An axiomatic approach to fuzzy set theory

INFORMATION SCIENCES

52,75-83

(1990)

75

An Axiomatic Approach to Fuzzy Set Theory DAN E. TAMIR CA0 ZHI-QIANG

ABRAHAM KANDEL Department of Computer Science and the Institute for Expert Systems and Robotics, Florida State University, Tallahassee, Florida 32306-4019

and

JOE L. MO’IT Department of Mathematics, Florida State Vniversify, Tallahassee, Florida 32306

ABSTRACT An axiomatic basis for the concept of “fuzzy set” is established, class theory developed by von Neumann and Bernays (VNB). The set the minimal set that enables a formal definition of fuzzy sets, without discourse, and requires fewer axioms than any other published formal

following the classical of axioms we exhibit is assuming a universe of definition of fuzzy sets.

1. INTRODUCTION Fuzzy set theory, developed by L. A. Zadeh [l], generalizes classical set theory (CST) [21 in a way that enables mathematical modeling and analysis of the actual vague and imprecise perception/description of the term “set” usually given by humans. Since the axiomatization of fuzzy set theory (FST) is problematic, some researchers choose to restrict their attention to fuzzy subsets of a crisp set [3, 61. The advantage of this restriction is that it enables the immediate discussion of fuzzy subsets without previously laying a formal basis. But it has OElsevier Science Publishing Co., Inc. 1990 655 Avenue of the Americas, New York, NY 10010

0020-0255/90/%03.50

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the disadvantage that it prohibits using the semantically meaningful term “fuzzy set.” Moreover, without a formal basis for FST, another disadvantage arises, namely, Russell-like paradoxes that come from questions like “Does the collection of all fuzzy subsets exist?” and “Is the collection of all fuzzy subsets a set or a fuzzy subset?” Some researchers have noted the intuitive relation between fuzzy sets and class theory, e.g. [4, p. 1351: “Intuitively a Fuzzy Set is a class.” It is our aim in this paper to formalize this observation. Some authors have established a system of axioms for FST. Chapin 151 assumed only logic, and developed a system analogous to the Zermelo-Frankel (ZF) system. While he introduced only one primitive-the membership relation-his system actually includes three primitives, namely (1) sets, (2) the membership relation, and (3) degree of membership (which turn out-by the axioms-to constitute a fuzzy set). Chapin’s definition of the primitives complicates the FST he proposes, and his system has 14 axioms, while our proposed system has only 10. Goguen generalized the term “fuzzy set” using the theory of categories of V-sets, where I’ is a partially ordered set [6]. If V=[O, 11, then Goguen’s theory coincides with Zadeh’s FST. While this is an excellent basis for FST, it uses categories, and therefore assumes at least two additional axioms, and some additional definitions/theorems. Thus our proposed system has the advantage of being more compact than Goguen’s system. Fuhrmann [7] suggests that classical set theory is not sufficient for an axiomatic definition of FST. On the other hand, he claims that class theory is too broad and concludes that its use as a basis for axiomatic FST can lead to contradictions and paradoxes. We refer the reader to References [f&15,18] for further discussions on the foundations of fuzzy set theory. In this paper we will establish a well-founded axiomatic FST. The derivation we present is based on the extension of the ZF CST [16] using the concept of “class.” There are several axiomatic systems establishing the theory of sets and classes; among them is the system developed by von Neumann and Bernays (VNB) [17], which we use in this paper. We will show that class theory is sufficient for axiomatic FST, that it is the minimal system that has this property, and that the formal description of FST via class theory can be made in a way that is free from Russell-like paradoxes. One of the problems in establishing an axiomatic definition of FST is that the primitive term of CST-the membership relation, denoted “ E”-has to be redefined. Zadeh introduces the definition of fuzzy membership by means of a membership function. In Section 4 we suggest a rather different approach to the problem, and the fuzzy membership relation is embedded in our definition of fuzzy sets.

AXIOMATIC 2.

APPROACH

PRIMITIVES,

TO FUZZY

AXIOMS

SET THEORY

AND BASIC THEOREMS

77 OF THE FST

The VNB axiomatic system of CST (also known as the Godel-Bernays system) will be our basis for the formalism of the FST, i.e. for the definitions of fuzzy sets and fuzzy operators (Section 3). The VNB system and our proposed system presume logic. The primitives, the syntax, and the semantics of logic are the building blocks for the formalism. PRIMITIVES

OF FST

The primitives of the proposed FST are the VNB system primitives “class,” and “membership relation” (“ E “1.

