SUO: Re: Foundation Ontology
Jim,
Yes, I have been thinking about submitting the Foundation Ontology, but it
looks like I need a couple more weeks. In the discussion below, I give an
overall picture of the ontological framework that I envision, I discuss some
of the conceptual problems being addressed, and I suggest a path for review
and editing. Because of my expanded response, I am cc-ing this to the list
at large.
____________________
The *overall picture* I have for an "industrial strength" ontological
framework goes is follows:
1. There is a basic or core IFF ontology that interfaces the new KIF logical
language with the metalevel ontological framework. This would be close to
that expressed as Chris Menzel's Basic Ontology, but perhaps a bit simpler
since only a few things are needed -- class, subclass, function, signature,
pairing '[-]', and projections '(- 1)', '(- 2)' etc.
2. There is a (rather large) metalevel (= structural level) ontology that
breaks up into two essential parts corresponding to the large/small
distinction in set-theoretic and category-theoretic foundations.
2.1 The upper metalevel represents the "large" notion of foundations. This
is where the Foundation Ontology resides; the other components in this upper
metalevel are the Category Theory Ontology and the large Classification &
Concept Lattice Ontology, the latter allowing us to represent the *truth
classification* and *truth concept lattice*, an overall goal for this
effort. The Foundation Ontology essentially represents set-theoretic classes
and their functions -- thus modeling the "large" notion of foundations.
2.2 The lower metalevel represents the "small" notion of foundations. A
specific goal of the lower metalevel is to represent some central notions of
model theory -- models (first-order structures), expressions and
satisfaction. In the approach that I am suggesting, models are 2-dimensional
structures composed of small classifications along one dimension and small
hypergraphs (or spans) along the other; here classifications represent the
instance-type distinction, whereas hypergraphs represent the entity-relation
distinction. The lower metalevel makes heavy use of the upper metalevel --
indeed, a goal in modeling the lower metalevel is to abide by the
CATEGORICAL PROPERTY (see [http://suo.ieee.org/ontology/msg02399.html]).
3. The object level is where the content ontologies reside. These could be
very generic, such as a 4D ontology, or specific, such as ontologies for
government or higher education. It should be noted that the object level
satisfies a representation property similar to the categorical property
satisfied by the lower metalevel -- the ontological language used is not
KIF; instead it is the terminology defined and axiomatized in the lower
metalevel (terms such as 'subtype', 'instance', 'expression', 'model',
'ontology', 'relation', 'entity', 'role', etc.).
____________________
Here is a discussion of some of the *conceptual problems* being addressed:
One reason for the delay in submitting the Foundation Ontology is the
following. Initially, I had thought that the Foundation Ontology, which as I
mentioned above models set-theoretic classes and their functions, would
encode the axioms for a *Cartesian-closed category* (and that the Set
Ontology, which models small sets and their functions, would encode the
axioms for a *topos*). To test this I was using lower metalevel ontologies,
such as the Classification Ontology. But recently, in order to satisfy the
categorical property, I seemed to need "power classes" (see
[http://suo.ieee.org/ontology/msg02381.html]) in order to represent some
aspect of fibrations (specifically, the category of classifications and
infomorphisms is fibered over the category of sets and function along the
"instance functor"). See McLarty's suggestion to use Benabou's theory of
fibrations and definability
[http://www.risc.uni-linz.ac.at/research/category/risc/catlist/goedel-cat].
My current belief is that fibers can be represented in the Foundation
Ontology, without a full-blown representation of fibrations a la Benabou. In
order to do this however, we seem to require that the Foundation Ontology
encode the axioms for a *topos* (see McLarty's first order expression of the
theory of a well-pointed topos with natural numbers and choice
[http://www.math.psu.edu/simpson/fom/postings/9802/msg00072.html]. These
axioms are now incorporated in the Foundation Ontology. Perhaps this version
of the Foundation Ontology would partially satisfy Feferman's
representational needs, as expressed in the FOM list Toposy-turvey thread
[http://www.math.psu.edu/simpson/fom/postings/9801/msg00164.html].
