John F. Sowa wrote: (01)
>Rich, Ian, and Chris M,
>
>
>...
> (02)
>CM>> Seems pretty unlikely to me that any real world ontology is
> >> going to be interested in declaring that there exists some specific
> >> finite number of things.
>
>IB> I may have missed the context of this statement, but it seems
> > rather at odds with my experience. Singleton and doubleton classes
> > are not that unusual in ontology, and their membership is finite.
>
>Yes, the context for interpreting a sentence is another source
>of confusion. In some examples in previous notes, there was
>a discussion of a universe or domain D with a fixed, finite number
>of elements. Chris made the point that such a limit is unlikely
>in most practical applications.
>
>
This discussion arose from a technical comment modifying a claim that
two theories can not
be inconsistant unless they agree on something  the meaning of some
terms in the ontology.
The exception to such a general rule was that two theories actually CAN
be inconsistant in
such a case if one has an axiom stating that there is a certain
cardinality of things, while the
second theory has an axiom stating that the cardinality of things is
different and does not
intersect (the example had 1 vs. >1 things).. (03)
Yes, it is pretty unlikely that a real world ontology specifies the
existance of some specific number
of things. That's why this was a merely <i>technical</i> modification
of a general rule. (04)
If there is a disagreement among theories about the cardinality of
elements of some class,
e.g. Unicorn, then the theories agree on the existance and meaning of
that class. A "real
world" theory might state that the cardinality of that class is 0, while
a theory that is being
used for calculations in a video game might have a different cardinality. (05)
>But it is also true that the elements of the domain D may frequently
>contain small finite sets such as singletons or doubletons, and
>the set R of relations will be specified by sets (possibly infinite)
>whose elements are finite ntuples.
>
>Bottom line: In order to specify a standard with sufficient precision
>that all implementations are exactly compatible, we must use formal
>languages whose semantics are specified in terms of formal models.
>
> (06)
The semantics of the language(s) for expressing axioms and defining
terms need to be so
specified. But that does not mean that the semantics of the terms in
the FO standard have
to (or can) be expressible by formal models. Their semantic grounding
must be outside
the FO and the language for expressing it. (07)
Little about semantics of an ontology can be specified in terms of
formal models. Setlike
relations (subset, membership, disjointness, partitioning, ...) relating
type to type or instance
and interpredicate relations (subrelation, inverse relation, transitive
closure, ...) can be formally
defined, but this does not provide semantic grounding for the set of
terms being defined.
[I say "setlike" above because a distinction needs to be made between
Sets (whose membership
is unchanging) and classes/types whose membership changes with time and
context.] (08)
Imho, it is necessary to define the terms of a fundamental ontology by
discussion & concensus,
with axioms which are processed by formal models, although the meanings
of the terms are
not defined by the formal models. Very basic concepts such as length,
mass, and some
concepts relating to time (e.g. TimePoint + TimeLine +Second) would be
grounded in agreed
upon definitions, but axioms about them would be included in the FO.
Note that different and
conflicting theories could be expressed in the FO about such concepts 
which suggests
that subclasses of these concepts (e.g., InfinitesimalTimePoint and
QuantizedTimePoint)
would be useful, with the axioms being stated using the more specific
concepts. (09)
When there are multiple theories for some terms, there seem to be two
different ways of
handling them. One would be to make the theories into different
contexts, allowing an
external ontology to select the theory that it uses as well as the FO.
A second method
would be to create subclasses of the terms, with axioms relating to the
subclasses, instead
of the more general classes. However in such cases, the creation of
distinct contexts may
be warranted. (010)
 doug (011)
>John
>
============================================================= (012)
doug foxvog doug@xxxxxxxxxx http://ProgressiveAustin.org (013)
"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
 Dr. Martin Luther King Jr.
============================================================= (014)
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