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Re: [ontolog-forum] orthogonal

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Ed Barkmeyer <edbark@xxxxxxxx>
Date: Tue, 04 Mar 2008 12:16:41 -0500
Message-id: <47CD83F9.20807@xxxxxxxx>
Alan Ruttenberg wrote:
> As used in the OBO Foundry, orthogonality is better first understood  
> on a term by term basis. Adding a term that is trivially redundant  
> with another one, by denoting the same thing, is the first thing to  
> avoid. Less trivially, if there is a way to logically define the new  
> terms in terms of existing ones, then not doing so leads to a  
> situation where two users might denote the same thing in two  
> unconnected ways: Using the new term, or using the compound of  
> existing terms. This is also to be avoided, if possible.
> The reason to avoid these situations is that one typically uses an  
> ontology to mediate queries and it is desirable to have any query  
> return all relevant answers. Having two ways to say the same thing  
> means that the user needs to know both ways to ask the question, and  
> this puts a higher burden on learning and using the ontology.
> Generalizing to orthogonality between ontologies, we'd understand two  
> ontologies as being orthogonal if no term in one is orthogonal to the  
> other ontology, in the senses above.    (01)

With all due respect, I don't see anything above that looks like a 
definition of "term is orthogonal to ontology".    (02)

It appears that the intended definition may be something like:
A term T in a given ontology O is said to be 'orthogonal to' another 
ontology P iff:
  (a) T is not a synonym for any term in P, and
  (b) T does not have a definition in the terms of P    (03)

And then two ontologies O and P are said to be 'orthogonal' iff every 
term in O is orthogonal to P and every term in P is orthogonal to O.    (04)

Is that what you mean?    (05)

If so, I would first observe that there is an issue of the 
expressiveness of the logic languages used to formulate the ontologies O 
and P.  If the languages are different, and their expressiveness is 
different, does "have a definition in the terms of P" mean a definition 
in the language of P or the language of O?  It may be that a term T from 
O "does not have a definition in terms of" P only because the language 
used to formulate P is not sufficiently expressive, rather than because 
the relevant underlying concepts are not all present in P.    (06)

And there is the question of what kind of expertise is required to 
determine that T "does not have a definition in P".  Much of physics, 
and most of category theory, is built on the observation that some 
cherished primitive concept T actually has a definition in terms of 
another ontology (theory); it just took a special insight to realize it.    (07)

Going further in this vein, "orthogonal" in this sense doesn't mean 
"unrelated".  It just means that it would require a third theory to 
relate them.  But that third theory might need to add only a single 
concept to enable the unifying definitions.  Or it may need to introduce 
a formal underlayment to enable the nominally "primitive" terms of both 
ontologies to be "grounded" in more fundamental concepts.  If you are a 
believer in "reference upper ontologies" (e.g., a Cyc-otic or a SUMO 
wrestler) you might say that "orthogonality" in this sense requires both 
of the ontologies O and P to be "inadequately grounded".    (08)

If you weaken the definition of orthogonality by requiring only that the 
"primitive" terms of O and P are orthogonal, you might then characterize 
as "orthogonal" two separate axiom/postulate systems that produce 
different consistent theories to describe some of the same complex 
concepts and prove some of the same properties.  In effect, where the 
worlds they describe overlap, they embody different views of the 
intersection.  (This is the current problem in many of the medical and 
bioscience ontologies.)    (09)

Finally, I am trying to guess what the *goal* of "orthogonality" might 
be.  Being sure that two ontologies have no apparent commonality might 
seem to be attractive in forming the "collected body of all knowledge", 
but it also seems to defeat the purpose, by ensuring that the knowledge 
they contain is "ungrounded".  If we invent the ontological underlayment 
that identifies the boxes to put them in, can we be sure that the boxes 
are still disjoint?  It may be nice that the ontology for cheese is 
'orthogonal' to the ontology for chalk, but where does that take us?    (010)

-Ed    (011)

Edward J. Barkmeyer                        Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263                Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263                FAX: +1 301-975-4694    (012)

"The opinions expressed above do not reflect consensus of NIST,
  and have not been reviewed by any Government authority."    (013)

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