|From:||Ali Hashemi <ali.hashemi@xxxxxxxxxxx>|
|Date:||Wed, 11 Feb 2009 10:31:18 -0500|
What you describe resonates a lot, but there are some minor quibbles.
First, by focusing on direct axiom reuse, we may limit ourselves to syntax matching. In my mind, it makes more sense to focus on the models that arise from the axioms as opposed to whether a particular matching axiom formulation has been implemented.
For example, restricted to lattices, the set of models for "meet" and "join" are the same as those for"minimal upper bound" and "maximal lower bound."
I would imagine that we would want this lattice of theories to be able to accommodate these types of equivalences.
Another point - i'm not sure whether you are actually saying this or not, it seems to be implied. But it appears you are suggesting that the level of granularity for "acceptance" as an ontology would be axiom level.
Imagine I have two modules, in M1 I have two sentences P and Q vs another module M2 where I have the single sentence S = P&Q
Are they two axioms or one? Further might P alone constitute an ontology? What're the criteria for the ontology "worthiness" of an axiom or a set of axioms? Practical use i would guess.
Imagine the first theory i see is that for "addition." It shares nothing with what I want but the property of associativity. Does this suggest that there is a unique theory called "associativity"? Is that an ontology? How exactly does the axiom (module? / ontology?) of "associativity" make sense de-contextualized from it's usual cohort axioms of transitive, reflexive etc.?
One practical question is also how to organize the resultant explosion of theories or ontologies.
Instead of organizing all these theories or modules in a single layer or hierarchy, we could go orthogonal. Using planes / layers seem to be a promising way to declutter the combinations and making traversing this "space" more efficient. If we know the content of modules, i.e. these modules are about geometries, these axioms are about mereotopologies, etc. - we might then use representation theorems or mapping axioms, to get us out of a particular plane and into another. We do this since, beyond axiom matching, they establish the semantic equivalence between axiom sets which differ syntactically and possibly even methodologically (mlb vs join).
Does that make sense? :P
On Wed, Feb 11, 2009 at 8:59 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
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