____________________
Here is one possible *path for review* and edit of the Foundation Ontology:
A. Submit to the SUO list for review, comments and criticisms. Beyond
set-theoretic critiques, also needed are comments about the new KIF and
concepts used from Chris Menzel's Basic Ontology.
B. Ask a select number of category-theorists to review. I have in mind the
following people:
- Ross Street [http://www.maths.mq.edu.au/~street/], a category-theorist
who seems to have some interest in applications (see his link to Category
Theory at The Boeing Company, which refers to the Specware software
mentioned by Michael Uschold [http://suo.ieee.org/ontology/msg00375.html]);
- Colin McLarty [http://www.cwru.edu/artsci/phil/mclarty.html], who gave
the challenge topos axioms mentioned above, and has written a topos-theory
book "Elementary Categories, Elementary Toposes"
[http://www.oup-usa.org./docs/0198514735.html];
- Bill Lawvere [http://www.acsu.buffalo.edu/~wlawvere/], who is one of
the primary creators of topos theory, and whose interest in applications
(although a theorist) is indicated in the very detailed response
[http://merwan.sunyerie.edu/~alsani/ctmail01-04,01/msg00014.html] that he
gave to a question of Mike Healy's on the category list.
C. Submit to the category list at large for review, comments and criticisms.
____________________
Robert E. Kent
rekent@ontologos.org
----- Original Message -----
From: "Schoening, James R CECOM DCSC4I"
<James.Schoening@mail1.monmouth.army.mil>
To: "'Robert E. Kent'" <rekent@ontologos.org>
Sent: Monday, May 21, 2001 3:00 PM
Subject: RE: Foundation Ontology
> Robert,
>
> Any further thought of submitting the Foundation Ontology as a
> starter document in SUO? As you know, the group needs one or more
documents
> to focus its efforts on.
>
> Jim Schoening
>
> -----Original Message-----
> From: Robert E. Kent [mailto:rekent@ontologos.org]
> Sent: Thursday, March 08, 2001 1:44 PM
> To: Schoening, James R CECOM DCSC4I
> Subject: Re: Foundation Ontology
>
>
> Jim,
>
> Possibly eventually -- not for merger with Ian's effort, but as an
> alternative approach. However, the IFF Foundation Ontology is still being
> worked on, with much feedback from several of the other IFF ontologies
such
> as Spans, Categories, Sets and Classifications (there is a lot of
synergism
> taking place right now). Although the fundamental idea is reasonably firm,
> the KIF code (at the ontologos site
[http://www.ontologos.org/IFF/IFF.html])
> is still in a flux. Perhaps in a month or so it will stabilize.
>
> Robert E. Kent
> rekent@ontologos.org
>
> ----- Original Message -----
> From: "Schoening, James R CECOM DCSC4I"
> <James.Schoening@mail1.monmouth.army.mil>
> To: "'Robert E. Kent'" <rekent@ontologos.org>
> Sent: Sunday, February 25, 2001 7:26 AM
> Subject: RE: Foundation Ontology
>
>
> > Robert,
> >
> > I forget if I've already asked you this.
> >
> > Is this Foundation Ontology something the SUO WG could use as a
> > starting point to develop a completed SUO document? If so, would you
like
> > to discuss the option of submitting this as a starter document. If so,
> > let's discuss some more.
> >
> > Jim
> >
> > -----Original Message-----
> > From: Robert E. Kent [mailto:rekent@ontologos.org]
> > Sent: Tuesday, February 20, 2001 5:09 PM
> > To: SUO
> > Subject: SUO: Foundation Ontology
> >
> >
> > All,
> >
> > I am working on a Foundation Ontology. I would appreciate any comments
and
> > criticisms. This is located at:
> >
> > http://www.ontologos.org/IFF/Metalevel/Foundation%20Ontology.pdf
> >
> > Robert E. Kent
> > rekent@ontologos.org
> >