John:
Can you provide references to more specific applications and/or examples of the use of a Lindenbaum Lattice.
From a quick overview it appears that the referenced techniques are used to create the "structure of an ontology" in any specific context.
Take care and have fun,
Joe
On Thu, Dec 15, 2011 at 6:47 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
Michael et al.,
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.
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.
1. Assume some fixed logic L for all the ontologies.
2. Let V be the total non-logical vocabulary that is used in all
the ontologies (it may be countably infinite).
3. Define an ontology as a collection of axioms stated in L
with some finite subset of V as its vocabulary.
4. Define a theory as the closure of an ontology.
5. Define two ontologies to be equivalent if their theories
are identical.
6. Define the Lindenbaum lattice as the lattice of theories
with entailment as the partial ordering.
7. Define a hierarchy as a finite subset of the Lindenbaum lattice.
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:
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.
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.
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.
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.
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.
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.
Point #5 would require considerably more discussion, and
it would require a separate paper on its own.
John
--
Joe Simpson
Sent From My DROID!!
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