29 Mar 2017

Kaufmann (preface) Introduction to the Theory of Fuzzy Subsets, “Avertissement” / “Preface”


by Corry Shores


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[The following is summary. Unless otherwise noted, boldface is my own. Page citations refer to the French edition first / then the English. I apologize in advance for my distracting typos or other mistakes, because proofreading is incomplete.]




Summary of


Arnold Kaufmann


Introduction à la théorie des sous-ensembles flous

à l’usage des ingénieurs

(Fuzzy sets theory)

1. Eléments théoriques de base


Introduction to the Theory of Fuzzy Subsets.

Vol.1 Fundamental Theoretical Elements




(by Arnold Kaufmann)



Brief summary:

The elements of fuzzy subsets are members in an “uncertain fashion” rather than in the certain fashion of classical sets where elements either are or are not in the set. The theory worked out here is about fuzzy subsets and not fuzzy sets, because “the reference set will always be an ordinary set, that is, [...] a collection of well-specified and distinct objects” (xii/xiii-xiv). This theory of fuzzy subsets will prove especially useful for designing intelligent machines so that they can handle fuzzy information, like human minds can.






Kaufmann notes that our scientific knowledge of the world is limited to the models, representations, “more or less true” laws, and “acceptable approximations in the state of our knowledge” that we use to study the world (ix/xi). [I am not sure about his next point, so I will quote it. It might be that the only confirmation we have of one model is that made by means of another model, and although they continue to correct one another, there will not be a perfect representation of the world, at least until some great revolution in ideas allows for a better means of representation. Here is the quotation:]

And the model of something for one is not exactly the same model of this thing for another; the formula may remain the same, but the interpretation may be different. The universe is perceived with the aid of models that are indeed perfecting themselves through embodying one in another, at least until some revolution in ideas appears, no longer permitting a correct embodyment.



But human thinking, unlike computer cognition, is fuzzy. This is partly because we use global or parallel reasoning which is necessarily fuzzy (ix/xi). And there is a lot of room for alterations and adaptation in human learning (ix/xi).


Kaufmann then wonders how we might introduce this real fuzziness into our mathematical systems (ix/xi).


Kaufmann then distinguishes classical and fuzzy membership.

For a mathematician, what does the word fuzzy signify (or synonymous words)? This will mean that an element is a member of a subset only in an uncertain fashion; while, on the other hand, in mathematics we understand that there are only two acceptable situations for an element: being a member of or not being a member of a subset. Any normal logic, boolean logic, rests on this base: membership or nonmembership in a subset of a reference set.

(x/xi, italics his)


L.A. Zadeh’s innovation was to allow for “weighted membership. An element may then belong more or less to a subset, and, from there, introducing the fundamental concept, that of a fuzzy subset” (x/xii).


The multivalued or n-ary logics of Post (1921), Lukasiewicz (1937), and Moisil (1940) opened the way for fuzzy logic. The two schools of fuzzy logic that emerged are of Zadeh and Moisil (x/xii).


One objection to fuzzy logic is that what it accomplishes can be accomplished by other systems. But this objection can be raised for almost any important system. [That in itself does not diminish the value of any system, so it should not diminish the value of fuzzy logic.] (x/xii)


The theory of fuzzy subsets should be of great interest to scientists who study fuzzy systems like language and thought, but also to “the literati and artists, those who construct truth and beauty with fuzziness” (x/xii).


This book is designed to be as accessible as possible for those with a technical interest in the field (x/xii).


Kaufmann added many examples, although for some readers that might make the book too lengthy (xi/xii-xiii).


Fuzziness is here limited to variables and configurations, but one could extrapolate from this presentation other conceptual aspects of fuzziness. So, many disciplines could derive value from this material. (xi/xiii).


Linear computation machines will be able to handle the fuzzy problems we deal with in this book (xi-xii/xiii).


Kaufmann then addresses the question, why do we use the term fuzzy “subset” and not fuzzy “set”? He explains that this is because “the reference set will always be an ordinary set, that is, such as one defined intuitively in modern mathematics, that is again, a collection of well-specified and distinct objects. It is the subsets that will be fuzzy, as we shall see” (xii/xiii-xiv).


Volume 1 presents the theory, while Volume 2 applies it to such areas as “fuzzy languages, fuzzy systems, fuzzy automata, fuzzy algorithms, machines and control, decision problems in a fuzzy universe, recognition of forms, problems of classification and selection, documentary research, etc.” (xii/xiv).


Kaufmann then thanks a number of people who helped with the production of this book (xii/xiv).


He especially thanks his son Alain for his corrections (xii/xiv).


Kaufmann notes that the human mind will “remain fuzzy and creative” (xii/xiv). [The French Avertissement ends here, and then there begins another one for the second edition. Part of that is found in English edition as a continuation of its Preface.]


Kaufmann then notes that he corrected a number of errors for this second edition (xiii/xiv). In the French edition he mentions some features of the text, like the extensive Bibliography, also his new volumes, and he calls upon his readers to work together on furthering our knowledge by means of these findings (xiii).










Kaufmann, Arnold. 1975 [1973]. Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs (Fuzzy sets theory). 1: Eléments théoriques de base. Foreword by L.A. Zadeh. 2nd Edn. Paris: Masson.


Kaufmann, Arnold. 1975. Introduction to the Theory of Fuzzy Subsets. Vol.1: Fundamental Theoretical Elements. Foreword by L.A. Zadeh. English translation by D.L. Swanson. New York / San Francisco / London: Academic Press.




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