On Wednesday 31 January 2007 16:13, Michael Gruninger wrote:
> Hi Chris,
>
> Quoting Christopher Menzel <cmenzel@xxxxxxxx>:
> > On 31 Jan, at 10:44 , Pat Hayes wrote:
> > >> Hi Everyone,
> > >>
> > >> ... I do think, though, that some
> > >> measure of correction of logical constructions is probably also
> > >
> > > necessary,
> > >
> > > Amen to that. But it is very hard to see how this is to be done. I
> > > REALLY wish there were a nontrivial and useful notion of how to
> > > measure 'correctness' of an ontology. It is not enough to just say,
> > > it is correct if it "fits the facts" in some sense, since ontologies
> > > may be based on very different, possibly mutually contradictory,
> > > conceptualizations, and yet both fit the facts perfectly well.
> >
> > Yes, exactly. And even the idea that there are theory/ontology-
> > independent "facts" relative to which an ontology can be deemed
> > correct is a HIGHLY dubious notion.
>
> I'm a little puzzled by these comments.
> Although there may be no absolute notion of correctness, there are
> several possible relative notions of the correctness of an ontology. (01)
Yes, of course, but my understanding of the desired requirement in the
original post was that it had to do with absolute correctness -- that there
be a way of verifying in some sort of ontologically neutral manner that a
given ontology is objectively true, that it "corresponds to the world",
or "fits the facts". My only point was that any way of trying to demonstrate
this would already require framing "the world" or "the facts" in an
ontologically loaded way. So it seems to me that there is no useful,
non-question-begging way to establish the correctness of an ontology in this
absolute sense. (02)
> First, an ontology may be correct with respect to the intended
> interpretations of its nonlogical lexicon. The intended interpretations are
> specified as some class of structures, and the ontology is correct with
> respect to these structures
> if every one of the intended structures is a model of the axioms of the
> ontology.
> An ontology will be complete with respect to these intended structures if
> all of the models of the axioms are isomorphic (or possibly elementarily
> equivalent) to the intended structures. (03)
Yes, sure, but correctness of this sort -- the likes of which I agree can be
quite important -- is again very far from absolute correctness, right? It
guarantees only that the constraints imposed by the axioms of an ontology
narrow down the class of interpretations to those that have the right sort of
*structure*. But (as you know, of course) declaring a certain class of
interpretations to comprise your intended interpretations leaves entirely
open the question of whether you've identified the *right* class of
interpretations. For instance, suppose I could axiomatize Newtonian physics
in such a way that it is complete in your sense (though I reckon that's
impossible given that it would have to contain elementary arithmetic)
relative to a class of intended structures. Nonetheless, the theory is just
false in general; it fails when objects get too large, too small, or too
speedy. So the intended interpretations don't themselves accurately reflect
the physical world as it is. So while completeness (or relative consistency
or whatever) might suffice to show that a given theory is internally coherent
in some robust sense, it doesn't seem to me to offer anything like what the
original poster was asking for. Once again, I of course *like* the sorts of
formal notions you are talking about, and believe that the sorts of things
they can be used to demonstrate about an ontology are very important. We
just need to be clear about how much they buy us (not that you are even
suggesting they buy us more than they do). (04)
> A second approach is that an ontology may be correct with respect to some
> software system that can be considered as solving some inference problem. (05)
Of course, but again you are focussing on a perfectly good, practical notion
of correctness that has nothing to do with the sense I was addressing in my
post. I believe you are identifying notions and tests that spell out what I
had in mind (as you seem to suggest) by: (06)
> > * Logical coherence: Are the various concepts of an ontology (when
> > rigorously spelled out) consistent, both individually and jointly?
> >
> > * Empirical adequacy: Relative to some assumed collection of facts
> > -- and therefore relative to some assumed underlying O -- is a given
> > extension O' of O compatible with that collection of facts?
> >
> > * Practical applicability: Does the ontology serve its intended
> > purpose? (07)
-chris (08)
ps: My remark above about arithmetic and Newtonian physics above raises a
general question about your notion of completeness -- won't any
(axiomatizable) ontology containing just a bit of arithmetic be incomplete in
your sense, since (if consistent) it will have models that are not
elementarily equivalent to one another? (And isn't the possibility of
isomorphic models always ruled out for first-order ontologies simply by the
Loewenheim-Skolem theorem? Or am I not understanding your definition?) (09)
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