28 Mar 2017

Zadeh (Foreword) in Kaufmann Introduction to the Theory of Fuzzy Subsets, “Préface” / “Foreword”


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 L.A. Zadeh



Brief summary:

Fuzzy sets are “classes with unsharp boundaries in which the transition from membership to nonmembership is gradual rather than abrupt” (Zadeh ix). The reliance on classical sets in studies of human life and in human or artificial cognition has limited these efforts, because the real world and human thinking involve fuzziness.






The theory of fuzzy subsets tries to bring together precise mathematics with the “pervasive imprecision of the real world” (Zadeh v/ix). This is also an effort to better understand mental cognition.


At the time of this writing, artificial intelligence science has been unable to replicate the many types of human cognition (v/ix).


The reason for this is that human cognition has the ability to process imprecise data, while computers do not (v-vi/ix).


“The fundamental concept in mathematics is that of a set – a collection of objects” (vi/ix). However, Zadeh thinks that most human cognition uses fuzzy sets or subsets:

We have been slow in coming to the realization that much perhaps most, of human cognition and interaction with the outside world involves constructs which are not sets in the classical sense, but rather “fuzzy sets” (or subsets), that is, classes with unsharp boundaries in which the transition from membership to nonmembership is gradual rather than abrupt. Indeed, it may be argued that much of the logic of human reasoning is not the classical two-valued or even multivalued logic but a logic with fuzzy truths, fuzzy connectives, and fuzzy rules of inference.



Because we have sought precision in our scientific endeavors, we have tried to make the real world fit into mathematical models that leave no room for fuzziness. We have even tried to use such precision to understand human individual and social behavior. Zadeh thinks this is a doomed project (vi/ix).

In our quest for precision, we have attempted to fit the real world to mathematical models that make no provision for fuzziness. We have tried to describe the laws governing the behavior of humans, both singly and in groups, in mathematical terms similar to those employed in the analysis of inanimate systems. This, in my view, has been and will continue to be a misdirected effort, comparable to our long-forgotten searches for the perpetuum mobile and the philosopher’s stone.



Instead, Zadeh argues that we need to incorporate fuzziness into our concepts and techniques for studying reality and human life (ix).

What we need is a new point of view, a new body of concepts and techniques in which fuzziness is accepted as an all pervasive reality of human existence. Clearly, we need an understanding of how to deal with fuzzy sets within the framework of classical mathematics. More important, we have to develop novel methods of treating fuzziness in a systematic – but not necessarily quantitative – manner. Such methods could open many new frontiers in psychology, sociology, political science, philosophy, physiology, economics, | linguistics, operations research, management science, and other fields, and provide a basis for the design of systems far superior in artificial intelligence to those we can conceive today.



Ladeh then praises Kaufmann’s text. It is thorough and lucid, and it is the “first systematic exposition” of fuzzy subset theory (vii/x).


This text will deal with the mathematical aspects of fuzzy subsets, and it should prove useful to engineers and artificial intelligence programmers, because among other things, it details the notion of fuzzy algorithms (vii/x).


Zadeh thinks this book will prove highly influential (x).







L.A. Zadeh’s “Préface” / “Foreword”  in


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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