the category of concept lattices as the lattice of all theories (was: Re: SUO: Re: Ballot Comment - 3D versus 4D.)
----- Original Message -----
From: "John F. Sowa" <sowa@bestweb.net>
To: "Chris Partridge" <chris_partridge@csi.com>
Cc: "Adam Pease" <apease@ks.teknowledge.com>;
<standard-upper-ontology@ieee.org>; "West, Matthew R SITI-GREA-UK"
<Matthew.R.West@is.shell.com>; "'pat hayes'" <phayes@ai.uwf.edu>
Sent: Monday, August 27, 2001 4:42 PM
Subject: Re: SUO: Re: Ballot Comment - 3D versus 4D.
> I believe that there are strong arguments for both sides (and maybe
> there are even more than just 2 options on this and many related
> issues). The lattice of all theories very nicely accommodates
> all of these views; it can show exactly what axioms are common
> to both, and what axioms are contradictory.
>
> All the effort spent in arguing over these issues could have been
> much more profitably spent in making a clean division of the
> axioms for both approaches and giving developers a choice.
[technical comments]
I have proposed, starting in the message
[http://grouper.ieee.org/groups/suo/email/msg03839.html] and continuing in
the discussion on slides #5 and #6 of my PowerPoint presentation at the SUO
workshop [http://reliant.teknowledge.com/IJCAI01/Kent.ppt], that the
*category of concept lattices* be used in the SUO as a representation for
John Sowa's "lattice of all theories." The formalism is now being
incorporated in the IFF Core (sub)Ontology, and will be released in version
2.0 of the IFF Foundation Ontology that I will post at the SUO site early
next month.
I wanted to make a few comments here as preview. The objects in this
category are large concept lattices, and each concept lattice has an
underlying large classification as its base and on which it is built. By
large I mean that the collections of instances, types and formal concepts
are classes, and not just sets. Thus, this category can accommodate the
truth concept lattice for a 1st-order language, whose instance collection is
the class of all models for that language, and whose formal concepts have as
their intents the (closed) ontologies for that language.
The morphisms in this category are called concept morphisms, and each
concept morphism has an underlying infomorphism as its base and on which it
is built. In addition, each concept morphism has a pair of adjoint monotonic
functions mapping between the source and target concept lattices. Being
adjoint, one function preserves the joins and the other function preserves
the meets of formal concepts. In terms of the truth concept lattices, they
preserve the joins and meets of ontologies.
The essential property of the *category of concept lattices* is that it is
categorically equivalent to the underlying *category of classifications and
infomorphisms*, which is complete/cocomplete. So the *category of concept
lattices* is also complete/cocomplete. This means that we can construct both
limits and colimits. But it is probably the colimits that are of most
interest here.
We can think of the truth concept lattices as being at the start of a build
process. By taking coproducts (sums) and coequalizers (quotients), we can
build more interesting concept lattices. However, this may involve the
definition of "interesting" infomorphisms underlying the concept morphisms
that link concept lattices. Formalizing and axiomatizing the methodology
that went into the "merge" construction of the SUMO might help define such
"interesting" infomorphisms.
Robert E. Kent
rekent@ontologos.org