Lotfi,
You brought up the problem of assessing the truth of the statement,
"Robert is German", when he is half German and half French. You argued that
probability theory could deal with uncertainty but not partiality of truth
(that is, he is German or not German, but we don't know which), while fuzzy
logic could deal with partiality of truth (i. e., the truth of "Robert is
German" is 0.5).
To correctly apply probability theory to a situation, the domain of
the random variable describing the real but unknown truth must cover all
possible scenarios. In this case, because probability theory is ``bivalent''
you assume that the variable, call it G (Robert is German), can only assume
two values, true or false. But since there are situations where it is
neither true or false, the probability of it being true (the sum of the
probabilities of all worlds in which it is true) plus the probability of it
being false would not add up to one. In this case it is necessary to assume
it has a non-binary domain, for example all the numbers from zero to 100 (0
means he is not at all German, 50 means he is half German, and 100 means
completely German). Then its distribution will satisfy the basic laws of
probability and it is possible to apply the standard calculus.
In general, partiality of truth can be addressed by a similar
method. Instead of quantifying the uncertainty over a binary variable, which
may not describe all possible states of the world, one can introduce a
variable with a non-binary domain. In this case, we assume the ``truth'' is
not necessarily yes or no, but it ranges over various possibilities. In
general, if we have a fuzzy set (say, "tall" in the Swedes problem) in which
membership is quantified by a non-binary value, the state of every potential
member can be modeled by a random variable that ranges over all the possible
degrees of membership. It is certainly true that the member has a precise
state (say, its membership is one half), so it is not a question of
partiality of truth, but rather of uncertainty. So in the Swedes example a
random variable ranging over the degrees of membership in "tall" describes
the true height of Mr. X, with the most likely degree being say 0.75, and
then the task becomes to calculate the expected degree of Mr. X's tallness.
Probability calculus can be applied as usual, and the result would be
perhaps a distribution over possible truth degrees, which can then be
translated into concrete values (say, centimeters) by a separate
interpretation of the semantics of the fuzzy set.
David Larkin
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of
Lotfi Zadeh
Sent: Monday, November 17, 2003 10:36 AM
To: [EMAIL PROTECTED]
Subject: RE:[UAI]causal_vs_functional models?
Dear Peter:
Thank you for not following Wittgenstein's advice in your
message :" That about which one cannot talk, one should remain silent
about." With characteristic humility and insight, you touch upon
issues which play basic roles in law and legal reasoning. Looking from
the outside, to clarify these issues it is necessary to differentiate
between partiality of truth and partiality of certainty. The problem
with the concept of factuality is that it may involve a mixture of the two.
A simple example which clarifies the difference is one that
I used before. If Robert is half- German and half- French, and I say
that Robert is German, then what I say is half-true, with no uncertainty
involved. On the other hand, if I am not sure whether Robert is German
or not German, then the probability that he is German may be 0.5. In
the first case, we have partial truth and no uncertainty ,while in the
second case we have partial certainty of full truth.
In the realm of law and legal reasoning, most assertions are
both partially true and partially certain.
In bivalent logic and bivalent- logic-based probability
theory, partiality of truth is not addressed. This is the reason why,
in my view, bivalent logic and bivalent- logic - based probability
theory do not provide adequate tools for formalization of legal
reasoning. In this sense, Ron Allen is right.
Probability theory has been in existence for over two
centuries. What is amazing is that in the enormous literature of
probability theory the fact that probability theory does not address
partiality of truth has not been articulated. Perhaps the reason for
benign neglect is that there was no machinery for this purpose. Fuzzy
logic provides this machinery because in fuzzy logic, truth, certainty
and everything else are, or are allowed to be, a matter of degree.
Perceptions play a pivotal role in law and reasoning. Much
of legal reasoning is perception-based. Perceptions are intrinsically
imprecise. Can standard probability theory, call it PT, operate on
perception-based information?
Following are a few simple test problems.It is understood
that imprecise terms such as most, tall, etc. require precisiation in
one form or another.
l. The tall Swedes problem: Most Swedes are tall. What is
the average height of Swedes?
2. Usually it is not very cold ,and usually it is not very
hot in Berkeley. What is the average temperature in Berkeley?
3. The Robert example: Usually Robert returns from work at
about 6 pm. What is the probability that Robert is home at about 6 :l5 pm?
4. The balls- in- box problem: A box contains about 20
black and white balls. Most are black. There are several times as many
black balls as white balls. What is the probability that a ball drawn at
random is white?
In probability community it is an article of unquestioned
faith that standard probability theory is adequate for dealing with all
problems involving uncertainty and imprecision.What is unrecognized is
that, unfortunately, this is far from being true. However, probability
theory can be generalized, making it capable of dealing with both
partiality of certainty and partiality of truth. The generalized
probability theory, which I call perception-based probability theory,
PTp, is outlined in the paper cited in my message of ll/l0/03. The
correct title is "Toward a Perception-based Theory of Probabilistic
Reasoning with Imprecise Probabilities."
Thank you for contributing so importantly to a
clarification of basic issues in the realm of law and legal reasoning.
Actually, these issues play pivotal roles not just in law and legal
reasoning but, more generally, wherever partiality of truth and
partiality of certainty are important factors, as they are in the
conception, design and utilization of intelligent systems.
Warm regards.
Sincerely,
Lotfi
- - - - --
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)
