Pirates & Revolutionaries: Kaufmann. Arnold Kaufmann

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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

Ch.1

Notions de base

Fundamental Notions

1.3

Le concept de sous-ensemble flou

The Concept of a Fuzzy Subset

Brief summary:

A fuzzy subset is one where the members admit of a variety of degrees of membership, ranging from 0 for not at all a member to 1 for fully a member, with all the decimal values between for the varying degrees of membership. We use a wavy line under the set name to designate it as a fuzzy subset, or we may use the wavy line under the set inclusion symbol.

We can define the members by assigning to them the set of values, partial or full, from a set M.

Fuzzy membership could also be written using the wavy line under the membership symbol.

We can also designate the degree of membership by writing it under the membership symbol.

Fuzzy subsets allow us to define imprecise concepts, like the fuzzy subset of integers very near 0. As we move away from 0, the membership values will decrease.

Summary

[In the prior section 1.2, Kaufman had an example of two sets:

E = {x₁, x₂, x₃, x₄, x₅}

A = {x₂, x₃, x₅}

(Here the E seems to operate like a domain of discourse). We can notate this using a valuation function μ which assigns 1 when a member is in a set and 0 when it is not, so:

μ_A(x) = 1 if x ∈ A

= 0 if x ∉ A

We can then designate the memberships for A as:

A = μ_A(x₁) = 0, μ_A(x₂) = 1, μ_A(x₃) = 1, μ_A(x₄) = 0, μ_A(x₅) = 1

We can also write these as pairs in this way:

A = {(x₁, 0), (x₂, 1), (x₃, 1), (x₄, 0), (x₅, 1)}

(page 2).] In the example [see above] from the prior section, we designated members of E as either belonging to A or not belonging to A. The valuation function μ (or “characteristic function” as it is called here) only takes one of two values, 0 and 1. (p.4/4)

Kaufmann then has us think that the valuation function μ can take any value between 0 and 1. This allows for partial membership in the subset.

Imagine now that this characteristic function may take any value whatsoever in the interval [0, 1]. Thus, an element x_iof E may not be a member of A (μ_A= 0), could be a member of A a little (μ_A near 0), may more or less be a member of A (μ_A neither too near 0 nor too near 1), could be strongly a member of A (μ_A near 1), or finally might be a member of A (μ_A = 1). In this manner the notion of membership takes on an interesting extension and leads, as we shall see, to very useful developments.

(4/4)

Such subsets are called fuzzy subsets, and they are notated by either placing a wavy line under the subset name or under the subset symbol. The degree of membership is then notated by writing the value under the membership symbol. [Perhaps we might say that we deal with fuzzy subsets rather than fuzzy sets, because we always need a domain of discourse whose members are established, and then these fully established members in the domain can then be said to be partial members of fuzzy subsets.]

(4/4)

So consider the fuzzy subset 3.1 above. We see that some members are only partially so. [His next point seems to be that we can use fuzzy subsets when dealing with vague predicates where there is still a scale of fittingness for that predicate. Let me quote.]

Thus, the fuzzy subset defined by (3.1) contains a little x₁, does not contain x₂, contains a little more x₃, contains x₄ completely, and a large part of x₅. This will allow us to construct a mathematical structure with which one may be able to manipulate concepts that are rather poorly defined but for which membership in a subset is somewhat hierarchical. Thus, one may consider: in the set of men, the fuzzy subset of very tall men; in the set of basic colors, the fuzzy subset of deep green colors; in the set of decisions, a fuzzy subset of good decisions; and so forth. We shall go on to see how to manipulate these concepts that seem particularly well adapted to the imprecision prevalent in the social sciences.

(4/4)

Kaufmann then gives Zadah’s rigorous definition, Z1. [Let us look first at the example, as it can help understand the structures.

