Professor Zadeh,
With standard Bayesian techniques it is possible to reason
probabilistically about variables with non-binary domains. For example, in a
standard belief network one might find a node labeled "Temperature" which
can take on the values "Low", "Medium", and "High". A fuzzy-logic based
interpretation of the variable would amount to a simple relabeling. The node
could be renamed, "The object is hot", taking a truth value of "0", "0.5",
or "1". Mathematically, there would be no difference. You would say,
according to bivalent logic, "The object is hot" can only be true or false,
and we can only reason probabilistically about the chance that it will be
true or false, and therefore partial truth cannot be addressed. But standard
belief networks are not bivalent (i. e., the variables are not binary), but
in general multi-valent. Vague and perception-based degrees of truth, such
as the temperature example given above, are commonly used to describe the
states of variables.
David Larkin
-----Original Message-----
From: Lotfi A. Zadeh [mailto:[EMAIL PROTECTED]
Sent: Wednesday, November 19, 2003 12:30 AM
To: [EMAIL PROTECTED]
Cc: R.J.Allen<[EMAIL PROTECTED]>.P. Tillers; D.Larkin; Lotfi Zadeh
Subject: RE:[UAI]causal_vs_functional models?
Dear Professor Allen:
Thank you for your detailed and authoritative comments on
the issues under discussion.
First, a point of clarification. It appears that you view
probability theory, fuzzy logic and rough set theory as alternative
methodologies. In fact, they are complementary rather than competitive.
Thus, you can achieve more by using these methodologies in
combination,rather than in a stand-alone mode. As I have stated on a
number of occasions, the principal limitation of standard probability
theory is that it does not address the issue of partial truth -- an
issue that has a position of centrality in law and legal reasoning. Let
me illustrate my point by an example which may be more illuminating than
my "half -German" example. I witnessed an accident involving drivers A
and B. I listen to A's description of the accident to a police
officer. I ask myself the following question. On the scale from 0 to l,
to what degree is A's description true? What I might say is 0.8 or ,less
precisely, high. My answer would be the truth value of A's description.
Note that no uncertainty is involved.
In another scenario, I assume that I arrived at the scene
after the accident has happened. I listen to A's description. I am not
sure that A is telling the truth because I did not witness the accident.
What is the likelihood that A is telling the truth? This likelihood,
expressed as a number between 0 and l, say 0 .8, is my subjective
probability that A is telling the truth. Note that 0.8 in the first
case has a meaning that is altogether different from its meaning in the
second case.
More generally, we may have a mixture of partial truth and
partial certainty. For example, in the second case, I may feel that it
is likely that A's description is more or less true. Such
characterization is an instance of what I call "bimodal distribution,"
with bimodality signifying that we have a mixture of partial truth and
partial certainty. I believe that the concept of bimodality could serve
a useful purpose in formalization of legal reasoning.
In my comments regarding formalization of legal reasoning,
what I had in mind is the extensive literature on this subject in the
field of AI and legal reasoning. If you search for "formalization of
legal reasoning" using Google, you get 7570 results. Prominent among
contributors to this field are Professors Len McCarthy and Edwina
Rissland. A very informative article on the subject is "Exploring the
Limits of Formalism: AI and Legal Pedagogy," by Professor Rudolf Peritz,
New York Law School.
Much of the literature in question is based on the use of
bivalent logic, with little if any use of probability theory. The
substance of my comment is that although significant progress has been
made, what is achievable today falls short of providing a realistic
model for the kind of legal reasoning that is used in the courts of
law. To move beyond what is achievable today, it is necessary to
abandon bivalence in both logic and in probability theory. I realize,
of course, that the tradition of basing scientific theories on bivalent
logic is much too deep-seated to be abandoned without a prolonged debate
and determined resistance.
In your comment, you point out that statistics is employed
extensively in the courts of law, while rough set theory and fuzzy logic
have little or no visibility. The reason for this state of affairs is
that statistical techniques have been around for a long time and are
familiar to all. By contrast, rough set theory and fuzzy logic are
recently developed methodologies which generally are not taught in
universities at this juncture. But what is certain is that it is only a
matter of time before these and related methodologies gain wide acceptance.
Regards to all.
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)
Address:
Computer Science Division
University of California
Berkeley, CA 94720-1776
[EMAIL PROTECTED]
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433
Fax (home): (510) 526-5181
http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
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