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Re: [ontolog-forum] Ontological correctness

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Jakub Kotowski <jakubkotowski@xxxxxxx>
Date: Thu, 01 Feb 2007 02:19:50 +0100
Message-id: <45C14036.5010303@xxxxxxx>
Hello,    (01)

Chris Menzel wrote:
> 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.
> 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)

I think that the original post didn't ask for correctness in absolute
sense but rather for some useful notion of correctness: (a quote) "I
REALLY wish there were a nontrivial and useful notion of how to measure
'correctness' of an ontology."    (03)

And I think that the two notion proposed below are very useful.    (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     (05)

It seems to me that on the contrary... that it is quite close to
absolute correctness. Because I understood the "intended interpretations
of its nonlogical lexicon" as actually the indended meaning... nothing
formal, no formal interpretation. (or at least I don't see what should
be interpretation of a *nonlogical* lexicon).    (06)

I would paraphrase this notion like "How nicely does our ontology
capture the concepts that we use to think/talk about our domain of
interest." Here I mean the human concepts - the indended meaning. So
this is a kind of philosophical correctness, isn't it? (as opposed to
the more formal second suggested notion).    (07)

Anyway I think that this is exactly what is very difficult to check
because who knows what the indended meaning is... or should be. I mean
what it is *exactly*. Everyone has an idea about the meaning but that is
usually far from being formal and actually ontology is a way of
formalizing it. So my point is that correctness can be even inherently
impossible to check in this case. Because in some cases we may not be
even sure what the intended meaning is exactly... but that may well
crystalize when trying to formalize it.    (08)

> 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.     (09)

Yes, but the problem that the theory itself may be wrong is not a
problem here, is it? Because, at least as I see it, we are trying to
check how well the ontology about Newtonian physics describes "what we
think about Newtonian physics", not the correctness of the Newtonian
theory itself. And also I think this is not the best example because
Newtonian physics is well formalized and conclusions based on it can be
("easily") verified and that is in big contrast to what we are usually
trying to capture by an ontology I think.    (010)

> 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).
> -chris
> 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?  
I think it is because of the, in my opinion, informal uses of the word
"interpretation" and "intended structures". So it doesn't talk about
equivalence of models of the axiomatization to one another but rather
about isomorphism to some external (informal?) structures.    (011)

>(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?)    (012)

Anyway I think that the most useful thing is to look for the practical
notions of correctness of an ontology. So that we have similar
theoretical tools to the ones we for relational databases - like the
normal forms. And luckily there is some ongoing research in this
direction but unfortunately more in the form of suggesting best
practices verified in real use rather than building a formal theory.    (013)

Jakub    (014)

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