IEEE Systems, Man and Cybernetics Magazine - July 2018 - 52

Remembering Lotfi Zadeh (continued from page 51)
theory, does not permit; and the third through the non-

The sixth concept, formulated by Prof. Zadeh many years

realistic precision imposed on questions and answers in

after 1965, once again clearly entailed entering into the

opinion polls. All of them were, actually, related with certain

domain of representing plain language, where "x is P" is

aspects of plain language.

true does not have the same meaning as "probable" or

ET: After his 1965 paper "Fuzzy Sets," 53 years later, there are

"possible." These constrained statements are submitted to

six concepts that undoubtedly deserve to be linked with Prof.

be computed by means of both the membership function

Zadeh's name [S14]. The first of these is the concept of fuzzy

mP and the measures for true, probable, possible, and so on.

sets as a nebulous entity giving extension in a given universe

In short, Prof. Zadeh took a step further along the pathway

of discourse to an imprecise predicate and contextually

embarked on by George Boole in the mid-19th century to

represented by a specifically designed membership

represent linguistic statements, surpassing earlier attempts

function. The second is the algebra of fuzzy sets reached

such as those of Menger and Black.

through their membership functions. The third is the concept

AS: Returning to the second concept for a while, this would

of a linguistic variable. The fourth is the compositional rule of

also mean the penetration of fuzzy sets into plain language,

inference. The fifth is the principle of extension, and the sixth

where even the simplest connectives "and," "or," and "not"

is the idea of a generalized constraint. Without them, it is

are not universal, as is the case with precise words (e.g.,

difficult to imagine how fuzzy sets could have been applied

the intersection, union, and complement of sets). Rather,

to technology as successfully as it was and still is, and, even

they vary contextually. In a long statement with several

less so, how Prof. Zadeh could have come up with the ideas

occurrences of such connectives, it cannot be assumed that

in CWP [S13]. Let's say something on each one of these six

they can all be represented by the same function. In fact,

basic concepts.

in his 1965 paper, "Fuzzy Sets" [S11], Prof. Zadeh introduced

The first and the second concepts constituted the starting

several possibilities for representing these connectives, such

point for everything that followed in applying fuzzy set theory

as, e.g., and for the case of the conjunction, the functions

to both theoretical and practical problems, and they also

minimum and product, and the Lukasiewicz operation,

meant a departure from earlier attempts at the very idea of

which were subsequently generalized to open up further

a fuzzy set. For instance, it was Karl Menger who introduced

possibilities for representing imprecise statements.

the concept of a hazy set as a cloud of points with positive

ET: The third and sixth concepts help greatly in dealing

probability of belonging to a classical set, but Prof. Zadeh

with common sense reasoning expressed in plain language,

expanded on the notion (of which he was unaware in

since the concepts involved therein, such as age, are

1965) by liberating it from probability, since not all cases

often not handled by means of numerical values but by

of uncertainty appearing in plain language correspond

linguistic values, such as "young," "old," "middle-aged,"

to a probabilistically measurable event; not all linguistic

"very young," "not young," "not old," and so on, which can

uncertainties are random and, hence, the supposition of the

be constructed from a principal value (e.g., young), its

additive law is, at least, problematic. Additionally, Max Black

antonym (e.g., old), and negation (e.g., not young), and

introduced the compatibility, or profile function, to represent

combining them by using semantic hedges like "very." To

an imprecise word, but neither Menger nor Black ever

sum up, these concepts assist in common sense reasoning

introduced an algebra with the new elements, as Prof. Zadeh

by shading the taken linguistic principal values. The concept

did in his first 1965 paper. These algebras, further extended

of a linguistic variable was essential for Prof. Zadeh's proposal

with more representations of the involved connectives than

CWP in the late 1990s (i.e., the establishment of a calculus

those initially presented by Prof. Zadeh, opened up the

allowing representation and reasoning, with precise and

possibility of fuzzy sets being operative in many different

imprecise complex statements in either a plain or semi-

situations and depending on the contextual meaning of the

artificial language, and based on fuzzy logic). It constitutes

words involved.

a step further along the path of Leibniz's famous dictum,

In fact, Prof. Zadeh accomplished what John von Neumann

"Calculemus!"

called for back in the 1950s [S10], which entailed considering

The fourth and the fifth concepts are technical in the

the semantics of concepts, not only the syntactical formulas

sense of allowing two important operative problems to

of a presumed universal validity. In addition, Prof. Zadeh

be solved: obtaining logical consequences from a set of

introduced the armamentarium of mathematical analysis

imprecise premises and extending classical operations to

into these problems, as had also been called for by von

the fuzzy field, respectively. The first is essential for the problem

Neumann. It should not be forgotten that Prof. Zadeh came

that underlies fuzzy control (i.e., how to obtain the output

from the time of cybernetics, where mathematical analysis
played an important role.

52

IEEE SyStEmS, man, & CybErnEtICS magazInE July 2 0 1 8

(continued on page 53)



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