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RE: singleton sets

From: Obrst, Leo J. <lobrst@mitre.org>
Date: Tue, 12 Aug 2008 18:07:48 -0400
Message-ID: <9F771CF826DE9A42B548A08D90EDEA8003681C38@IMCSRV1.MITRE.ORG>
To: "Richard H. McCullough" <rhm@pioneerca.com>, "Pat Hayes" <phayes@ihmc.us>
Cc: "KR-language" <KR-language@YahooGroups.com>, "Semantic Web at W3C" <semantic-web@w3.org>, "Adam Pease" <adampease@earthlink.net>

Actually, it's not so cut and dried as Pat has stated. In fact during
the development of OWL, there was quite a bit of discussion when the
Class as Instance notion was put forth by Chris Welty, with the usual
example about species. Chris argued that there was a real engineering
need for this, going back to his earlier paper [1]. 

Personally, I do think many folks have problems with this, even though
the formal semantics of the language states what is meant, because it
seems alien to the usual modeling notions. It seems as though you are
equating Universals and Particulars, ontologically speaking. And if I
recall, in those OWL discussions there were questions about equating
two separate universes of discourses, i.e., potentially quantifying
over both classes and instances, which struck many as also moving to a
second-order logic. For example, object level classes as being
instances of a meta-level Class. How do you synchronize the Class
Elephant with the Instance Elephant? In fact, Welty's original paper
considered that we were in the realm of second-order logic.

I think Dick is highlighting that this is a problematic notion for him
and for others. 


[1] C. Welty. 1995. Towards an Epistemology for Software
Representations. Proceedings of KBSE-95, The Tenth Knowledge-Based
Software Engineering Conference. IEEE Computer Society Press. Pp.
148-154. November, 1995.

Dr. Leo Obrst       The MITRE Corporation, Information Semantics 
lobrst@mitre.org    Information Discovery & Understanding, Command and
Control Center
Voice: 703-983-6770 7515 Colshire Drive, M/S H305 
Fax: 703-983-1379   McLean, VA 22102-7508, USA 

-----Original Message----
From: semantic-web-request@w3.org [mailto:semantic-web-request@w3.org]
On Behalf Of Richard H. McCullough
Sent: Tuesday, August 12, 2008 5:05 PM
To: Pat Hayes
Cc: KR-language; Semantic Web at W3C; Adam Pease
Subject: singleton sets

Here's someone else who doesn't like singleton sets,
and hence doesn't like classes which are individuals.

John Barwise & John Etchemendy (1992), "The Language of First-Order
Third Edition, Revised & Expanded, Center for the Study of Language and

Information, Stanford, Page 212

    Suppose there is one and only one object x satisfying P(x).
to the
Axiom of Comprehension, there is a set, call it a, whose only member is
That is,
a = {x}.  Some students are tempted to think that a = x..  But in that 
direction lies,
if not madness, at least dreadful confusion.  After all, a is a set (an

abstract object)
and x might have been any object at all, say Stanford's Hoover Tower. 
Hoover is
a physical object, not a set.  So we must not confuse an object x with
set {x},
called the singleton set containing x.  Even if x is a set, we must not

confuse it with
its own singleton.  For example, x might have any number of elements in
but {x}
has exactly one element: x.

Dick McCullough
Ayn Rand do speak od mKR done;
mKE do enhance od Real Intelligence done;
knowledge := man do identify od existent done;
knowledge haspart proposition list;
Received on Tuesday, 12 August 2008 22:08:53 UTC

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