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One approach I tried is to apply fuzzy logic, so that the
membership function goes from 0 when one is definiely in one region (say the
town) and 1 when one is in the other (the country side).
Applying a quantization function, say Q:[0..1] ---.>
{T, F} s.t. (Q(x) = T if x < t otherwise F) defines a threshold t at whch one
(arbitarily) decides the transition occurs. One might then compare the effects
of selecting different threshold values, for example to identify properties that
do not vary with t.
A variation on this is to define a three valued quantization
function R, R:[0..1] --> {in, on, out} with two thresholds t, u (and
typically, u = 1-t), s.t. (Q(x) = in if x < t, Q(x)=on t =<x < u, out
otherwise) with the interpretation of on is that it represents a thick
boundary. Again one can compare the effect of say, having
the widest possible boundary to having a line boundary. At least, it was quite
useful in computational geometry for proving theroums over the floating-point
numbers. which do not have even the basic property of being a metric
space.
S. Barker
Bristol
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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
(Dr. Ravi Sharma)
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