Re: Fuzzy vs. Probability: Just give it to me in plain English

S. F. Thomas (sthomas@decan.com)
Mon, 20 May 1996 12:29:28 +0200


Teeni Moddy (moddy@admonit.weizmann.ac.il) wrote:
: Things are not as simple as that. You might devise an experiment to find if
: Bill Clinton is 'tall' - just ask everyone if he is, find that 28% of
: American citizens think so, and name him '0.28 tall'.

If an eyewitness were to testify in court that the perpetrator
was ".28 tall", I think the jury would be entitled to wonder
whether this eyewitness had just alighted from another planet.
So, I would disagree with the idea of bringing a metalanguage
construct (the 0.28 bit) *into* the object language, at least
not until the robots take over, when I hope no longer to be
around. Apart from that reservation, the very natural
significance of the result of your survey would be that the
height value for which Mr. Clinton stands as exemplar could
be assigned a grade of membership of 0.28 in the fuzzy set
corresponding to the term "tall". I see nothing wrong with
such a conception, simple though it may be. If you have
another way to operationalize the concept of "grade of
membership" that does more than assign it a psychological
origin intrinsic to the user(?)/listener(?), I'd like to
hear it. If you insist that the notion of grade of
membership is fundamentally psychological, I'm afraid
I don't buy it, because I agree with Wittgenstein that the
meaning of a word is its use in the language, and such use
must transcend the psychology of individual users/listeners
if communication is to take place. And language *does*
succeed in facilitating communication, however imperfectly,
therefore a theory seeking to explicate the fuzziness
in natural language terms must proceed fundamentally from
the position that language-use is an observable--albeit
chance--phenomenon susceptible of objective characterization.

: You may even find the
: probability that a man who was called 'tall' by a random UCLA graduate is over
: 1.80 . I think it is meaningless.

Not meaningless, just uninteresting in the present context.

: Fuzziness is not a collection of yes/no questions. It is the fact that height
: is not a strict formula, even if you ask one person. Is a 1.76m man tall ? I
: don't know! He is taller then I am. I might call him 0.45 tall, which only
: indicates that he is shorter than 0.57m tall man.
: The thing is, some questions are not yes/no questions. If I want to work with
: them, I need fuzzy.

I think it is fruitless to attempt to say what fuzziness is
or is not without first grounding the discussion in the
empirical reality that it presumably attempts to model
or explicate. Yes, one can develop an uninterpreted mathematical
theory that is called "fuzzy logic" or some such, but the
issue of this thread is not the uninterpreted theory, rather
precisely the interpretational question what is and is not
fuzzy. To my way of thinking, the place to start is with
the imprecision of terms in natural language, and ask how does
such imprecision come about, and how may it be explicated.
I assert that the imprecision comes about for no other reason
than that the usage convention is not perfect. Speaker A
may describe Mr. Clinton as "tall", and Speaker B may say
not, both quite properly within the language convention.
If one samples the population, this would very quickly be
borne out, as I think you agree. But
leave Mr. Clinton aside, and consider now an unknown person.
If such a person is described as "tall", and one wants to
infer from this description what the height of this unknown
person could be, one finds oneself in a set-up exactly
analogous to that of statistical inference, in which, as
we know, probabilistic sample data give rise to uncertain
hypotheses over probability-model-or-parameter space. Here
we have the same. Probabilistic yes/no data as to actual
patterns of usage with respect to the term "tall" give
rise to uncertain hypotheses over the space of height
values for the unknown person. There is thus now a
notion of a "semantic likelihood" function induced by
the imperfect usage convention. I now argue that the notion
of the semantic likelihood explicates the imprecision in
the term "tall" -- it can be mapped, objectively -- as
it provides a basis for inferring what the relative
possibilities might be for the range of heights consistent
with the description. The notion of the membership function
of a fuzzy set sets out to achieve the same thing.
Therefore I identify one with the other, and voila',
I have my notion of what fuzzy is. No recourse to undefined
psychological concepts is necessary.

Now, this conception is more general than might
first be supposed. If the listener chooses to make
inference after first allowing for the speaker's
height, or for some sub-population to which the
speaker belongs that might affect his usage pattern,
that can readily be incorporated. For example,
basketball players may use the term differently
from jockeys, in which case separate calibration
exercises could conceptually be done for whatever
sub-population is deemed relevant. Even a single
individual--the ultimate sub-population--may conceptually
be subject to a calibration experiment seeking to map
his usage patterns.