(6/6)

Here we see that E is like the domain of discourse, because it has all the possible members. M is the set of membership degrees or values that any member can take. We then define a fuzzy subset A by giving ordered pairs, where the first of the pair is the member name, and the second is the assigned partial value. For some reason, each pairing is enclosed in parentheses and separated by a vertical line. As you will see, there is a special symbol for the function that assigns the partial value to the fuzzy subset member. But I think I am misreading the notation for (3.4). The second member of the pair is ∀x∈E. But I do not know exactly what that would be, as something you could designate as a second member of an ordered pair. It does not seem to show in the examples. Please consult the text below to interpret it for yourself.]

(5/5)

[Kaufmann then restructures the above definition for Boolean functions. I am not sure what the important differences are.]

(5/5)

Of course:

(5/5)

Kaufmann then notes that “Thus, the notion of fuzzy subset is linked with the notion of a set and allows one to study, using mathematical structures, imprecise concepts” (5/5).

He next gives some examples for such imprecise concepts:

the fuzzy subset of numbers x approximately equal to a given real number n, where n ∈ R (R being the set of reals);

the fuzzy subset of integers very near 0;

let a be a real number and let x be a small positive increment given to a; then the numbers a + x form a fuzzy subset in the set of reals;

let H be an element of a lattice; the elements most near to H in the order relation form a fuzzy subset in the set of elements of the lattice.

(5/5, boldface his)

Kaufmann will use boldface to designate sets or subsets, and the wavy line under them to designate fuzzy subsets.

(5/5)

(6/6)

Kaufmann then uses these symbols to notate fuzzy membership.

(6/6)

We may also write the degree of membership under the membership symbol.

(6/6)

[Kaufmann then illustrates with three examples. The first one we already examined above. The second one is interesting, because it might remind us of Nolt’s example of a vague predicate, where each iteration of a statement in which we increase a figure by 1 we also decrease the truth value by a small amount. See Nolt Logics section 16.1.]

(6/6)

[Example three introduces some alternate notation. See p.7/7.]

From:

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.

Kaufmann (1.2) Introduction to the Theory of Fuzzy Subsets, “Rappel sur la notion d’appartenance” / “Review of the Notion of Membership”

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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

Ch.1

Notions de base

Fundamental Notions

1.2

Rappel sur la notion d’appartenance

Review of the Notion of Membership

Brief summary:

Set A being a subset of E is written:

A ⊂ E

x being a member of A is written:

x ∈ A

We can also describe membership using a valuation function μ:

μ_A(x) = 1 if x ∈ A

= 0 if x ∉ A

The complement of a set is notated by writing a line above its symbol. We can calculate whether or not some element is a member of a union or intersection of sets by using Boolean operations.

μ_{A ∩B}(x) = μ_A(x) . μ_B(x)

μ_{A ∪B}(x) = μ_A(x) + μ_B(x)

Summary

[Kaufmann will define membership and subset inclusion. See especially Suppes’ Introduction to Logic sections 9.1-9.2 and section 9.3 for another great introduction to the concepts and notation.]

We suppose that E is a set and A is a subset of E. We then would write:

A ⊂ E.

(Kaufmann p.1/1). [This symbol is used to mean a proper subset, in Suppes’ text; (if all the members of one set are included in a second, but not all of the second are included in the first, then we call the first a proper subset of the second)]

Suppose E has an element x. If it is a member of A, we write it:

x ∈ A.

[The next idea seems to be that we can describe membership using a valuation function.]

In order to indicate this membership one may also use another concept, a characteristic function μ_A(x), whose value indicates (yes or no) whether x is a member of A:

μ_A(x) = 1 if x ∈ A

= 0 if x ∉ A.

(p.1/1)

Kaufmann then gives an example. Here we have two sets with members:

E = {x₁, x₂, x₃, x₄, x₅}

A = {x₂, x₃, x₅}

(As you can see, the E seems to function like a domain of discourse).