“set,”

A23OMS FOR FST

The axioms for FST are the VNB axioms. A detailed list of these axioms can be found in [16] or [17]. Nevertheless, let us briefly review the list of axioms we are assuming in this section. The primitives are those listed above. Moreover, we assume that A E B is a statement if A and B are classes. We thus assume the following axioms: (1) Axiom

of equality.

Let A, B be classes. Then

[A=B] (21 Axiom of sets.

-

[Vx,x=A

= XEB].

Let A, B be classes. Then [A isaset]

e;r [3B, AEB].

(3) Axiom of separation. Let P(x) be a statement scheme. exists a class A such that a E A if a is a set satisfying P(x).

Then

there

Note that the counterpart for this axiom in the ZF system is a theorem and not an axiom. The class version of the separation axiom is more general than the set version. The set version demands that a statement scheme will specify a subset of a given set. The class version constructs a class that contains all the sets that are solutions to a statement scheme. This way of class construction has caused some misunderstandings, since natural language (NL) sentences may seem like statement schemes even though they are not. These sentences cannot be used to construct classes. Fuhrmann [7] claims that classes can be

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defined via NL. However, his claim is not accurate, schemes can be used and not arbitrary NL sentences.

since only statement

(4) Axiom of the null class. Let 0 denote the statement scheme Wx E 0 x f xl (the null class). Then (5) Axiom of infinity. There exists an infinite set. (6) Axiom of unions. Let the class lJ B be the statement scheme [3A such that x E A and A E B]. If a set.

solution class of the 0 is a set.

The above axioms suffice for the definitions axioms use this definition.

solution class for the B is a set, then lJ B is

of mappings.

(7) Axiom of replacement. Let f be a function (f : A - B). If A is a set, then B is a set.

The following

mapping

A onto

B

The power class of a class A, denoted by [A], is the class of all subclasses of A. Formally the power class is constructed using the axiom of separation. (8) Axiom of the power class. If A is a set, then so is [A]. (9) Axiom of regularity. If A is a class and a E A, then a is a set. (101 Axiom of choice. There exists a choice function for any class A # 0. Thus, our new system adds four axioms to the ZF system: The axiom of separation and the axiom of replacement (which were theorems in ZF), and the axiom of sets and the axiom of regularity (which have no counterpart in ZF). All the axioms of ZF are present in the VNB system with slight changes. It should be noted that the definition of fuzzy sets demands the use of some CST compound terms such as “functions” and the set “[O, 11.” These terms are defined in the CST, and their properties are developed using the assumptions, theorems, and definitions of CST. 3.

NOTATION The following notation

(1) “17”). (2) (3) (4) (5)

An element

will be used throughout

of a set will be denoted

the article:

by a small Greek letter (other than

A set will be denoted by a lowercase Latin letter. A class will be denoted by a capital Latin letter. The symbol “ E ” denotes “belongs to a set” (e.g. (YE x). The symbol “7” denotes “belongs to a class” (e.g. x n A).

Note that our system of axioms assumes only one primitive membership relation. In the axioms we used the same primitive (,‘ E “1 to denote set and

AXIOMATIC

APPROACH

TO FUZZY

79

SET THEORY

class membership. We introduce different notations for set membership and for class membership in order to increase readability and differentiate the usage for classes (where it is necessary to assume a class) from the usage for sets. Some axiom systems [17] distinguish between class and set membership and therefore must have two different primitives-one for each relation. We follow the VNB approach, which uses the same primitive for both. (6) A mapping from a set to a set (function) will be denoted domain of f by df, and the range of f by rr. (7) A mapping from a class to a class (functor) will be denoted domain of F by D,, and the range of F by R,.

4.

THE EXTENSION

TO FUZZY

by f, the by F, the

SETS

In this section, the axiomatic system for FST, described in Section 2, will be used to define the term “fuzzy set” and also to define some of the operations on fuzzy sets. Let W denote the class of all sets. Using the class version of the separation theorem, we can construct the class of all functions (say T) as a subclass of W. Again using the separation theorem, we can construct a subclass of T: the subclass of all functions over the [O, 11 interval. Thus, we can have the following definitions. DEFINITION 1.

interval (A = order 0.

Let A denote the class of all functions over the [O,l] = [0,11). Then A is defined to be the fuzzy class of

f 17Tlr,

DEFINITION 2.

