John and Ali

 

Appreciate the explanation of model theory. Suppose we replace true and real these two words with a prefix "approximately", what opens up as additional attributes that are required to complete model theory or a model of whatever, and the knowledge seeker can quantify the adjective "approximately" for example elementary particle scattering model to approximately (10**-13 cms or 1 Fermi distance) between particles.

 

Although I am also wanting to understand model theory, what I am after are the concepts that are "irreducible must have" for a model and what are other tests such as measurements to verify that the model is reasonable. Then next step is addressing "Semantic" models that would have notion of what is more relevant, predicate ordering by importance etc.?

John - FYI the online tutorial link is broken in your Tarski paper referenced above.

 

Thanks.

Ravi

 

On Sun, Jan 24, 2010 at 10:35 PM, Ali Hashemi <ali.hashemi+ontolog@xxxxxxxxxxx> wrote:

On Sun, Jan 24, 2010 at 9:58 PM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:

Ravi,

<<snip>>

 

But people have criticized model theory because there is more to say
about the meaning of a statement than just its truth conditions:

 1. For example, the following three statements are true in
    every possible model, but the fact that they talk about
    different subjects indicates that they are not synonymous:

     a) Every cat is a cat.

     b) Every dog is a dog.

     c) Every unicorn is a unicorn.

 2. Not all true statements are equally important, but model
    theory has nothing to say about importance or relevance.

 

These objections don't imply that model theory is wrong for

what it does.  But they indicate that there is more to meaning
than just the truth conditions for a statement.

John

Just an observation on weakness #1 as identified by John above. This "weakness" also provides some concrete advantages vis-a-vis ontology alignment and mapping. 

Specifically, if one accepts the premise that much of our structured thinking (say in a formal logic) reuses the same patterns or "logical building blocks," then these model structures provide a very nice way to identify agreement / conflict between ontologies in the same domain, and/or to highlight interdisciplinary borrowing of concepts, and/or to help clarify metaphors.

By model structure, i mean the set of permissible models attained for a given theory, but decontextualized from the universe of discourse --- so in the trivial example above, the (logical) synonymity holds for the models allowed by "Every A is A" - i.e. if we strip the models from their referent to specific objects, then the sets of the models for the above theories are isomorphic.

Of course, this captures only logically synonymous formulations, and as it has been pointed out, there's more to meaning than that. But this type of synonymity is a very useful insight that can go a long way in alleviating the problems of multiple, competing ontologies.

Ali

 

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