Dear Jason:
Thank you for the comment. However, your perception of the
relationship between standard probability theory, call it PT, and fuzzy
logic, FL, is in need of discussion.
The principal difference between PT and FL is this: In PT, only
certainty is a matter of degree. In FL, everything, including certainty,
is--or is allowed to be--a matter of degree. However, PT can be
generalized to perception-based probability theory, PTp. (See "Toward a
Perception-Based Theory of Probabilistic Reasoning with Imprecise
Probabilities," Journal of Statistical Planning and Inference, Vol. 105,
233-264, 2002, Downloadable at:
http://www-bisc.cs.berkeley.edu/BISCProgram/default.htm under
Sponsors/Project Titles.) In PTp, as in FL, everything is--or is allowed
to be--a matter of degree. This includes such basic concepts as
randomness, independence, stationarity, etc. In PTp, as in FL, bivalence
is abandoned. Let me be more specific.
A concept which has a position of centrality in human
cognition is that of partiality. We have partial knowledge, partial
understanding, partial belief, partial doubt, partial solution, partial
ability, partial control, partial membership, etc. Among the many kinds
of partiality, there are three that stand out in importance: partiality
of truth (verity), partiality of likelihood (probability) and partiality
of possibility. What is not realized to the extent that it should, is
that these concepts are distinct. To clarify, let me add a few examples
to those given in earlier messages. First, let me focus on the
difference between partiality of truth and partiality of likelihood.
I know Robert very well. Someone is asking me: On the scale
from 0 to 1, how would you rate Robert's honesty? My answer may be, say,
about 0.9. There is no randomness and no uncertainty. Thus, "about 0.9"
may be interpreted as my perception of the truth value of the
proposition "Robert is honest," or, equivalently, as my perception of
the grade of membership of Robert in the fuzzy set of honest men. My
perception is subjective, context-dependent and imprecise. This, in a
nutshell, is how humans assess degrees of truth. We have been doing this
for years when we filled out survey forms with questions such as: How
would you rate quality of service: excellent, good, fair, poor? What
should be stressed is that there is nothing that is probabilistic in
these examples. The same applies to Peter's questions: Is a wheelchair a
vehicle? Is a motorized wheelchair a motorized vehicle? What I said
above relates to Paul's question about partial truth.
Now let us consider another scenario. I am acquainted with
Robert but do not know him well. Someone is asking me: On the scale from
0 to 1, how would you assess the likelihood that Robert is honest? My
answer may be: about 0.9. In this instance, "about 0.9" is my subjective
probability that Robert is honest, with the understanding that honesty
is treated as a bivalent attribute. For humans, assessment of truth is
intrinsically much easier than assessment of likelihood. Note that
likelihood and certainty are closely related but are not identical. Both
truth (verity) and probability are subjective. The widely accepted
thesis that subjective probabilities are definable via betting behavior,
is indefensible because betting behavior is influenced by many factors
other than probability.
What about partiality of possibility? Here is a simple
example drawn for my 1978 paper, "Fuzzy Sets as a Basis for a Theory of
Possibility," (Fuzzy Sets and Systems, 1, 3-28). (For an up-to-date in
depth discussion see the special issue of Artificial Intelligence on,
"Fuzzy Set and Possibility Theory-Based Methods in Artificial
Intelligence" Elsevier, Vol. 148, Issues 1-2, Pages 1-424, 2003, edited
by: D. Dubois and H. Prade.) What is the possibility that Hans may eat n
eggs for breakfast? For n=5, say, it may be about 0.3, on some scale of
ease of attainment, and for n=1, it will be 1. Note that there is
nothing that is probabilistic. On the other hand, the probability that
Hans may have one egg for breakfast may be about 0.1, and the
probability that he may have more than two eggs for breakfast is zero.
In bivalent logic, in modal logic and, more generally, in mathematics,
possibility is bivalent. Thus, when I stipulate that a variable, X,
takes values in a set, A, what I am defining is the set of possible
values of X, or equivalently, its bivalent possibility distribution.
