Richard, Ronald, Alex, and Jon,
I'll start with the point that Alex made:
> for me, the definition is an element of a particular theory.
I agree that every definition from Aristotle to the present is
a statement within some theory T. But you would have to add
more detail about syntax and truth conditions.
> I use the Aristotle definition: genus and differentia.
Yes. That is a method of stating those conditions in a convenient
way, but there are other equivalent ways.
> it is a combination of different definitions from different contexts.
> I define the ambiguity of a concept hierarchy as the sum of log(number
> of genera) over its concepts.
In a system of logic, you might have many equivalent definitions
in the same theory. But if that theory had two non-equivalent
definitions for the same term, it would collapse in contradiction.
If you're talking about natural languages, you get into all
the complexities of lexicography. See, for example, the article
"I don't believe in word senses" by the lexicographer Adam Kilgarriff:
> I grow nervous in Ontolog company, because I have yet to discover
> what kind of bridges you build between signs and reality...
> our research was always conducted under Popper’s Refutationist rules,
I very strongly agree with that observation. The bridges I have
been building are based on Peirce's philosophy of signs and science.
Popper is a special case of Peirce. Popper wrote his major books
before he discovered Peirce, but when he did, he admitted that
Peirce had expressed the same observations in somewhat different
terminology. For more about them, google "Peirce and Popper".
> In some early math course I learned a fourfold scheme of Primitives
> (undefined terms), Definitions, Axioms, and Inference Rules. But
> later excursions tended to run the axioms and definitions together...
Yes. The choice of which terms to designate as primitive is
often arbitrary. It's more general to say that a theory is
the deductive closure of a set of axioms -- and treat the
definitions as axioms, perhaps in a specialized syntax.
> And later still I learned correspondences between axioms and
> inference rules that blurred even that line...
That distinction can be drawn much more sharply. But it would take
a fair amount of time and effort to explain it in an email note.
> the important theme running through all these variations is...
All the issues about what is a logic and how to use various
logics have been thoroughly analyzed. For example, look at the
attached diagram dol1.jpg. (This is from the OMG standard for DOL.)
But those issues about logic should be clearly distinguished
from the issues of ontology (what assumptions about existence are
we expressing in some logic), epistemology (what can we know, how
can we know it), and philosophy of science (epistemology applied
to experimental and theoretical methods in science).