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Re: [ontolog-forum] Ontology, Analogies and Mapping Disparate Fields

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Thu, 15 Dec 2011 09:47:57 -0500
Message-id: <4EEA089D.3070708@xxxxxxxxxxx>
Michael et al.,    (01)

The paper is nice, but 42 pages is more than twice the space needed
to define the those terms.  You can shorten it to much less than 20
pages with a huge improvement in clarity, generality, and readability.    (02)

The key generalization that cuts through the muck is something I've
been talking about for years:  The Lindenbaum Lattice for a given
logic and vocabulary.    (03)

  1. Assume some fixed logic L for all the ontologies.    (04)

  2. Let V be the total non-logical vocabulary that is used in all
     the ontologies (it may be countably infinite).    (05)

  3. Define an ontology as a collection of axioms stated in L
     with some finite subset of V as its vocabulary.    (06)

  4. Define a theory as the closure of an ontology.    (07)

  5. Define two ontologies to be equivalent if their theories
     are identical.    (08)

  6. Define the Lindenbaum lattice as the lattice of theories
     with entailment as the partial ordering.    (09)

  7. Define a hierarchy as a finite subset of the Lindenbaum lattice.    (010)

Furthermore, once you introduce the Lindenbaum lattice, you can
define all kinds of transformations and relationships among
ontologies and their theories as walks through the lattice:    (011)

  1. The AGM operators for belief revision (expand, contract,
     and revise) specify walks through the lattice:  expand
     moves down, contract moves up, and revise moves sideways.    (012)

  2. Deduction stays within a given theory.  If the logic is
     complete, it can be used to derive any statement in the
     theory from any equivalent ontology.    (013)

  3. Induction is a revision by expansion that can reduce the
     number of axioms by deriving generalizations.  It may
     enlarge the theory to a proper superset, but it does
     not increase the size of the vocabulary.    (014)

  4. Abduction is a revision by expansion that can also reduce
     the number of axioms, but it introduces new vocabulary
     and axioms that are not generalizations of statements
     in the theory.    (015)

  5. All versions of nonmonotonic reasoning can be defined
     as similar walks through the lattice.  This requires
     more discussion that would go beyond 20 pages.  But it
     shows the power that come with the Lindenbaum L.    (016)

The first four points above are more general and easier to
explain than the discussion of extensions etc. in that 42
page paper.  Most of the proofs in that paper become trivial
corollaries.  You can define all of that clearly and simply
with much less hairy notation than the present paper contains.    (017)

Point #5 would require considerably more discussion, and
it would require a separate paper on its own.    (018)

John    (019)

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