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Re: vocabulary for traditional sets

From: Dan Connolly <connolly@w3.org>
Date: Wed, 07 Nov 2001 09:10:35 -0600
Message-ID: <3BE94EEB.C7663694@w3.org>
To: Graham Klyne <GK@ninebynine.org>
CC: www-rdf-comments@w3.org
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.

> >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
> >         }.
> 
> Rather than trying to define this broad notion of equivalence,

it's actually identity.

[...]
> 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

-- 
Dan Connolly, W3C http://www.w3.org/People/Connolly/
Received on Wednesday, 7 November 2001 10:10:20 UTC

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