On Tue, May 03, 2005 at 07:25:48AM -0700, Duane Nickull wrote:
> >Well, if a definition of a concept *does* make use of concepts that are
> >axiomatized in terms of concept being defined, then it is just a bad
> >definition.
> >
> Chris:
>
> This is what I wanted to explore. Look at the english dictionary - it
> uses all the words that are defined in the dictionary to define the
> words in the dictionary. A great deal of care is taken to avoid direct
> inclusion of terms in circular references however most of the words
> defined by the dictionary are probably used in definitions of other
> words. (01)
Yes, this is of course inevitable. (02)
> Is that really an ontology? (03)
It is not an ontology (though of course it could be useful in building
one). (04)
> Are there formulas that state the number of levels a word must be
> reasonably not used in a set of definitions until it is used again?
> (05)
A good question, to which I think there is a good answer. Any ontology
will have to contain a set of primitive, *undefined* terms. But we
don't want to confused "undefined" with "meaningless". Genuine
definitions are in principle eliminable; they introduce new terminology
in terms of the primitives of one's ontology, and in every context in
which they are used they could, in principle, be replaced by their
definitions. The meanings of our undefined primitives are captured, not
by definitions, but by *axioms*, sentences accepted without proof that
(when properly formulated) express the logical properties and logical
interconnections of our chosen primitives. (06)
As always, mathematics provides some clear, simple examples. Consider
basic arithmetic, which is a basic ontology of the natural numbers.
This ontology contains five primitive terms: "number", "0", "successor",
"sum", and "product". ("number" is necessary only if we are in a
context in which we are talking about other things as well, which is the
usual case in an ontology.) These terms are undefined, but are of
course intended to signify the number zero, the successor operation,
addition, and multiplication, respectively. We therefore axiomatize
these concepts so as to capture their intended properties: (07)
A1. 0 is a number.
A2. 0 is not the successor of any number.
A3. Every number has a unique successor.
A4. Distinct numbers have distinct successors.
A5. Anything true of 0 and, when true of a number n, is true of the
successor of n, is true of all numbers.
A6. For numbers n, the sum of n and 0 is n.
A7. For numbers n and m, the sum of n and the successor of m is
the successor of the sum of n and m.
A8. For numbers n, the produce of n and 0 is 0.
A9. For numbers n and m, the product of n and the successor of m is
the sum of n and the product of n and m. (08)
None of A1-A9 is a definition, but they still provide a very clear
account of the intended meanings of our primitives. (09)
> Example:
>
> 1. A Company is a military unit, typically consisting of 100-200 soldiers
>
> 2. A Battalion is an army unit usually consisting of a headquarters and
> three or more companies
>
> 3. A Division is an military unit large enough to sustain combat
>
> 4. A Regiment is a military unit, larger than a company and smaller than a
> division
>
> In the definition of Regiment #4, we have used words to explain it that
> were just defined themselves #1,3. (010)
Right, so you'd have to choose your primitive here and take one or more
of the definitions above as axioms. Seems to me the natural thing here
would be to take "military unit", "soldiers", and some number theory
(with "larger than" as a defined notion) and whatever auxiliary notions
we need (e.g., "consisting of", "combat", assumed to be axiomatized
elsewhere) as primitive. If we're willing to remove the vague terms
"typically" and "usually", #1, #2 could then be taken as definitions;
otherwise "Company" and "Battalion" can be taken to be primitives, and
#1 and #2 can be taken as (somewhat vague) axioms. Similarly, #3 will
have to be an axiom unless we want to clarify "large enough" and
"sustaining combat" precisely. As for #4, I don't see any problem with
circularity looming here, since "Regiment" was not used in any of the
preceding sentences. It can be taken as a definition if we have so
taken #1 and #2 and if we're willing to say precisely (1) how big a
company is, (2) how big a division is, and (3) how much bigger than a
company and how much smaller than a division a Regiment must be.
Otherwise it can be taken to be an axiom. (011)
> Sorry to once again be the loose cannon ;-) (012)
Fire away! :-) (013)
Chris Menzel (014)
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