Relating groups of axioms within different ontologies by the lattice
relation would probably be useful - I agree. I would like to see examples
of that, starting with existing foundation/upper ontologies. It would help
make the lattice discussion more concrete - the better to evaluate its
utility for our purposes. (01)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
> bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F. Sowa
> Sent: Wednesday, January 14, 2009 9:33 AM
> To: [ontolog-forum]
> Subject: [ontolog-forum] Lattice of theories
> I received an offline note with some objections to the lattice
> of theories as a useful framework for the Foundation Ontology.
> Following is a slightly edited version of my reply.
> John Sowa
> -------- Original Message --------
> The lattice is a purely theoretical structure that embodies
> all possible generalization and specialization relations among
> theories. Every implementation of any special case is an
> implementation of that theory.
> > I am not convinced that the lattice of all possible theories
> > is the most efficient solution to the problem.
> That is like saying that you don't like integers because there
> are infinitely many of them and some functions over the integers
> are difficult to compute.
> The fact that the theory of lattices or the theory of integers
> embodies a very wide range of useful relationships is good.
> The fact that some relations may be hard to compute is not an
> argument against the theory. If you don't need them, you don't
> have to compute them. If you do need them, the lattice is not
> a hindrance, and it can be a help.
> > I am concerned, for example, about the relation of the Cyc and
> > SUMO and BFO and DOLCE. I don't think that any one of the
> > relations applies to any two of those, as whole theories.
> Of course it applies. It says that they are cousins, not supertypes
> or subtypes of one another. But the lattice also shows how to find
> common supertypes:
> 1. If you can find any subset of axioms that is common to all three
> of them, it is automatically the axiomatization of a theory that
> is a common supertype (or "core") of all three.
> 2. If you can find axioms common to two out of the three, it defines
> a common supertype of those two.
> 3. If you can't find any common axioms (or can't find all you'd
> like to find), you might find some set of simpler axioms that
> imply different axioms in each of the three. That set of
> simpler axioms is also a common supertype.
> 4. Any core that you can propose is guaranteed to be a common
> supertype of any theory derived by adding axioms to that core.
> This is an illustration of how the theory shows you how to think
> about the problem. Any core you propose is going to belong to
> cases #1, #2, #3, #4 above or some variation of them. Some of
> the common axioms may be easier to find than others, but the
> methods of testing them to see whether they are indeed common
> are all based on the relationships embodied in the lattice.
> All of those techniques plus many others are implementations
> of that theory. Some of them may be easier to implement than
> others. But the lattice displays all the possible relationships
> among theories. Any implementation of any subset of those
> relationships counts as an implementation of the lattice.
> If you ignore the lattice, that is like playing with integers
> without any theory about how they are related to one another.
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