John,
I find rather worrying your description of the structure of the
lattice in terms of the axioms used in an ontology. From the data
exchange perspective, I have noted the use of standard data models to
exchange incompatible data - incompatible not because the data
structures were different, but because the groundings of the model
were
different. (01)
What you are proposing looks more like a lattice of ontology
structures than a lattice of ontologies. Not that this is not useful -
mathematicians generally stop exploring algebraic structures when they
can show they are isomorphic to some known structure. Theorem solving
could be reduced to searching the lattice for an ontology with the
required structure, and searching from there to the known theorems of
that structure. (02)
However, to make the lattice one of ontologies, one would need
to identify the grounding of the ontology. I suppose we might also end
up with a lattice of groundings, though since the specification of
groundings seems to be poorly understood, I don't have any suggestions
of how to do it. (03)
Sean Barker (04)
Bristol, UK (05)
-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F.
Sowa
Sent: 14 January 2009 14:33
To: [ontolog-forum]
Subject: [ontolog-forum] Lattice of theories (06)
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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. (09)
John Sowa (010)
-------- Original Message -------- (011)
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. (012)
> I am not convinced that the lattice of all possible theories is the
> most efficient solution to the problem. (013)
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. (014)
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. (015)
> 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. (016)
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: (017)
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. (018)
2. If you can find axioms common to two out of the three, it defines
a common supertype of those two. (019)
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. (020)
4. Any core that you can propose is guaranteed to be a common
supertype of any theory derived by adding axioms to that core. (021)
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. (022)
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. (023)
If you ignore the lattice, that is like playing with integers without
any theory about how they are related to one another. (024)
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