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Re: [ontolog-forum] Lattice of theories

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Patrick Cassidy" <pat@xxxxxxxxx>
Date: Wed, 14 Jan 2009 11:18:53 -0500
Message-id: <096201c97663$cb0e9460$612bbd20$@com>
John,
  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)

Pat    (02)

Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx    (03)


> -----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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>     (04)


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