Well, it depends on the level of details that the paper is meant
to address. For example, an article in a popular science magazine
about, say, quantum mechanics, will hardly have any formula;
general ideas can be made comprehensible by plain text
and intuitive illustrations. Conversely a paper in a specialized
journal devoted to quantum mechanics will be much compacter
with mathematical formalism and, at the same time, much more
precise.
But the same does not apply to fuzzy logic. Indeed, the latter is
less a _descriptive_ theory than a design method supposed to be
especially convenient to _use_. Hence too mathematical papers
unfortunately fail to address their best potential "customers":
the users (e.g., engineers).
On the other hand, one could argue that fuzzy logic is a "new
scientific discipline" which requires mathematical foundations.
However, the fervor with which it is sometimes claimed that
"fuzzy logic is an area distinct from probability theory"
remains extremely puzzling for statisticians. In particular, the
argument that "fuzzy membership functions are not to be mixed up
with probabilities" is only valid for the _frequentist_
definition of probability. By contrast, in the (subjective)
_Bayesian_ interpretation, probability may be viewed as a degree
of belief in (or a degree of relevance of) a given proposition.
This interpretation makes in practice often more sense than the
frequentist one, but surprisingly few people seem to be aware
of it. In fact a "membership function" corresponds to a
Bayesian _conditional_ probability. Yet the "ad hockery"
present in fuzzy logic (such as in the choice of the form of
the membership function, or the combination of several ones)
usually makes statisticians skeptical. Of course, there is
nothing wrong with heuristics when it reveals efficient in
practice; but it fails to _guarantee_ consistent answers,
whereas an old, solid and mathematical discipline does it already:
probability theory.
ULF>So, anyway. I suppose what I'm trying to say is:
ULF>Don't complicate things. Allow fuzzy logic to remain as
ULF>simple as it really is.
I understand the adjective "simple" of the above statement in
a positive sense, and I fully agree.
Andreas Poncet
Institute for Signal Processing and Information Theory
Swiss Federal Institute of Technology (ETH)
Zurich, Switzerland
poncet@isi.ee.ethz.ch