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Re: [ontolog-forum] confounded models

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Wed, 18 Jul 2007 14:18:33 -0400
Message-id: <469E5979.1020608@xxxxxxxxxxx>
Sean,    (01)

To see the commonality among various positions, it is helpful
to map them to a common terminology.    (02)

> Alternatively, one could treat the taxonomy not as a model,
 > (these terms 'are an image of' reality) but as a system of
 > differentia (these criteria map reality into this set of terms).
 > The question is then not whether one system was more truthlike
 > than another, but whether the differentia are adequate for our
 > purposes.    (03)

In predicate calculus, for example, each differentia is represented
by a monadic predicate, such as isAnimate(x) or isPhysical(x).    (04)

The taxonomy is a lattice formed by conjunctions of differentiae.
For any set of differentiae, d1, d2, ..., dN, a new type in the
taxonomy is defined by their conjunction:    (05)

    t(x) = d1(x) & d2(x) & ... & dN(x).    (06)

This is basically the approach used for Aristotle's syllogisms,
which are essentially a subset of OWL.  It is also used in FCA
(Formal Concept Analysis) to define a lattice of concepts in
terms of a set of differentiae.    (07)

The taxonomy is definitely *not* a model in Tarski's sense
because it only defines monadic predicates or types.  It does
not say anything about which types are true or false of anything.
It is also a very limited logic because it does not support
relations with 2 or more arguments.    (08)

A Tarski-style model for such a system could be defined by
specifying some set D of individuals, and for each individual
x in D saying which differentiae are true or false of X.    (09)

Then Tarski's definitions would determine how the truth of
any first-order statement defined in terms of the base
differentiae could be evalated to T or F.    (010)

John Sowa    (011)

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