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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Tara Athan <taraathan@xxxxxxxxx>
Date: Tue, 18 Jan 2011 09:40:36 -0800
Message-id: <4D35D094.30809@xxxxxxxxxx>
Thanks for weighing in on this rambling discussion. And I look forward to reading your book.
We have definitions for four kinds of sets - fuzzy, probabilistic,
rough, and crisp sets.  How many more KINDS of definitions
(i.e., aspects like fuzzy, probabilistic, ..) are there in principle?
That was actually Rich's question, not mine.

I do have some confusion regarding your definition of class:

I define a type as a method of grouping specified by some monadic
predicate, which is true of everything in the type and false of
everything not in the type.  I define a set by the usual criteria
that two sets with the same elements are identical.

But then I define a class as a set that is determined by some type.
This means that the identity criteria for classes are not the same
as the identity criteria for sets:  A company may have many employees,
some or all of whom are drivers.  The type Employee and the type Driver
are distinct, and therefore the class Employee and the class Driver are
distinct.  But if all employees happen to be drivers, the sets are the
What happens when an Employee is fired or a new Employee is hired. The type doesn't change. But the set of Employees becomes a different set - does the class change? Or does the class disappear, to be replaced by a new class? Or are we talking about the set of all Employees, past, present and future?

The example I frequently see used to illustrate this point is the classes "three-sided polygon" and "three-angled polygon", which have the same extension but different definitions, so they are different classes. But there is no element of time in that example, so the temporal issues don't come up to cloud the issue. Plus there is a difference in the examples that it is possible to prove that every three-sided polygon is always a three-angled polygon, where in your example the extensions just happen to be the same in some possible world, and are different in others.


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