Hi Chris, (01)
Quoting Chris Menzel <cmenzel@xxxxxxxx>: (02)
> > 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.
> 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. (03)
Sorry, I saw the new thread title without looking at it's genealogy. (04)
> > 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.
> 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. (05)
My standard analogy is with differential equations.
For a given differential equation, mathematical techniques
can guarantee that a particular function is a correct solution.
The question of whether or not a particular differential equation
was the correct one to represent a particular physical problem
is a question for the use. A linear differential equation is not
the right one to use to represent heat diffusion, and a second-order
partial differential equation is not the right one to represent
the behaviour of a spring. Solutions to the differential equation
are given some physical interpretation and used to make predictions
about some physical system; if the predictions are wrong, the original
differential equations are wrong. (06)
Guaranteeing that the models of the ontology are intended interpretations
is analogous to guaranteeing that a particular solution for the
differential equation are correct. (07)
The *right* class of interpretations is analogous to the *right*
differential equation to represent the physical system. (08)
These are two distinct problems. (09)
> 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?)
I did generalize the completeness condition from isomorphism to
elementary equivalence, so the L-S theorem doesn't cause concern.
This also means that an ontology can be complete even though the
class of structures are not first-order definable.
For example, the class of connected linear orderings is not first-order
definable, but you can provide a theory whose models are
elementarily equivalent to connected linear orderings. (011)
However, the more serious problem is of course with an ontology
that has unintended models that are not elementarily equivalent
to each other.
I agree that any such ontology would not be not complete. (012)
On the other hand, we need to be careful when we say "an ontology containing
just a bit of arithmetic". For example, it is tempting to represent "age" as a
natural number satisfying the axioms of Peano Arithmetic, but we
should resist this temptation unless absolutely necessary.
In an ontology with the concept of age, we would be comparing
ages of things and perhaps using addition to say one
thing is so much older than another, but we would not need to multiply ages.
In such a case, we would at most need
Pressburger arithmetic, which is complete.
Now you might say that you need the ontology to represent statements
like "Alice's age is the product of half of Bob's age and twice Carol's age",
and in that case your intended model might well indeed be the standard model
of PA, but I still think that such statements are so artificial that they would
to be captured. (013)
The class of intended structures should be as weak as possible while still
the inuitions about the concepts. (014)
- michael (015)
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