Re: fuzzy logic and probability

S.F. Thomas (
Thu, 11 Jul 1996 12:21:11 +0200

Subject: Re: fuzzy logic and probability
Organization: DECAN Corp.
Keywords: fuzzy,probability

On Tue, 9 Jul 1996, Reza Langari wrote (to
with copy to me in private email):
[see below]

I thank Mr. Langari for his comment, but I must disagree.
He would seem to want to have it that Mr. Jakobs' question
is somehow a priori indecidable, and therefore it is
foolish to address it. Since Goedel, we need to take claims
of the indecidability of certain propositions seriously, but
not in this case. There is nothing a priori "foolish", or
indecidable, in saying what one means, and being clear about it,
especially when advancing a theory supposedly capable of capturing the
essentials of an observable real-world phenomenon.

I think Zadeh's insight was a brilliant one, and he deserves
all the accolades that have come his way since his seminal
paper introducing the concept of the fuzzy set. I am not one
of those chauvinistic probabilists who dismiss his contribution
as nothing new, because falling ultimately under the ambit of the
probability theory. Those who so insist have missed the point,
which lies in the new way of thinking allowed by such derivative notions as
that of the linguistic variable and approximate reasoning, which
allow a whole new, simpler, and more powerful way of looking at,
for example, problems of automatic control of complex systems.
That's breakthrough in itself, and I'm not persuaded, by those
who show that for every fuzzy control system there is an equivalent
probability-based system that can do the same thing, that it
therefore follows that Zadeh's contribution is for that reason

Equally, however, we have known at least since Gaines (1975)
that, technically at least, the fuzzy set theory may proceed
starting from an interpretation of the notion of grade of membership
as being a probability. Gaines showed that the min/max rules
in particular, among many others, could as easily be derived
starting from this probability interpretation, as without it.
Therefore, as a matter of strict formalism, it is futile to assert that
the grade of membership is not or cannot be probability.
If there is such an insistence, which there has been, and which
Gaines has ascribed to "tutorial exagerration", it has been to
protect what is in fact fresh semantics from too-quick dismissal
by those who would say, "well, it reduces to probability after all,
therefore the old (eg. Bayesian) methods are all we need, and
we can safely ignore these fuzzicists as they go about inventing
a new theory to address problems of uncertainty already
addressable using existing (eg. Bayesian) methods."

But too many have taken this tutorial exagerration as gospel,
and it has now become an article of faith among now chauvinistic
fuzzicists, no doubt in reaction to the caustic reception that
Fuzzy received from the chauvinistic probabilists. But the
insistence that the notion of grade of membership is *not* probability,
since it cannot, after Gaines, be an objection of formalism, must
be an objection of interpretation. Yet here too we find that there is
an immediate and obvious probability interpretation to the notion
of grade of membership as it relates to the application domain
of natural language semantics, which after all provided important
motivation for Zadeh's (1965) introduction of the notion of fuzzy sets
in the first place. That interpretation is the probability that
a random user of the language (presumably competent) would use
the label for which the fuzzy set in question stands as model,
to describe the point attribute value whose grade of membership
in the set is in question, for example that age 28 is describable
as "young".

Such an interpretation, rather than being limiting, as might at
first be feared, in fact is helpful in eliminating a number of
foundational difficulties from which the fuzzy set theory otherwise
suffers. Hisdal (various) has addressed some of these issues,
in addition to myself. What in addition I discovered, in
attempting to integrate the fields of fuzzy and probability,
is that a variety of foundational issues involving the *latter*
also appeared amenable to resolution when the fresh Zadehian semantics
was brought to bear. At bottom, the extended likelihood uncertainty
calculus which eluded Fisher and generations of statisticians since,
may at last, through identification with the fresh Zadehian semantics,
be strengthened sufficiently to perform the variety of inferential tasks
so far clumsily performed by classical methods, and with
questionable justification by Bayesian methods.

