On Jul 8, 2012, at 12:41 PM, John F Sowa wrote: (01)
> On 7/8/2012 11:11 AM, Pat Hayes wrote:
>> OWL Full does indeed treat classes intensionally. OWL-DL, however, treats
>> them extensionally. See the 'direct semantics' (which is normative) for
>OWL-DL
>> inhttp://www.w3.org/TR/owl-semantics/direct.html, where the interpretation
>> of a class name is simply a subset of the universe. That is an extensional
>> meaning for classes.
>
> In every version of model theory for every version of logic, the
> denotation of a relation is a subset of the universe of discourse. (02)
Not exactly. In RDF, RDFS, OWL-Full and Common Logic, the denotation of a
relation symbol (a class or property name in RDF/OWL, a name in a relation
position in CL) is an *individual* which has an associated relational
extension, which is a set of tuples (in general: in the RDF/OWL languages it is
a set of individuals or a set of pairs of individuals, i.e. 1- and 2-tuples.)
It is this semantic distinction between the actual denotation and the
associated extension which makes the semantics able to handle intensional
identity between classes and relations. The OWL-DL semantics does not have this
device (which AFAIK was original in the Common Logic standard, and was invented
by Chris Menzel, from whom I copied it and incorporated it into
RDF/RDFS/OWL-Full), and instead treats class names as directly denoting sets of
individuals, with no intervening individual. This more conventional
construction forces the language to be extensional. (03)
>
> But two different relations (or OWL classes) can be specified by axioms
> that are not logically equivalent. They might have the same extension
> in one universe, but different extensions in a different universe. (04)
Of course, but that is not what makes the language intensional. This is true
for both extensional and intensional languages. (05)
>
> The direct model theory (by Patel-Schneider & Horrocks) acknowledges that:
>
> From http://www.w3.org/TR/owl-semantics/direct.html
>> Definition: A collection O of abstract OWL ontologies and axioms and
>> facts entails an abstract OWL ontology or axiom or fact O' with respect
>> to a datatype map D if each interpretation with respect to map D that
>> satisfies each ontology and axiom and fact in O also satisfies O'.
>
> They are talking about different "interpretations" in which the same
> a given set of axioms can specify a class that may have different
> extensions in different universes.
>
> It is much cleaner and simpler just to say that every OWL class
> has a defining relation (specified by axioms) (06)
I am not sure what you mean by a "defining relation". If it means what it seems
to mean, then not all OWL classes need have one. (07)
> and a extension that
> varies from one universe of discourse to another. (08)
Well, yes, it does vary from one universe to another, but thats how model
theory works :-) (09)
Pat (010)
>
> John
>
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