IEEE Systems, Man and Cybernetics Magazine - July 2018 - 54
Remembering Lotfi Zadeh (continued from page 53)
References
sense, trivialized how to compare systems. Prof. Zadeh rejected
[S1] M. A. Boden, "Modelling creativity: Reply to reviewers," Artif.
the assumption that the performance of a system can be
Intell., vol. 79, no. 1, pp. 161-182, 1995.
measured by a single number and argued for a vector
[S2] S. Guadarrama, E. Renedo, and E. Trillas, "A first inquiry on
description and corresponding multiobjective optimization.
semantic-based models of And," in Accuracy and Fuzziness, R.
He relaxed the strong assumption of a total order over systems
Seising, L. Argüelles, Eds. New York: Springer, 2015, pp. 331-350.
to a mere partial order. This elegant little paper launched its
[S3] S. Ovchinnikov, "My journey to fuzziness at Berkeley," in On
own scholarly light cone of Pareto and other techniques for
Fuzziness, vol. II, R. Seising, E. Trillas, C. Moraga, S. Termini, Eds. New
multiobjective optimization.
York: Springer, 2013, pp. 503-506.
The more famous example is Prof. Zadeh's seminal formulation
[S4] R. Seising, "On fuzzy sets and the precisiation of meaning-
of fuzzy set theory in his 1965 paper, "Fuzzy Sets." The paper
An interview with Prof. Dr. Lotfi A. Zadeh," Archives Philosophy His-
boldly extended bivalent sets to multivalent sets. Prof. Zadeh's
tory Soft Computing, vol. 1, pp. 1-18, 2013.
deep insight was akin to that of Henri Lebesgue's when he
[S5] E. Trillas, "Lotfi A. Zadeh: On the man and his work," Scientia
reformulated the integral in 1902: he focused on the range of
Iranica, vol. 18, no. 3, pp. 574-579, 2011.
a function rather than its domain. The function here was a set's
[S6] E. Trillas, On the Logos: A Naïve View on Ordinary Reasoning
indicator function that states whether a given element does
and Fuzzy Logic. New York: Springer, 2017.
or does not belong to the set. Prof. Zadeh saw that, in general,
[S7] E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo, and E. Casti-
these indicator values could be any number in the continuum
ñeira, "Computing with antonyms," in Forging New Frontiers, vol.
of the unit interval, so set membership was now a matter of
II, M. Nikravesh, J. Kacprzyk, L. A. Zadeh, Eds. New York: Springer,
degree, and fuzzy sets were born.
2007, pp. 133-153.
No one who knew Prof. Zadeh or his work doubts that he
[S8] E. Trillas, P. P. Bonissone, L. Magdalena, J. Kacprzyk, Eds., Com-
would have won a Nobel Prize if they gave such a prize for
bining Experimentation and Theory: A Homage to Abe Mamdani.
the information sciences. A giant has passed.
New York: Springer, 2012.
[S9] E. Trillas and L. Eciolaza, Fuzzy Logic: An Introductory Course
Madan M. Gupta
for Engineering Students. New York: Springer, 2015.
Fuzzy logic, a new mathematical tool for the analysis and
[S10] J. von Neumann, "The general and logical theory of
management of uncertainty that arises from human cognitive
automata," in Cerebral Mechanisms in Behavior: The Hixon Sym-
processes such as thinking, reasoning, and perception. It was
posium, L. A. Jeffress, Ed. Hoboken, NJ: Wiley, 1951.
introduced in 1965 by Prof. Zadeh, "The Father of Fuzzy Logic."
[S11] L. A. Zadeh, "Fuzzy sets," Inf. Control, vol. 8, no.3, pp. 338-353,
Over the last five decades, this new mathematical logic, unlike
1965.
Boolean (binary) logic, has revolutionized relevant tools and
[S12] L. A. Zadeh, "Fuzzy logic and approximate reasoning (In
computing methods for the analysis and management of
memory of Grigori Moisil)," Synthese, vol. 30, no. 3, pp. 407-428, 1975.
cognitive uncertainty. More recently, fuzzy logical tools have
[S13] L. A. Zadeh, Computing with Words: Principal Concepts
been used to introduce virtual intelligence into the household,
and Ideas. New York: Springer, 2012.
entertainment, automobiles, and space industries. Fuzzy
[S14] L. A. Zadeh, "Fuzzy logic-A personal perspective," Fuzzy
mathematics have been extensively explored to develop a
Sets Syst., vol. 281, pp. 4-20, Dec. 2015.
new class of expert systems for potential applications in fields
such as medical diagnosis, the control of complex processes,
Bart Kosko
manufacturing processes, art, architecture, music composition,
Prof. Zadeh was a consummate intellectual of the old school.
mining and space exploration, management science, court
You could talk or argue with him about anything. No subject
of law, and stock market evaluation. In general, this innovative
was out of bounds or likely to give offense. The goal was
method of mathematics has proven to be very useful in the
always deeper insight into the matter at hand. The method
creation of a new class of intelligence-virtual intelligence-
was simply giving reasons for claims.
that can solve a category of problems where human
This helps explain why Prof. Zadeh could make so many
intelligence is intimately needed.
profound intellectual contributions in so many fields. He
The theory of fuzzy logic was introduced to me at its very
thought issues through from their foundations, and he did
early stages of inception by its founder, Prof. Zadeh, during
it alone. He usually worked and published as a sole author
the second International Federation of Automatic Control
and struck research gold this way many times.
(IFAC) Symposium held in Dubrovnik, Yugoslavia (formerly),
A good example is Prof. Zadeh's 1963 reformulation of
on 26-31 August 1968. I recall my first, pleasant meeting with
multiobjective nonlinear optimization in IEEE Transactions on
Fay and Lotfi at the breakfast table in the hotel on the first
Automatic Control. He took a fresh look at optimal control
theory and found it unduly restrictive. The theory had, in a key
54
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