We can then designate the memberships for A as:

A = μ_A(x₁) = 0, μ_A(x₂) = 1, μ_A(x₃) = 1, μ_A(x₄) = 0, μ_A(x₅) = 1

We can also write these as pairs in this way:

A = {(x₁, 0), (x₂, 1), (x₃, 1), (x₄, 0), (x₅, 1)}

(2/2)

Kaufmann next has us consider Boolean binary algebra. [Recall the following from Suppes’ Introduction section 9.5:

Certain operations can be performed on sets. If we find all the members shared in common between two sets, we are finding their intersection (∩):

(x)(x ∈ A ∩ B ↔ x ∈ A & x ∈ B)

When two intersecting sets share no members in common, that is, when they are mutually exclusive sets, their intersection is the empty set. The set containing all the members in total from two sets is their union (∪):

(x)(x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B)

All the members in set A that are not in set B is the difference (∼) of A and B.

(x)(x ∈ A ∼ B ↔ x ∈ A & x ∉ B)

And recall the following from section 9.6. A domain of individuals (also called a domain of discourse) is a specific set. We use the symbol “V” to denote a domain. Suppose we have a domain V and a set A. The complement of A relative to the domain are all those items in the domain that are not in A. We symbolize it either as V∼A or just ∼A. In section 9.9 he noted a number of identities (here Λ is the empty set and ~ is the complement):

] [In the following, I cannot duplicate the notation using typographical symbols, so I will paste images for certain parts. A set’s complement has all the members in the domain that the first set lacks. The intersection of sets are all those members shared by both sets. Thus the intersection of a set and its complement will be empty. The union of sets are all those found in either set. So the union of a set and its complement will be all those in the domain.]

(2/2)

Kaufmann next will examine intersections in terms of Boolean products. We first consider two subsets, A and B, and their intersection A ∩ B. We thus can make the following determinations:

μ_A(x) = 1 if x ∈ A

            = 0 if x ∉ A ,

μ_B(x) = 1 if x ∈ B

            = 0 if x ∉ B ,

μ_{A ∩ B}(x) = 1 if x ∈ A ∩ B

            = 0 if x ∉ A ∩ B.

(2/2)

[The next piece of notation is a ‘.’, which seems aligned not above the base-line but right at it, like a period. I am not sure what it means at this point. I first I thought it was like arithmetical multiplication, which would hold in this case, but the ‘+’ operation to follow does not add the 1 values to get 2. It instead seems to be an operator that simply should be understood as conjunction, and + as disjunction.]

(2/2)

(3/3)

We then define the union of two subsets A and B by using the ‘+’ or Boolean sum operator [which seems to correspond to disjunction.]

μ_{A ∪B}(x) = 1 if x ∈ A ∪ B

= 0 if x ∉ A ∪ B

μ_{A ∪B}(x) = μ_A(x) + μ_B(x)

or

(3/3)

[Kaufmann then gives more examples of these operations and also adds to them complementarity. See page 3/3.]

From:

Kaufmann (1.1) Introduction to the Theory of Fuzzy Subsets, “Introduction”

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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

Ch.1

Notions de base

Fundamental Notions

1.1

Introduction

Brief summary:

We will deal with fuzzy subsets and not fuzzy sets, starting first with a review of sets.

Summary

Kaufmann will first review basic notions regarding sets, because we will apply or modify many of these notions when describing fuzzy subsets (1/1).

Kaufmann will proceed slowly for those less adept with mathematics (1/1).

The reader can check their understanding by examining the examples. But Chapter 1 will not be the challenging part. It gets difficult starting with the second chapter (1/1).

As we will see, we will deal with fuzzy subsets and not fuzzy sets. This theory is useful. Although what this theory does can be accomplished with other concepts, it is most effectively expressed in terms of fuzziness (1/1).

From:

Kaufmann (symbols list) Introduction to the Theory of Fuzzy Subsets, “Liste des principaux symboles” / “List of Principal Symbols”

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[The following presents the symbols Kaufmann uses in his book. It is presented here for the sake of commentary on his text (see the other entries here), to be used for research and teaching. As the symbols do not exist typographically, they are reproduced photographically here. Owners please contact me if you do not approve of this usage.]

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

Liste des principaux symboles

List of Principal Symbols

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

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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

Avertissement

Preface

(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.

Summary

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.

(ix/xi)

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).

From:

28 Mar 2017

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

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