Jakub Kotowski wrote:
> 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). (01)
hmm..., I see it now...trivial error..:( so, I'm sorry for confusions.
But then I'm also a puzzled by the definition of completeness now... (02)
>
> 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).
>
> 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.
>
>> 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.
>
> 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.
>
>> 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.
>
>> (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?)
>
> 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.
>
> Jakub
>
>
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