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Re: singleton sets

From: Richard H. McCullough <rhm@pioneerca.com>
Date: Wed, 13 Aug 2008 11:04:03 -0700
Message-ID: <D5ED452F24F941B48F4838A0BDDFF06F@rhm8200>
To: "Frank Manola" <fmanola@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>

Hi Frank
I should have explained my thoughts better.
You're right, singleton set and class-individual are not the same.
(Jacek Kopecky also asked if I was conflating them.)

But there's an analogy there.
In both cases, an abstract group (set or class)
is being equated to something which is a member of a group.

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;

----- Original Message ----- 
From: "Frank Manola" <fmanola@acm.org>
To: "Richard H. McCullough" <rhm@pioneerca.com>
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>
Sent: Wednesday, August 13, 2008 7:59 AM
Subject: Re: singleton sets

> 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.
> --Frank
Received on Wednesday, 13 August 2008 18:29:02 UTC

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