On Sun, Mar 12, 2006 at 08:57:04PM -0800, John Sowa wrote:
> 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. (01)
I sort of figured you did! (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).
>
> Actually, the partial ordering is not subset, but implication
> or entailment (which are equivalent for FOL theories). (03)
I was taking theories to be closed under implication/entailment. Is not
subset the arc relation in the lattice under that understanding? (04)
> Most subsets and supersets of a theory are not theories, and the union
> of two theories is usually not a theory. (05)
Of course. (06)
> 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. (07)
That's interesting. I'll have to take a closer look at your definition. (08)
> 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.
>
> I agree that relative interpretability is important. But who uses the
> word "subsumption" for it? (09)
Probbly no one; but -- ignoring the fact that the term is deeply
entrenched in AI/KE literature -- conceptually speaking, "subsumption"
is surely a reasonable term for relative interpretability; surely there
is a reasonable sense in which, say, ZF subsumes Peano Arithmetic. (010)
> The word I use is "analogy", which I define as a renaming of one or
> more entities (usually predicates) of a theory. (011)
Well, I think I'd quibble with you on that, but we have more important
things to do than quibble -- not that that has stopped us in the past. :-) (012)
> 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.
>
> 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. (013)
I can see that it would, esp in light of your comments above. My main
point is that we need to get people clear about the fact that it is
of (perhaps significant) heuristic/pedagogical value only. (014)
-chris (015)
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