> Mr. Thomas raises an important issue in connection with fuzzy logic.
> He points out that:
> > You can of course claim fuzzy logic to be whatever you want it
> > to be, axiomatized in whatever way seems preferred by two or more
> > fuzzicists at the moment. However, to the extent that fuzzy logic
> > attempts to model certain aspects of natural-language semantics,
> > there is an external reality out there that exists quite independently
> > of any received theory of the moment. It is by reference to its application to
> > that empirical domain that fuzzy logic as theory, as opposed to
> > axiomatic construct, must ultimately be judged.
> However, the notion that ".... there is an external reality independent
> of any received theory of the moment." is itself a theory about that
> "external reality" and as such only constitues a metaphysical
> speculation.

The same could be said of *any* theory, but it obviously hasn't deterred
Newton or Einstein, nor prevented the successful application of their
theories in various ways. I don't see any speculation,
metaphysical or otherwise, in establishing as the grade of membership
of age 28 in "young", the proportion of competent users of the language who
would use that label to describe that age in a given context.

> Moreovoer, the idea that "... by reference to its
> application to that empirical domain that fuzzy logic... must
> ultimately be judged" hinges on the belief that empirical validation of
> formal systems is even possible [...]

Not at all. As to formalism, by which I assume Mr. Langari means
uninterpreted formalism, there is no argument, as I have
explicitly stated. A (uninterpreted) formal axiomatic system
may be judged only on the basis of its internal consistency.
The issue at hand though is one of interpretation... is there
a sense in which the grade of membership may be construed as a
probability? Once interpreted, the fuzzy set theory (of semantics)
is no longer mere formalism. It becomes material-axiomatic theory,
and it may be judged by how well its axioms and theorems accord
with the reality it purports to model.

> [...] which in turn hinges on prior belief in
> the validity of probability theory as a means of validation of
> empirical knowldge, [...]

If probability is what emerges from a process of
classifying and counting occurrences with respect to certain
attribute(s) over a population of some sort in the real world,
then such probability *is* empirical knowledge, and
the notion of "prior belief in the validity of probability
theory" really does not usefully arise. Poll the presumably
finite population of speakers, and count those concurring. The
proportion which that count bears to the total population is
the "probability" that we want. I don't see that "prior belief
in the validity of probability theory as a means of validation
of empirical knowledge" comes into play at all. If it helps,
simply think of the proportion just defined as being the grade
of membership sought. Separately, such a proportion may *also*
be construed as a probability, as primitively, if one so chooses,
as the notion of grade of membership sought to be elaborated.
Neither notion is necessarily prior to the other.

> which in turn hinges on the validity of probability
> theory itself, and so one enters an infinite regression.

I see no infinite regression in conducting a simple poll. Nor do
I see any conceptual difficulty in identifying a population *proportion*
obtained from exhaustive polling, with the "probability" of the same
result on "random" selection of an individual from the population,
as what probability, at least of the "frequency" kind, primitively

> These ideas as
> might be expected have kept many a great philosophical mind busy since
> at least the time David Hume published his Enquiry Concerning Human
> Understanding on through the heyday of logical positivism and remain,
> perhaps forever unresolved.

A good philosopher tries to say what he/she means, as best he/she can,
within the limits imposed by language. It is clear that language
is in some sense not plastic enough to follow, curve for curve,
every contour of an infinite reality. Nor is it plastic enough even always to
allow the "logical clarification of thoughts" (Wittgenstein) *about*
possible realities. But, that acknowledged, neither is it the case that
language is not adequate to the task of describing what
one means by the notion of grade of membership of a fuzzy set.

> The idea that one may successfully debate these issues is itself, seems
> to me at least, rather folly but nonetheless intellectually stimulating

I think the greater folly would rest in attempting to proceed
with the material application of a theory while leaving undefined,
or unexamined, the core foundational concept on which the theory rests.

> and I agree with Mr. Thomas that one should not necessarily ignore the
> issue or worse, condemn those who raise it.

Yet that would exactly be the result if we took your indecidability
position seriously.

> Reza Langari, Assistant Professor
> Department of Mechanical Engineering
> Texas A&M University
> College Station, TX 77843-3123
> Phone:(409)845-6918
> Fax: (409)845-3081

S. F. Thomas