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Re: [ontolog-forum] Ontology of Rough Sets

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Tue, 18 Jan 2011 10:33:53 -0500
Message-id: <4D35B2E1.1000603@xxxxxxxxxxx>
Tara,    (01)

Thanks for the note of support:    (02)

> Well, if we are going to develop a formal ontology, there
> a number of methodologies out there to follow.
> I'd be interested in trying out John Sowa's proposal from
> http://www.jfsowa.com/pubs/rolelog.pdf    (03)

I am now in the process of writing a book with the title
"Principles of Logic and Ontology".    (04)

In the first chapter, I begin with "Knowledge Design Patterns",
which are the basis for various methodologies.  Then I show how
those patterns are expressed in different versions of logic.    (05)

For example, Section 1.2 is "The Aristotelian Subset", which
introduces:    (06)

   KDP 1. A hierarchy of categories.
   KDP 2. Definition by genus and differentiae.
   KDP 3. Reasoning about monadic predicates.
   KDP 4. Categories, types, classes, and sets.    (07)

> 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?    (08)

In principle, infinity.  But they can be grouped in various major
kinds with an open-ended number of variations.    (09)

For example, under KDP 4, I start with Aristotle's term 'category'.
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.    (010)

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
same.    (011)

The criteria for membership in a collectio9n (which I use as a
general term for any kind of grouping, including sets) are related
to the criteria for determining the truth of a predicate in various
circumstances.  This gets us into a very wide range of KDPs for
nonmonotonic reasoning, open worlds vs. closed worlds, situations,
contexts, possible worlds, uncertainty, probability, fuzziness,
roughness, hypotheses, etc.    (012)

> I imagine there are some statisticians who disagree. And the
> disagreement hinges on the ambiguity of "probability"; the
> Bayesian and the frequentist definitions.    (013)

Yes.  And I'm sure that many more variations will be invented
and used during the coming centuries.  For some purposes, any
of them might be useful.  The only point that is certain is
that any finite upper bound will be an inadequate approximation
to infinity.    (014)

Much of the material in the rolelog.pdf paper will be incorporated
into the book, but with more detail and examples.    (015)

John    (016)

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