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Re: singleton sets

From: Ian Emmons <iemmons@bbn.com>
Date: Wed, 13 Aug 2008 15:20:53 -0400
Message-Id: <6E8D9A7D-468D-400C-8525-C6B6A6C2711E@bbn.com>
To: Semantic Web at W3C <semantic-web@w3.org>

Frank,

Thanks for making this distinction clear -- I had a suspicion that the  
discussion was confusing these concepts, but I skipped several of the  
intervening messages, and I wasn't sure.

This may or may not be a useful interjection, but the level of  
abstraction in this discussion is a little high, so I thought a  
concrete example might help.  I encountered a situation where "X   
type  Y;  X  subClassOf  Z;" seemed to be useful.  Our application  
modeled transportation assets, and it also drew its data from a  
relational database.  So, we had a table of vehicles that had (in the  
ugly manner of an RDB) a column VehicleType, which contained one of an  
enumerated list of possible types (1 for truck, 2 for cargo plane, 3  
for containerized cargo ship, etc.).  Once a row from this table was  
translated into RDF, we wanted to classify the type of vehicle via an  
RDF class, so we created a taxonomy of vehicle classes with an entry  
for each of the enumerated vehicle types.  This yields the following:

     x type V;  V subClassOf Vehicle;

Where things got messy was when we realized that our RDF  
representation needed to retain the enumerated vehicle type code.  To  
do this we added the type code as a data type property of the vehicle  
class, like so:

     x type V;  V subClassOf Vehicle;  V hasTypeCode t;

As we usually do, we gave the hasTypeCode property a domain  
(EnumeratedVehicleType), and so the obvious inference yields the  
following:

     x type V;  V subClassOf Vehicle;  V hasTypeCode t;  V type  
EnumeratedVehicleType;

In particular:

     V subClassOf Vehicle;  V type EnumeratedVehicleType;

I'm sure we could have modeled this differently so as to eliminate the  
subclass-and-type pattern, but this seemed to be the nicest way to  
handle it to us.  Hopefully it's a compelling use case for this  
discussion.

Cheers,

Ian



On Aug 13, 2008, at 10:59 AM, Frank Manola wrote:

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 19:21:36 GMT

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