*Tom
> [...]
> > Bernard, that is very astute of you! You are suggesting, are
> > you not, that there should be one "master" strict hierarchy
> > association type (designated by a PSI)? I could support such
> > a thing. I think it would be desirable to provide a partial-
> > ordering-only association type as well. Would you agree?
> > Any suggestions for a name for a virtual hierarchy association type?
>
> Tom,
>
> I certainly believe Bernard is more qualified to answer this
> than I (his mathematical chops are better than mine), but I
> think you may be misinterpreting "partial", which isn't the
> converse of "strict".
This is not so much of a qualification, but let's go for it.
- "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").
> 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.
> 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 :))
>
> Tautologies
> ________/ | \________________
> / | \ \
> / | \ \
> Antisymmetry | \ Symmetry
> \ | \ /
> \ Transitivity Reflexivity /
> \ / \ | /
> \ / \ | /
> \ / \ | /
> PartialOrdering Equivalence
> / \
> / \
> Trees Lattices ______________
> \ / | \ \
> \ / | \ \
> LinearOrdering Theories Types Collections
> / \ / \
> / \ / \
> Sequences Numbers Sets Mereology
> / | \
> / | \
> Discrete Lumpy Continuous
>
>
> 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.
Cheers
Bernard
Bernard Vatant
Senior Consultant
Knowledge Engineering
Mondeca - www.mondeca.com
bernard.vatant-eA+GF8qqnh1BDgjK7y7TUQ@xxxxxxxxxxxxxxxx
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