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Re: Why do you want to do that?

From: Sampo Syreeni <decoy@iki.fi>
Date: Mon, 11 Aug 2008 04:08:10 +0300 (EEST)
To: Pat Hayes <phayes@ihmc.us>
cc: "Richard H. McCullough" <rhm@pioneerca.com>, Semantic Web at W3C <semantic-web@w3.org>, KR-language <KR-language@YahooGroups.com>, Adam Pease <adampease@earthlink.net>
Message-ID: <Pine.SOL.4.62.0808110321170.25511@kruuna.helsinki.fi>

On 2008-08-08, Pat Hayes wrote:

>> 1. X subClassOf X; A neat mathematical property, right?
> Im not sure why you call it 'mathematical'. It follows from the usual 
> definition of subClass, which is that A subClass B just when every 
> member of A is also a member of B. Given this, its obvious that A is a 
> subClass of A.

Yes. I would also add that, well, if a statement like that is *so* 
self-evident, shouldn't it *per force* apply in your logic? Perhaps it 
feels sort of redundant. But is that a problem? Not really: after all 
everything in any given theory except the fundamental axioms is *fully* 
redundant, in the sense that all sound theorems can at least in theory 
be deduced from the axioms. From that starting point at least, logical 
redundancy shouldn't be a problem.

And if you then go further, and consider semantics as well, quite a 
number of tautologies hold real value, as well. Here we're talking about 
the fact that X subClassOf X is asserted by default. From the human, 
semantically aware point of view that is rather stupid: of course 
classes/sets are their on subclasses/subsets.

But then, from the logical and automated inferencing point of view quite 
the contrary holds: no computer knows about X's being sets, and even 
less about any concept of subsets of sets. We have to declare such 
things and operationalize the means of dealing with the concepts before 
a machine can help us deal with them.

As such, is it not so that classes/sets being *declared* as being their 
own subclasses/-sets constitutes an *essential* part of the formalized 
definition of what sets/classes *are*? Does not such a trivial 
declaration actually constitute an easily formalizable means of 
*defining* a *fundamental* property of sets and classes, *per se*? In 
the sense that it tells us something very real, and easy to reason 
about, on how those funny, formal concepts *behave*, and as such about 
how they're defined/which real life models they could perhaps possess?
Sampo Syreeni, aka decoy - mailto:decoy@iki.fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
Received on Monday, 11 August 2008 01:09:02 UTC

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