“A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.”

—  Lotfi A. Zadeh

Source: 1960s, Fuzzy sets (1965), p. 338

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Lotfi A. Zadeh 18
Electrical engineer and computer scientist 1921–2017

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“When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence \underline {within} M of this set with the set of integers.”

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Set theory and the continuum hypothesis, pp. 19–20 https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA19
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Lotfi A. Zadeh photo

“A frequent source of misunderstanding has to do with the interpretation of fuzzy logic. The problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization of classical multivalued logics. But in a wider sense, fuzzy logic, FLw is almost synonymous with the theory of fuzzy sets.”

Lotfi A. Zadeh (1921–2017) Electrical engineer and computer scientist

In this context, what is important to recognize is that: (a) FL<sub>w</sub> is much broader than FL<sub>n</sub> and subsumes FL<sub>n</sub> as one of its branches; (b) the agenda of FL<sub>n</sub> is very different from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FL<sub>w</sub>
Zadeh (1995) in Foreword of George J. Klir Fuzzy sets and fuzzy logic: theory and applications.
1990s

Lotfi A. Zadeh photo

“A frequent source of misunderstanding has to do with the interpretation of fuzzy logic. The problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization of classical multivalued logics. But in a wider sense, fuzzy logic, FLw is almost synonymous with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLw is much broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLw</sub”

Lotfi A. Zadeh (1921–2017) Electrical engineer and computer scientist

Zadeh (1995) in Foreword of George J. Klir Fuzzy sets and fuzzy logic: theory and applications.
1990s

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Lotfi A. Zadeh photo

“Essentially, a fuzzy algorithm is an ordered sequence of instructions (like a computer program) in which some of the instructions may contain labels or fuzzy sets, e. g.:
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If x is small then stop; otherwise increase x by 2.”

Lotfi A. Zadeh (1921–2017) Electrical engineer and computer scientist

Source: 1970s, Outline of a new approach to the analysis of complex systems and decision processes (1973), p. 30

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https://capx.co/remain-campaign-is-misleading-voters-on-the-single-market/

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“It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum.”

Hermann Weyl (1885–1955) German mathematician

Riemanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie (1925), as quoted/translated by Erhard Scholz, "Philosophy as a Cultural Resource and Medium of Reflection for Hermann Weyl" (2004)

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