[topicmapmail] Hierarchy PSIs

[Murray Altheim]

Bernard Vatant wrote:
> *Tom
> 
>>[...]
> - "Partial ordering" is a reflexive, antissymetric and transitive relation.
> Generally, it's simply called "ordering" or "order relation", "partial"
> being added to make the point that such a relation is not necessarily
> "total".
> 
> - In a total (or linear) ordering all entities can be compared, IOW
> 
> For any (a,b)  (a < b)  or  (b < a)
> 
> Trees are defined by partial ordering, unless they are reduced to a single
> branch like bamboos, in which case the order is total.
> 
> - A "strict ordering" is derived from an ordering (partial or total) by
> striking all (a,a) couples from the relation graph. A strict ordering is
> not reflexive, so it is not an ordering at all (like in the Chinese story
> "A White Horse is not a Horse").

Bernard -- thanks. I read this stuff and understand it but don't
quite feel qualified to act as an expert, at least quite yet. :-)

>>In creating an 'ur-hierarchy', I believe
>>one must also create definitions for sets, collections,
>>classes, and the like, and define membership in terms of
>>intentionality or extensionality. Absent this people will
>>generally misinterpret the various things available to them,
>>e.g., conflating sets and collections.
> 
> Sure enough, that was my point. There are hierarchies and 
> hierarchies, and you'd better know which one you use.

Yes, I agree wholeheartedly. Because what we create is often
intended for use by others, and perhaps by others who've not
spent the time trying to understand it, I think we have the
responsibility to try to get it right. I.e., if there's any
pedagogical aspect to our work, we certainly should.

>>But I think we are onto something. Since I believe we've all
>>referenced Sowa, one look at Figure 2.14 (p.95 of KR) shows
>>a generalization hierarchy, and that PartialOrdering may be
>>the place we want to start. I'll try to reproduce a bit of
>>that diagram here:
> 
> Impressive! Unfortunately, the link between Reflexivity and 
> PartialOrdering is missing :))

I really must bow before the ASCII God Jon Awbrey, whose work
often reaches the aesthetic level of art. As for the mistake,
and just for completeness, I'll fix that:

   ~-----------------------------------------------------------------~
   |                                                                 |
   |                 Tautologies                                     |
   |         ________/  |  \________________                         |
   |        /           |   \               \                        |
   |       /            |    \               \                       |
   |   Antisymmetry     |     \          Symmetry                    |
   |       \            |      \             /                       |
   |        \   Transitivity  Reflexivity   /                        |
   |         \        /    \  /   |        /                         |
   |          \      /      \/    |       /                          |
   |           \    /       /\    |      /                           |
   |            \  /       /  \   |     /                            |
   |         PartialOrdering  Equivalence                            |
   |           /      \                                              |
   |          /        \                                             |
   |       Trees      Lattices                                       |
   |         \          /    \_______________                        |
   |          \        /      |      \       \                       |
   |           \      /       |       \       \                      |
   |      LinearOrdering  Theories  Types  Collections               |
   |        /    \               ___________/    \                   |
   |       /      \             /                 \                  |
   |  Sequences  Numbers     Sets              Mereology             |
   |      \      / \        /   \              /   |   \             |
   |       \    /   \      /     \            /    |    \            |
   |    Integers  Reals  ZF-Sets  VNGB-Sets  /     |     \           |
   |                                        /      |      \          |
   |                                   Discrete  Lumpy  Continuous   |
   |                                                                 |
   ~-----------------------------------------------------------------~
   Figure 2.14: A generalization hierarchy of theories
                from "Knowledge Representation", John Sowa (p.95)

>>This all has to do not with truth values but with theories, which
>>are (to my understanding) the basis of ontological commitments.
> 
> I'm not sure what you mean exactly here, Murray, but if you want to make
> the point that there are a variety of theoretical contexts (theories) that
> a topic map (or any other knowledge representation) can commit to, and that
> having a single PSI for "hierarchical relations" without precision of this
> context is too much fuzzy, then I agree completely.

Yes, what I'm looking for is a set of PSIs for that "ur" hierarchy
and its role types; that, and a bunch of other ones, for some of the
relations expressed in Sowa's figure. I'm particularly interested
in set and collection membership, and mereology (which is misspelled
in Sowa's diagram as "meterology", and for those unfamiliar with the
term, it's the various kinds of "part of" relations, a specialization
of the collection membership relation).

For an authoring ontology (as well as an enormous number of other
applications) mereological relations are very common and important,
and much has been written on the subject.

Murray

......................................................................
Murray Altheim                    http://kmi.open.ac.uk/people/murray/
Knowledge Media Institute
The Open University, Milton Keynes, Bucks, MK7 6AA, UK               .

    "The US Government has expanded its recall of beef after the
     discovery of its first case of "mad cow" disease. Officials
     had said the meat from the infected cow, slaughtered on
     9 December in Washington State, went to Oregon, California,
     Nevada and Washington. They have now determined that meat
     from the same facility, killed on the same day, went to
     Alaska, Hawaii, Montana, Idaho and the territory of Guam."
     http://news.bbc.co.uk/1/hi/world/americas/3352911.stm

   One cow distributed across nine states and territories. Maybe
   if we were buying meat raised and slaughtered locally, and not
   feeding meat to herbivores (to increase profit, not quality)
   this might not have happened.