On Jan 25, 2010, at 10:52 AM, ravi sharma wrote: 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
Good question. There is no definitive answer, but one that can (and has) been given, is that to be approximately true is to be true (in the original, exact, model-theoretic sense) of an approximation, ie of a world which is in some appropriate sense 'sufficiently like' the actual world (assuming that there is a notion of a precise actual world available, of course.) The great utility of this is that it keeps the semantic theory intact (and unchanged), and simply adds to it a notion of approximation or closeness between worlds. This is what a physicist would call a 'hidden variable' description, which of course we know to be inadequate as an account of the fundamentals of QT, but it is not bad for describing approximation in the macroscopic world, based on an underlying classical physical description; and it even works for QT, most of the time. This seems to model how we often think of approximate truth, eg as in your scattering example:
, 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.
It can also be used to account for 'fuzzy-boundary' terms like "outback" or "countryside". If you start in a city center and drive out, at what point do you enter such a region? One answer is that there is indeed such a point, but nobody knows exactly where it is, and in fact it may not be possible to say exactly where it is; nevertheless it exists, and can be reasoned about, and is a genuine boundary point. While this may sound silly, it gives a robust solution to what are otherwise very difficult problems about how to characterize an 'approximate' boundary.
Pat Hayes 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.
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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Thanks.
Ravi
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