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Re: Infinite cardinalities

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Mon, 02 Apr 2001 08:24:53 -0400
To: kenb@ccs.neu.edu
Cc: www-rdf-logic@w3.org
Message-Id: <20010402082453L.pfps@research.bell-labs.com>
Ken may have pointed out a problem with the KIF axiomatization for
DAML+OIL.  

However, even if we make the condition that properties have some local
finiteness built into them (and I'm not even sure if this does follow from
the KIF axiomatization, and it is certainly not in the model-theoretic
semantics), this does not mean that the entire domain is finite, nor does
it mean that a property (taken as a whole) need have a finite extension.

peter




From: Ken Baclawski <kenb@ccs.neu.edu>
Subject: Infinite cardinalities
Date: Sun, 1 Apr 2001 17:34:48 -0400 (EDT)

> Recently, Peter Patel-Schneider mentioned that DAML+OIL allows infinite
> models.
> 
> On Fri, 30 Mar 2001, Peter F. Patel-Schneider wrote:
> 
> > ...
> > DAML+OIL does not have a finite model restriction.  It is entirely possible
> > to have DAML+OIL models with an infinite number of objects that are not
> > datatype values.
> > ...
> 
> Furthermore, the preface to the DAML+OIL axioms makes this statement which
> seems to allow for models that have infinite cardinalities:
> 
> "This axiomatization is designed to place minimal constraints on the
> interpretation of the non-logical symbols in the resulting logical theory. 
> In particular, the axioms do not require use of a set theory, that classes
> be considered to be sets or to be unary relations, nor do they require
> that properties be considered to be mappings or binary relations." 
> 
> However, the axioms make frequent use of finite lists, which are much more
> restrictive than sets.  Because of this there are implicit finiteness
> assumptions in the axioms.  Consider for example Axiom 105: 
> 
> Ax105.(=> (and (PropertyValue onProperty ?r ?p)
>                      (PropertyValue minCardinality ?r ?n))
>                 (forall (?i) 
>                         (<=> (Type ?i ?r) 
>                              (exists (?vl) 
>                                      (and (no-repeats-list ?vl) 
>                                           (forall (?v) 
>                                                   (=> (PropertyValue 
>                                                         item ?vl ?v) 
>                                                       (PropertyValue
>                                                         ?p ?i ?v))) 
>                                           (>= (length ?vl) ?n))))))[9]
> 
> It follows from this axiom that restricting a property to have
> minCardinality of 0 has the side-effect of restricting the cardinality to
> be finite.  Is this what was intended?  Does DAML+OIL really support
> infinite cardinalities?
>
> Ken Baclawski
> College of Computer Science
> Northeastern University
> kenb@ccs.neu.edu
> 
> 
Received on Monday, 2 April 2001 08:26:07 GMT

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