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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "doug foxvog" <doug@xxxxxxxxxx>
Date: Fri, 21 Jan 2011 10:46:55 -0500 (EST)
Message-id: <61332.>
On Thu, January 20, 2011 0:27, John F. Sowa said:
> A question about types, sets, and classes arose on the AESIG
> mailing list, which is related to the earlier discussion in
> this thread.  See, in particular, the slides by Peter Aczel.
> John
> -------- Original Message --------
> Subject: Re: [architecture-strategy] Relationship between types, classes
> and sets
> Date: Wed, 19 Jan 2011 23:54:43 -0500
> From: John F. Sowa <sowa@xxxxxxxxxxx>
> To: architecture-strategy@xxxxxxxxxxxxxxx
> Cory,
>> The terms "type", "class" and "set" are used within many
>> modeling
>> languages, formal languages and natural language.  A precise
>> specification of languages involving these terms must have them
>> precisely specified.
> ...
>> Rick murphy referenced this paper:
>> http://www.cs.man.ac.uk/~petera/what-is-a-set-leeds-nov-2010.pdf
> These slides by Peter Aczel distinguish the terms 'type', 'class',
> and 'set' as used by logicians who talk about higher orders of
> infinity.  Most of that discussion is irrelevant to AESIG.
> ...
> Summary:
>   1. A set is extensional: it is uniquely determined by its elements.
>   2. Type is an informal notion has been formalized in different ways.
>      But a very common and useful way is to choose some predicate
>      that specifies the type.
>   3. A class is the extension of some predicate.
>   4. Every set is a class, but some classes could be too big to
>      be sets -- but those are hyper-infinite monsters that are
>      irrelevant to computer systems.    (01)

>> If an object could change types...
> No object can "change" types without becoming a different object.    (02)

I think that John refers to a mathematical object here.    (03)

> [Example of type Integer vs. floating point number]    (04)

However, for a temporal object, such as a person, the specific "type" of
that object can change.  For example, at one time Barak Obama was of type
HumanChild, while at another he was of type HumanAdult.  Of course, there
is some more generic type of which the object is an instance throughout
its existence.  But sets, classes, and types can be generated or defined
using narrower predicates.    (05)

A standard distinction between a set and a class, is that membership in
a class cannot change, while membership in a class can.  The set is either
defined extensionally or generated as the extension of a predicate in a
given context.  Once the set is generated, the extension of the predicate
in a different context (which might merely mean a different time) is no
longer necessarily the same set.    (06)

If the extension of the predicate is context-independent, membership in
the associated class is fixed.  An instance of a context-free type (such
as Integer) can not change whether or not it is an instance of that type.    (07)

-- doug f    (08)

> John    (09)

doug foxvog    doug@xxxxxxxxxx   http://ProgressiveAustin.org    (010)

"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
    - Dr. Martin Luther King Jr.
=============================================================    (011)

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