W3C home > Mailing lists > Public > www-rdf-comments@w3.org > October to December 2001

Re: vocabulary for traditional sets

From: Graham Klyne <GK@ninebynine.org>
Date: Wed, 07 Nov 2001 18:48:13 +0000
Message-Id: <5.1.0.14.2.20011107183817.033feec0@joy.songbird.com>
To: Dan Connolly <connolly@w3.org>
Cc: www-rdf-comments@w3.org
At 09:10 AM 11/7/01 -0600, Dan Connolly wrote:
>Graham Klyne wrote:
> >
> > At 04:02 AM 11/6/01 -0600, Dan Connolly wrote:
> > >But I would like to be able to say that
> > >X and Y are sets, which allows you to conclude,
> > >from the fact that their extensions/membership
> > >are  the same that they are identical.
> >
> > So far, OK.
> >
> > >So I propose
> > >         rdfs:Set rdfs:subClassOf rdfs:Class.
> >
> > I'm deeply suspicious of this.  It feels like a "layer violation" -- a
>[...]
> > So we have the denotation of a node with type rdfs:Set is some value x
> > such that ICEXT(x) are the members of the set.
>
>oops; yes, I meant
>         rdfs:Set rdf:type rdfs:Class.

Ah, that changes everything, I think...

> > >where
> > >         this log:forAll :x, :y.
> > >         { :x a rdfs:Set.
> > >         :y a rdfs:Set.
> > >         :x rdfs:subClassOf :y.
> > >         :y rdfs:subClassOf :x.
> > >         } log:implies {
> > >         :x ont:equivalentTo :y
> > >         }.

... but

     :x rdfs:subClassOf :y .

implies that :x and :y are *classes*, which is not entailed by:

     rdfs:Set rdf:type rdfs:Class .
     :x rdf:type rdfs:Set .
     :y rdf:type rdfs:Set .

Looking at this, I think you probably *did* mean:

     rdfs:Set rdfs:subClassOf rdfs:Class .

> >
> > Rather than trying to define this broad notion of equivalence,
>
>it's actually identity.

Which is, I think, about as broad a notion of equivalence as you can 
get.  (I think of equivalence as yielding the same truth under certain 
usage;  identity would yield the same truth values under all usages.)

>[...]
> > My intuition about a set as opposed to a class is that the set simply is
> > the denotation of a node, not "indirected" via ICEXT like a class.
>
>Yes, that's more intuitive, but what I'm proposing is equivalent,
>and it uses the existing model theory without mucking with it.
>
>
> > And a parting shot:  the construction of classes allows a class value to be
> > a member of its own extension.  Indeed, we have:
> >
> >      I(rdfs:Class) in ICEXT(I(rdfs:Class))
> >
> > Your proposal for rdfs:Set retains this structure, and nothing has been
> > said to prevent a set value being a member of its own class extension.
>
>Quite... I'd have to give the other axioms of sets, in addition
>to extensionality (sets with equal members are equals); from memory:
>         -- the axiomatic existence of the empty set
>         -- singletons (I mostly get these for free with rdf:type)
>         -- pairs
>         -- union
>         -- power set
>         -- well-formedness

OK.

I felt had to try the challenge, though even as I wrote it I didn't feel I 
could make it stick.

#g


------------
Graham Klyne
GK@NineByNine.org
Received on Wednesday, 7 November 2001 16:12:05 UTC

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