Hi All, (01)
Category Theory is a good foundation to start with on these particular
issues and having had some experience with the subject and Analogical
Reasoning in particular, my *belief* is that these may hold shed light
on a possible path forward. (02)
As a starting point, for example, consider metaphor and analogy as a
means to unify between disparate ontologies --- where an ontology is
viewed as a theory over some model. In metaphor and analogy, a problem
in one field can be transformed into a problem statement that is
understandable in another field. In order to do this, however, you need
an invertible function which provides the LIFT to different algebra and
this means that the kinds of properties that you need will be found in
the algebra of toposes (since their are varieties that have pullback
recovery) which is needed to guarantee scope across a broad range of
representations (here I use the term for categories). The idea of
"understanding by changing perspective" deals with a general methodology
of understanding related to "change of basis" and states that in order
to understand an object, you just have to walk round the object to get
another view (for example, in maths you change from rectilinear to polar
coordinates to solve certain kinds of problems). This trivial insight
was mathematically defined by Nobuo Yoneda [Yoneda 1954] as the Yoneda
lemma which states that any mathematical object can be classified up to
isomorphisms by its "functor" which is the system of all "views" or
"perspectives" of the given object from all other objects of the same
category (ie. in structural form). This statement even allows one to
construct objects via their functors alone (ie. using only the abstract
definitions of inter-relatedness to recover an object into its category
or *structural environment*). Hence, it is a deep an powerful concept,
but, the mechanics of going from a relation to its implied objects to
which it applies in still non-trivial to solve. (03)
However, these notions may be useful in abstracting ur understanding
from the morass of linguistics (and its attendant ambiguities) to
mathematics, wherein we may specificy domain specifc theories under
which various models (providing the interpretations, and, contexts) can
be placed. As noted in the short note on the insight of the Yoneda
Lemma, and connecting this the Chu Spaces mentioned by Rick Murphy
(below), requires the development of theories about functional
dependency and a focus on relational and relational derivatives, like
information flow etc... In terms of a concrete contribution beyond an
email posting, I suggest that there is a nice connection between Sowa's
UF and one possible methodology using Hirst's lexical chaining, by
lifting the idea to "conceptual" chaining - that may become something we
can model in Chu Spaces perhaps more effectively than within the rubric
of the linguistically motivated lexical chaining. (04)
The challenge resides in mathematization of the semiotics to the extent
that we can take an object or term and make the type/token distinction.
Without a principled way to make the type/token distinction, we will get
inconsistencies or amgiduities. One possibility is by defining the
environment space in which the distinction is resolved (hence we create
the objects as embedded, or embeddings). (05)
On a trivial level as a concrete example, the phrase "I went to the
bank" may not mean to a financial institution, but if uttered near a
rive, would imply a geographic feature. Hence, packaging the background
environment, I conjecture here, means that we need to work with
*embeddings* as a fundamental construction in ontological research and
not only the relations. Of course, embeddings are a non-trivial subject. (06)
Any reactions welcome - perhaps more concrete examples may be needed,
and, I would be happy to so do if requested. (07)
-Arun (08)
richard.murphy@xxxxxxx wrote: (09)
>Barry, Pat & All:
>
>What Cory's explaining in terms of his modeling challenge in Semantic Core
>has been formalized in the theory of information flow. Information flow,
>or channel theory provides a formal definition of context - there called
>classification and also known as Chu Spaces - as " A classification A = <
>A, SA, |=A> consists of a set A of objects to be classified called tokens
>of A, a set SA of objects used to classify the tokens, called the types
>of A, and a binary relation |= between A and SA that tells one which
>tokens are classified as being of which types. " See Barwise and
>Seligman.
>
>Classifications are more than arbitrary, they're valuable applications of
>John Sowa's principle of modularity in the UF.
>
>Best wishes,
>
>Rick
>
>office: 202-501-9199
>cell: 202-557-1604
>
>
>
>
>"Smith, Barry" <phismith@xxxxxxxxxxx>
>Sent by: ontac-forum-bounces@xxxxxxxxxxxxxx
>11/28/2005 10:05 AM
>Please respond to
>"ONTAC-WG General Discussion" <ontac-forum@xxxxxxxxxxxxxx>
>
>
>To
>"ONTAC-WG General Discussion" <ontac-forum@xxxxxxxxxxxxxx>
>cc
>
>Subject
>RE: [ontac-forum] Neutrality Principle
>
>
>
>
>
>
>At 02:22 PM 11/28/2005, you wrote:
>
>
>>This is the way I have started to approach context; Contextual statements
>>can be made, for example, OWL-Full by allowing statements about
>>
>>
>statements.
>
>
>>Given a class of context and an instance "car" we would have statements
>>about "steering wheel". "steering wheel" and associated axioms are "in
>>
>>
>the
>
>
>>context of" "car" (none exclusively). The same relation would hold for
>>statements in the context of "Cyc" (Or some Cyc microtheory). A
>>
>>
>computation
>
>
>>done outside of the context of Cyc would then not include those
>>
>>
>statements.
>
>
>>In the problems I was facing in merging forms of expression for
>>architectures as well as for expressing the often conflicting
>>
>>
>architectures
>
>
>>them selves (and reasoning about them), context seems necessary. It also
>>seems necessary for extremely common concepts.
>>
>>
>
>For example?
>
>
>
>> It would also seem a way to
>>get around the inevitable "single truth" conflicts and arguments that
>>
>>
>arise
>
>
>>when all things are absolutely true all the time.
>>
>>
>
>My suspicion is that it is a too easy way (analogous to the
>teenager's cry "Well, it's true for ME").
>
>BS
>
>
>
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