and coherent)
> >model of the two rules would satisfy it, simply because it would
> >either satisfy (citizen Joe) or (not (citizen Joe)), and in both
> >cases, as it satisfies both rules, it would also have to satisfy
> >(status-known Joe).
> 
> (I think you mean, as it satisfies *one of the* rules?). But merely 
> being satisfiable in a single interpretation is not sufficient to 
> trigger a rule.
 
I think you confuse something here: Triggering a rule is a different 
(proof-procedural) thing than the model-theoretic semantics of rules.
Again: Let M be a classical model of the two rules (that is, it 
satisfies their consequent if it satisfies their antecequent). Then 
either M satisfies (citizen Joe), in which case it has to satisfy 
(status-known Joe) because it satisfies the first rule, or it satisfies 
(not (citizen Joe)), in which case it has to satisfy (status-known Joe) 
because it also satisfies the second rule.

So, the author of this text on KIF seems to make a mistake here.
In fact, he unintentionally describes the behavior of rules in
extended logic programs where "not" corresponds to the strong
(monotonic) negation of partial logic, where models are partial
and the "tertium non datur" (= law of the excluded middle) does
not hold.

-Gerd


--------------------------------------------
Gerd Wagner        Email: G.Wagner@tm.tue.nl
http://tmitwww.tm.tue.nl/staff/gwagner

Eindhoven University of Technology
Faculty of Technology Management
Department of Information & Technology 
P.O. Box 513       
5600 MB Eindhoven    Tel: (+31 40) 247 26 17
The Netherlands      Fax: (+31 40) 243 26 12
Received on Friday, 12 October 2001 03:32:30 GMT

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