>Len, Paola, and Ron,
>Marvin Minsky made the point that if you only understand a subject
>from one point of view, you don't really understand it.
Einstein said it much earlier and better:
"We can't solve problems by using the same kind of thinking we used when we
created them." (03)
>There is no such thing as a one-size-fits-all textbook, tutorial,
>lecture, or course. Each one has a different point of view, and
>it's best to read several so that you can see the interconnections.
I agree. But to have a interdisciplinary discussion the interconnections are
the most important basis and have to be understood by all parties. May be you
can recommend a link that stresses interconnections, but I don't know of any.
In a meantime referring to one-sided definitions can add to confusion. (05)
>I recommended the tutorial by Norman Swartz because it covers
>a lot of ground in a short space from a point of view that is
>logically equivalent to, but slightly different from what is
>usually said in most of these email threads. (06)
I disagree. The reason I quoted the definitions below is to stress that
differences. The most problematic one is world "timeless" used in first
definition. Does it assume 4D prospective? Does everyone subscribe to 4D?
I am not sure. Later in the same text the author stresses fixed nature of
extension. Again, I am not sure that is acceptable assumption. (07)
>LY> For example, following explanation: "The extension of a term
> > or phrase is understood to be the timeless class of all things
> > which properly 'fall under' or are described by that phrase."
> > This may be workable for linguist, but not for computer scientists
> > or mathematicians (although I am not one of them).
> > Here is quite different definition of the same term in
> > http://en.wikipedia.org/wiki/Extension_(predicate_logic)
> > "The extension of a predicate ? a truth-valued function ? is
> > the set of tuples of values that, used as arguments, satisfy
> > the predicate. Such a set of tuples is a relation."
>The definition from the Wikipedia covers only relations. The
>definition by Swartz is more general because it covers terms
>that represent relations as well as other kinds of things
>-- including relations that might be called by other names,
>such as properties or predicates. (08)
As I said there are two separate definitions in Wikipedia, recognizing
different use of the term. However, the explanation of interconnections is
still missing. And I don't see definition by Swartz as simply more general
because of the assumptions it seems to have made. (09)
>I also recommend a definition of the distinction between
>extension and intension by the logician Alonzo Church.
>See Section 2 of his book on the lambda calculus:
>This is a very different, but highly illuminating insight
>into the same distinction. But it is more mathematical. (010)
Here is a quote from the link you gave above: (011)
"The foregoing discussion leaves it undetermined under what circumstances two
functions shall be considered the same. (012)
The most immediate and, from some points of view, the best way to settle this
question is to specify that two functions f and g are the same if they have the
same range of arguments and, for every element a that belongs to this range,
(fa) is the same as (ga). When this is done we shall say that we are dealing
with functions in extension." (013)
The difficulty I have with it is not that is more mathematical. The difficulty
is precisely in making interconnection with other two definitions. May be you
can explain it. (014)
>If you want a gentler introduction to these issues, see
>my tutorial on math and logic:
>LY> I recall several discussions on this forum that attempted to
> > connect the terms "intension", "extension", "class", "predicate"
> > etc. into coherent framework, but IMO failed to do so for the
> > very same reason - lack of interdisciplinary definition that
> > would satisfy the need of constructive discussion...
> > If anyone agrees with me I would suggest to begin with notions
> > of "intention" and "extension" as it relates to notions of "model"
> > and "theory".
>For more about those terms, see my tutorial on math & logic. (015)
For further illustration of my point here is the quote from your article above
(note that this is the first appearance of the term "extension"):
"A set specification that lists all elements explicitly is called a definition
by extension. A specification that states a property that must be true of each
element is called a definition by intension. Only finite sets can be defined by
extension. Infinite sets must always be defined by intension or by some
operations upon other infinite sets that were previously defined by intension." (016)
>PDM> ... each definition can be interpreted differently by different
> > people who place it in different context (relating to different
> > axioms) ...
>That may be true of some words. But for the terms 'intension' and
>'extension', all the definitions are equivalent. They are different
>ways of expressing the same distinction, and it is important to
>realize that the distinction is identical in each case.
>For other words used in ontology, there was a series of workshops
>during the late 1990s that was sponsored by the NCITS T2 committee.
>As a result of that workshop, I put together a glossary of many
>of the common terms. It was circulated at the workshops, and many
>people made comments and suggested additions and revisions. It
>was never officially voted and approved by the committee, but it
>is still a useful glossary. I appended it to the end of the
> Building, Sharing, and Merging Ontologies
>RW> If we are not willing to put the effort in, either nothing
> > will happen or someone else will.
I think most of us are willing to put the effort. What some of us are not
willing to do is to waste our efforts. Unfortunately there are two many
examples of that. I am not convinces that wiki is the answer to this concern. (018)
>If you want to start with the glossary in that article, you
>have my permission to use it. If anyone is interested in the
>discussions of the old X3T2 committee, their archive can be
>found on the Stanford web site:
>Reading these old archives brings to mind the French proverb,
>"Plus ça change, plus c'est la même chose." The only difference
>is that we were younger then, and some of us were still hopeful.
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