Chris and Leo, (01)
I agree with Chris that subsumption is only one of the
important relations among any collection of theories.
In fact, I do talk about other relations, including
relative interpretability. (02)
CM> The idea — induced simply by the fact that, for any two
> theories, one is a subset of the other or not (in which case
> the theories are not on a common branch of the lattice) — is
> meant only as a helpful image (I think John himself agrees),
> not as anything implementable (not that Leo is suggesting
> otherwise). (03)
Actually, the partial ordering is not subset, but implication
or entailment (which are equivalent for FOL theories). Most
subsets and supersets of a theory are not theories, and the
union of two theories is usually not a theory. (04)
Furthermore, the possible paths through the lattice happen to be
*identical* to the AGM operators for belief revision, which has
a very large and fruitful literature. In fact, *every* method
proposed for nonmonotonic logic has been shown to be equivalent
to a method of belief revision: in effect, all known methods of
nonmonotonic reasoning correspond to a walk through the lattice
of theories. (05)
CM> The really important subsumption relation between ontologies
> is relative interpretability, i.e., whether one ontology O1 can
> be mapped into another O2 in such a way that the content of O1
> is preserved in perhaps a conservative extension of) O2, albeit
> in the terminology of O2. (06)
I agree that relative interpretability is important. But who
uses the word "subsumption" for it? The word I use is "analogy",
which I define as a renaming of one or more entities (usually
predicates) of a theory. Analogy creates "jumps" across widely
separated parts of the lattice induced by the AGM operators.
I certainly agree that it is an important addition. (07)
CM> So it strikes me that it might be best to play down the lattice
> of theories, both because it is arguably too coarse and also
> because too many folks seize upon the image and think it buys
> something practical. (08)
Obviously, it doesn't buy anything practical since the infinite
lattice is never going to be implemented by anyone. However, I
have found it a convenient pedagogical tool for talking about and
classifying the operations of combining, extending, and revising
theories. (09)
John (010)
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