Re: SUO: *Date 15 Apr 2002 -- Theory Query
Jon Awbrey wrote:
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>
> SUO WG Members,
>
> With special reference to those speculating
> about sundry "lattices of theories" (LOT's):
>
> What is your definition of a "theory"?
>
Jon --
The way I am using "theory" is in
the sense of an ontological theory, which is
a "theory of the real world" -- one which
addresses the question of which formally
expressed concepts and relations
best represent the objects and processes of the
real world for the purpose of automated reasoning.
In order to be such a "theory", an ontology
must have specific widely recognized real-world
objects and processes and events which are
**instances** of some concept in the "theory".
Different ontological theories will have some of
the specific objects (e.g. my Honda, Albert Einstein,
the Second World War) defined as instances of different
formal concepts in the ontologies. The lattice will
need to specify at which point the ontologies
diverge.
This definition specifically excludes mathematical
theories which do not purport to represent objects
or events on the real world.
For the definition you propose:
> | A '(first-order) theory' T of $L$ is a collection of sentences of $L$.
> |
I would view this as a mathematical theory, unless
some of those sentences describe real-world objects
as instances of some concepts in the theory.
Two different ontologies will be inconsistent if,
for example, they each have some specific object
(e.g. the IBM corporation) as an instance of
different concepts that are specified as disjoint.
A subsumption lattice does not necessarily have
inconsistent theories, but it may. If the theories
are not inconsistent, they may be used as a convenient
way to organize a large ontology so that users may
create by selection of theories (modules, in this
case) the smallest ontology that will suit their purposes.
Pat Cassidy
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> More acutely to the purpose:
>
> What is a theory that it should latticed be?
>
> I have asked this question many times before,
> and all I've gotten is lots of gesticulation.
>
> In particular, I am skeptical that you can even find,
> without begging the question of intercommunicability,
> any definition of a lattice of theories that remains
> invariant over the choice of a language in which the
> theories are expressed, indeed, where the finding of
> the envisioned common language of comparison is just
> another way of stating the initial problem to be met.
>
> Consider this standard definition of a theory, the only one I know,
> such as makes sense within the favored frame of first-order logics:
>
> | T is said to be 'closed' iff it is closed under the |= relation. Etc.
> |
> | Chang & Keisler, 'Model Theory', page 36.
>
> Notice that the very definition of a theory is stated
> relative to a given first-order predicate language $L$.
> Until such a common language has been established, all
> talk of lattices or other orders is just so much Babel.
>
> Jon Awbrey
>
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>
--
=============================================
Patrick Cassidy
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735 Belvidere Ave. || (908) 668-5252 (if no answer)
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internet: cassidy@micra.com
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