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

 To: ontolog-forum@xxxxxxxxxxxxxxxx "John F. Sowa" Thu, 15 Dec 2011 09:47:57 -0500 <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) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J    (020) ```