Hi all, (01)
to be exact, the graph of all subsets forms a lattice only in set
theories that incorporate the empty set. If the empty set is not
included, the graph of all subsets is an irregular acyclic graph,
without the empty set in the bottom. The same goes when discrete
mereology is used as the model. (02)
Lainaus "John F. Sowa" <sowa@xxxxxxxxxxx>: (04)
> Pat, Bene, and Alan,
> This thread shows why trees are inadequate for representing hierarchies
> of ontological categories. Aristotle's syllogisms support lattices.
> Leibniz's Universal Characteristic, based on his Ars Combinatoria,
> supports lattices. The method of Formal Concept Analysis (FCA),
> which is widely used to verify that OWL definitions are consistent,
> supports lattices.
> If you use lattices and the FCA software (or other related techniques),
> you automatically solve the problems that normalization was designed
> to address. That includes the issues discussed in the cited paper:
> PH> ... the semantic model underlying OWL-DL is very straightforward.
> > So, ask how your vision can be re-stated in terms of classes
> > considered as simple sets.
> If you do that, you run into the same issues: For any set, the
> set of all subsets forms a lattice. If you have a formalism that
> can only support trees, you can't represent all possible relations
> among sets. The normalization methods that have been proposed are
> techniques for dealing with such issues.
> My recommendation is to use a system that supports full lattices
> instead of just trees. For further info, check Google for
> the terms "FCA OWL" -- but you might add the word 'concept'
> to the search list to avoid extraneous hits.
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