Hi Pat, (01)
Quoting Pat Hayes <phayes@xxxxxxx>: (02)
> Michael Gruninger <mudcat@xxxxxxxxxxxxxxx> wrote: (03)
> >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.
>
> I agree this is very nice definition of
> correctness and completeness, the straight
> model-theoretic view. Unfortunately, though, it
> doesn't seem (to me) to be very useful in
> practice, for several reasons. First, this kind
> of completeness is almost never obtainable,
> because of Goedel: you can't even get it for
> arithmetic. (04)
This is vastly overstating the case -- there is more to life than PA.
In fact, there are many theories for which we can prove completeness. (05)
In terms of real ontologies, this approach has been successfully
done with all of the core theories of the PSL Ontology, and is
almost done for the remaining definitional extensions of
that ontology. It has been done for a first-order axiomatization
of a computer vision domain for 2D object recognition.
It has been done for several ontologies in the domain of mereotopology. (06)
I'm not denying that there exist classes of structures that cannot
be axiomatized, or that any ontology containing PA is incomplete.
Nevertheless, as said in my earlier reply to Chris M., we often take the
easy route and use PA to axiomatize concepts that we could have
actually axiomatized with a weaker theory. (07)
> Second, even if we abandon
> completeness, its about as hard (harder?) to
> describe the class of intended models as it is to
> state the facts using the formal language itself. (08)
If you can't formalize your intuitions, what axioms
are you writing? Once you have written your axioms,
you still need to characterize the models of these axioms.
Once you have this characterization up to isomorphism,
you can then look through these models to see if they all
make sense (i.e., they are all intended). If they are not,
you add axioms to remove the unintended ones.
If you can't add axioms (i.e. there are continuum many
unintended models, as with PA), well, then we can give up,
and we will need to be satisfied with correctness and not completeness. (09)
> The relational structrures will look very
> different (though there are of course mappings
> between them) but they can both be 'fitted' onto
> the actual weather, one presumes, but 'fitted'
> differently.
>
> In a nutshell: reality isn't a relational
> structure. One can hallucinate a relational
> structure onto it in many different ways. Which
> of *these* is "right"? Does the question even
> make sense? (010)
I am not some wild-eyed Platonist who thinks that
there is One True structure for reality.
When it comes to ontology design and evaluation,
I am a pragmatist.
By the soundness and completeness of first-order logic,
there is a nice side-effect of showing that your ontology
axiomatizes the class of intended models -- any sentence
satisfied by all models will be derivable from the axioms.
You will have the "right" models if you can derive the
"right" consequences from the axioms.
What are the "right" consequences?
That's the question for the ontology designer. (011)
- michael (012)
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