It might be objected that an individual speaker
may as well be asked directly the question what
degree of membership he would assign to any given
exemplar of height value in the term "tall", rather
than asking him, yes or no, would he use the term
to describe that exemplar. I choose not to do
so, because utterances do not come in half-measures;
either they are made, or not made. If a metalanguage
construct is forced into the object language, and
we have a witness, say, in court, describing
a perpetrator as "tall (0.28)", then "tall (0.28)"
now becomes the whole utterance, and the jury
still has to figure out what exactly "tall (0.28)"
might mean, given that the usage patterns will
continue to be imperfect even after the attempted
precisiation. Neither does an attempted fuzzification
help. The hedged "more or less tall" also
would be a whole utterance, dilating the range
of hypotheses consistent with the description,
without affecting the principle that the fuzziness
derives from imperfect usage, and imperfect usage
is a matter of whole utterances, not utterances
to a degree.

: Probability always assume that the true answer exists somewhere, we just
: don't know it. This was the example of the land/sea problem: What does it mean
: to have a region which is 0.3 land ? that it is 70% of its area is sea ? or
: that 30% of the reports you got about it said it was 'land' ? Or the region
: is flooded 70% of the year ? or maybe it is just a big swamp ?

: I think any of these interpretations can be translated to some probability
: problem, but each of them is deferent. It depends on what you ask for, and
: anything you do with the datum depends on what you think it means. The only
: sure thing is that this are is 'less' land than a 0.32 land region.

Conceptually that goes without saying, I agree.
But the land/sea question cannot elucidate the conceptual
issue we have been addressing until it is itself
better defined. As I said in my earlier post, I
doubt whether a single pixel -- which I imagine as having
characteristics of color-mix and brightness, and not
much else -- may be characterized as land or sea.
The land/sea determination requires
standing back from the single pixel, and looking
also at the others surrounding it. It's a matter
of texture and pattern recognition. This
requires a model, from which will flow the notions
of uncertainty appropriate to the problem.
For example, suppose pattern recognition rules
look something as follows:

ALL type-A patterns in the vicinity of subject pixel
represent a region of LAND (1)

ALL type-B patterns in the vicinity of subject pixel
represent a region of SEA (2)

MOST type-C patterns in the vicinity of subject pixel
represent a region of LAND (3)

MOST type-D patterns in the vicinity of subject pixel
represent a region of SEA (4)

Probabilistic and fuzzy elements are mixed.
Realistically, I would expect that patterns could
only be imperfectly identified, and so would in
general be fuzzy, suggesting the implausibility
of rules like (1) and (2) containing ALL as a
quantifier. OTOH, wide stretches of land and
wide stretches of sea may be perfectly and easily
identifiable. Not being a subject-matter
expert, I cannot say. Rules like (3) and (4)
appear more achievable, with fuzziness (MOST)
appearing in the quantification, as well as in the
patterns, while the labels "LAND" and "SEA"
may in general be fuzzy in this context, thinking
of "wetlands" as an in-between variant that is
part LAND, part SEA, and of "coastal shoreline"
being part SEA, part LAND. But if we stick
with just the two labels LAND and SEA, it is
nevertheless clear that the fuzzy quantifier
MOST will ensure that in addition to the
fuzzy uncertainty in the consequent LAND, there
may be probabilistic uncertainty also, since
the rule will be wrong SOME of the time.
Consider a different example:

MOST RICH men are HAPPY (*)

Notwithstanding the fuzziness of the terms
RICH and HAPPY, we may nevertheless find, according
to this law, SOME men who are RICH yet
NOT HAPPY. In extension, over the population
of men, there is a statistical or probability
distribution over the (wealth X happiness)
sample space. Likewise, matching regions of an image,
with a priori pixel patterns, may MOSTly correctly
identify as LAND or SEA a pixel central to the region,
but equally may SOMEtimes be wrong. This is a mixture of
probabilistic and fuzzy uncertainty, analogous
to the rich/happy example above.

But enough. I'm no expert on imaging. I wish
only to point out that giving thought to the
model should take precedence over any a priori
notions of which form of uncertainty,
fuzzy or probability, should apply, as though
they are fundamentally in competition.
They are not.

: You may try this problem: If region A is 30% land, and region B is 60% land,
: how about the combined region of A and B ? it yields different results for
: each interpretation.

Possibly. But I'm certainly not one to argue
the primacy of fuzzy over probability, or
vice versa, if that is your apprehension.

: -----------------
: Piglet sidled up to Pooh from behind.
: "Pooh?" he whispered.
: "Yes, Piglet?"
: "Nothing," said Piglet, taking Pooh's paw, "I just wanted to be sure
: of you." -A.A.Milne ( thanks to Jo Ann )

: -Moddy (moddy@wisdom.weizmann.ac.il)

Regards,
S. F. Thomas

"No, I have never seen Mr.
Elton," she replied..."is he--is he
a tall man?"
"Who shall answer that question?"
cried Emma.
"My father would say 'Yes,'
Mr. Bentley, "No," and Miss Bates and I
that he is just the happy medium."
--Jane Austen, Emma.