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Re: singleton sets

From: Frank Manola <fmanola@acm.org>
Date: Wed, 13 Aug 2008 10:59:39 -0400
To: Richard H. McCullough <rhm@PioneerCA.com>
Message-Id: <9F4132D4-45EC-4D9A-B058-552D114200B9@acm.org>
Cc: "Pat Hayes" <phayes@ihmc.us>, "KR-language" <KR-language@YahooGroups.com>, "Semantic Web at W3C" <semantic-web@w3.org>, "Adam Pease" <adampease@earthlink.net>

On Aug 12, 2008, at 5:05 PM, Richard H. McCullough wrote:

> 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  
> Logic",
> 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).   
> According to the
> Axiom of Comprehension, there is a set, call it a, whose only member  
> is x. 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 the 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 it, but {x}
> has exactly one element: x.

Whoa!  What we were originally talking about wasn't singleton sets, it  
was the following question:

>>>>> 2. X  type  Y;  X  subClassOf  Z;
>>>>> Another neat property: X is an individual and a class.
>>>>> Now I can ... What?  I don't know.
>>>>> Why do you want to do that?

Wanting to be able to treat a class X as an individual may or may not  
be a good idea, but this isn't the same as wanting to treat a  
singleton set as *the same* individual as its only member.  To  
paraphrase your quotation above, in the direction of subtle subject  
changes like this lies, if not madness, at least dreadful confusion.

Received on Wednesday, 13 August 2008 15:00:27 UTC

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