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Re: [ontolog-forum] Fw: Context in a sentence

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Thu, 28 Jan 2010 09:01:11 -0500
Message-id: <4B6198A7.4080005@xxxxxxxxxxx>
Ali,    (01)

I'm violating my one-note-per-day recommendation because I wanted
to address the following point:    (02)

AH> The theories / ontologies in COLORE do not constitute one single
 > giant lattice.    (03)

Of course not, because the lattice is infinite.  It represents all
*possible* theories that can be stated in any given logic.  I use
the term 'hierarchy' for that subset of *currently defined* theories
that happen to be stored in any given repository.    (04)

AH> Unless I have misunderstood, whereas the lattice mentioned in
 > your work consists of a single relation "logical extension,"
 > there are multiple links in COLORE: specifically representation
 > theorems / definable interpretations etc.    (05)

In my KR book, I adopted the three basic operators of the AGM
axioms for theory revision:  expansion, contraction, and revision.
Then I added a fourth operator of analogy.  As a brief summary
of the four operators, see Figure 4 of the following paper:    (06)

    A Dynamic Theory of Ontology    (07)

All possible operations for transforming or relating any two
theories can be defined as combinations of those four operators.
Actually, you only need three operators, since revision can be
defined as a sequence of contractions and expansions.  Analogy
is the operator you can use for representation and interpretation.    (08)

You can explain all methods of theory revision, learning,
discovery, representation, and interpretation as methods
of walking or jumping through the lattice.  Expansion and
contraction are single-step operators used in walking, and
analogy introduces jumps.    (09)

You can also use the lattice to explain the difference between
monotonic and nonmonotonic reasoning.  Classical deduction starts
with some subset of a theory (called the axioms) and derives any
or all of the propositions of the full theory.  All versions of
nonmonotonic reasoning explicitly or implicitly involve expansion
or contraction that walk through the lattice to a theory that is
different from the one you start with.  Analogy is a high-speed
method for relabeling predicates in order to convert (or relate)
one consistent theory to another.    (010)

Finally, the lattice is the foundation for modularity.  Simple
modules are located in upper areas of the lattice, and consistent
combinations of modules are the infimum of the modules from which
they were derived.    (011)

I am *not* claiming that I discovered everything that could be
discovered about relating theories.  And I am certainly not
claiming that I anticipated what you wrote in your book.    (012)

The only point I'm making is that the lattice is a systematic
framework for showing how all the methods that have been invented
or could ever be invented are interrelated.    (013)

John    (014)

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