More generally, possibility may be epistemic, that is, defined by
constraints induced by knowledge. As an illustration, the proposition X
is A, where A is a fuzzy set, equates A to the possibility distribution
of X.
It is of some help to view partiality as a dimension. In
this sense, PT is one dimensional. FL is three dimensional. Natural
languages are three dimensional in the sense that, in general,
propositions drawn from a natural language, NL, involve partial truth
and/or partial likelihood and/or partial possibility. The mismatch
between dimensionalities of PT and NL is the reason why there are no
means in PT for understanding propositions drawn from NL.
Dear Paul:
Thank you for the comment. You refer to solutions to my test
problems. Actually, no acceptable solutions have been put forward. There
is a basic reason why my problems cannot be addressed through the use of
PT. What is absent in PT is a means of counting the number of elements
in a fuzzy set, that is, the concept of cardinality of a fuzzy set.
Suppose that I am in a room and am asked, "How many tall men are in this
room?" with the understanding that tallness is a matter of degree. Note
that in this context there is no randomness and no uncertainty. Can you
point to a discussion of this basic issue in a standard text on
probability theory? Without addressing the issue of cardinality, you
cannot define the meaning of propositions of the form Q A's are B's,
e.g., most Swedes are tall.
Manipulation of partial truths, partial likelihoods and
partial possibilities calls for different calculi and different
conceptual structures. To try to fit everything into the conceptual
structure of probability theory is unnatural and counterproductive.
Simple problems in one realm become complex problems in another realm.
It is a little like trying to eat soup with a fork. Try solving a system
of linear equations in which coefficients are random variables.
In perception-based probability theory, PTp, a default
assumption is that information is granular and is defined by what are
called generalized constraints. Special cases of generalized constraints
are veristic constraints, probabilistic constraints and possibilistic
constraints. Furthermore, a wide variety of combinations of these
constraints are allowed. In this setting, probabilistic constraints,
which are the province of standard probability theory, are a special case.
The much more general framework of PTp is needed to deal
with perception-based information described in a natural language. This
is what my test problems are intended to demonstrate. Here is an
additional example. I am in a hotel and see a sign saying, "Checkout
time is 1 pm." What does it mean? It is not possible to come up with a
realistic answer to this question if you choose to stay within the
conceptual structure of PT.
Viewed in a historical perspective, it is amazing that the
issue of partiality of truth--in the context of probability theory--has
not been raised--at least to my knowledge--long before. What we see is
that, in large measure, practitioners of probability theory are
satisfied with the status quo. But we have to remember that scientific
progress is driven by a questioning of dogmas, traditions and
conventional wisdom.
Regards to all,
Lotfi
P.S. Personal note. Sometimes I get the impression that some
participants in our discussions view me as an opponent of probability
theory. In reality, probability theory has always played, and is
continuing to play, a major role in my work. My first published paper
was entitled, "Probability Criterion for the Design of Servomechanisms."
(Journal of Applied Physics, 1949.) A paper published a year later was
entitled, "An Extension of Wiener's Theory of Prediction," Journal of
Applied Physics, 1950. Many others followed. A recent paper is, "Toward
a Perception-Based Theory of Probabilistic Reasoning with Imprecise
Probabilities," (Journal of Statistical Planning and Inference, 2002.) A
note scheduled for publication in the Journal of the American
Statistical Association is entitled, "Probability Theory and Fuzzy
Logic--A Radical View."
In summary, I am not an opponent of probability theory, nor
do I view fuzzy logic as an alternative to probability theory. What I do
see--and what is unrecognized and denied--is that standard probability
theory, PT, has fundamental limitations which are rooted in the failure
of PT to address the issues of partiality of truth and partiality of
possibility. These limitations can be removed by generalization of PT,
leading to perception-based probability theory, PTp.
A realm in which partiality of truth is ubiquitous--and yet
not a center of attention--is that of law and legal reasoning. Do any
approaches to formalization of legal reasoning address the issue of
partiality of truth?
- --
Lotfi A. Zadeh
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
[EMAIL PROTECTED]
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
BISC Homepage URLs:
URL: http://www-bisc.cs.berkeley/
URL: http://zadeh.cs.berkeley.edu/