Let f~ A. Then

f

is a fuzzy set of order 0.

(In this paper we treat only 0 order fuzzy sets.) Note that the relation of fuzzy membership is embedded in Definitions 1 and 2. At this point one may raise the question: Why should A be a class (and not a set)? We will answer this question in Section 5. The symbols “ = ” (equality), and “ c,” “ s” (containment) will preserve their classical meanings when applied to fuzzy sets. Next let us define some of the important operators on fuzzy sets. Using the concept of functor defined previously, we generalize the current definitions of these terms. DEFINITION 3.

(F: A X A --) A). If

Let

F

f 17R,,

be a mapping (functor) then f is a fuzzy operator.

from

A x A

to

A

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Definition 3 allows the development of fuzzy operators. We suggest here the commonly used operators. A thorough discussion of these operators and others can be found in [31. DEFINITION4. Let f, VA, f2q A, and define f, :df, n df2 -IO,11 (Va E df,) f+(a) = max{f,(cy),fi(a)l (fuzzy sum or union). DEFINITION5.

Let f1 q A, f2q A, and define f, :df, n dfi -[O,ll min{fi(~),f2(~)) (fuzzy multiplication or intersection). @‘a E df) fz+z C(u)= DEFINITION6. Let fv A, and f_(a) = 1- f(a) (fuzzy complement).

define

f_ : d, --) IO,11 by

by

by

(Vcu E d,_ )

It should be noted that in this case F: A X 1 + A, i.e., F : A + A. One of the important developments of FST is fuzzy relations. We propose here a definition that resembles the classical definition of relations. First we define a fuzzy Cartesian product. DEFINITION7. Let flea, f2~A, and define f,:fiXf2+[0,1] (Wa,P)e df, x df2) fx(a,P) = ~{max{fl(cY),f*(P)}+min(fl(“),fi(P)}}. f, is the fuzzy Cartesian product, f, X f2. Now it is possible to use the separation in the following way: DEFINITION 8. Let construct the class

fl q A, f217 A;

theorem

define

by Then

and define a fuzzy relation

fx(fl, f2) as

in Definition

7; and

R={f77AIdf~df,cf,,f~,,rf =[OJl}. Let f, TJ[R] (where [RI is the power class of RI. Then f, is a fuzzy relation from fl to f2. Therefore, a fuzzy relation is a fuzzy subset of a fuzzy Cartesian product.

5.

ANALYSIS

If we compare the new definitions subsets, then we note the following:

with the traditional

definition

of fuzzy

(1) The new definitions are based on an axiomatic approach. Thus the new fuzzy set system is less susceptible to paradoxes. (2) The new definitions are generalizations of the old ones in the sense that the operations between two fuzzy sets fi, f2 are always valid and are meaningful as long as df, cl df2 # 0. In the traditional approach the operations

AXIOMATIC

APPROACH

TO FUZZY

SET THEORY

81

were valid only over fuzzy subsets of the same crisp set and under a given universe of discourse. (31 In some cases the new approach is more restrictive than the traditional one. For example in Definition 8, the domain of a fuzzy relation is a classical subset of a fuzzy Cartesian product. This is done to maintain logical consistency (that is, to avoid contradictions and paradoxes). Yet the new definitions construct a system that is equivalent to the old ones, in the sense that all the important theorems or conclusions derived in the traditional fuzzy set theory can be derived in our system. SUFFICIENCY

The proposed system is sufficient for an axiomatic FST. Actually we showed this in Section 4, where we introduced the definitions of the term fuzzy set and of some of the important operators over fuzzy sets. These definitions were based on the axiomatic system of classes. MINIMALITY

Since FST is a generalization of CST, any FST system has to contain a set of axioms that is equivalent to the ZF CST system. Our system (VNB)’ includes four additional axioms that enable the formulation of classes. An examination of the proposed system will show that the term “class” makes two contributions to our system. (1) In the definitions of fuzzy set and fuzzy operators, we used the class A, which is a subclass of the class of all sets. Trying to avoid class theory would force A to be a set, but then we would encounter a contradiction, since by assumption A is the class of all fuzzy sets. But consider the function f : [A] + [O, l] (where [A] is the power set of A). This is also a fuzzy set, but from classical set theory P(A) G A. In other words, there are fuzzy sets not contained in A. A similar problem would arise with the class [R] (used in the definition of fuzzy relation). (2) This system enables a definition that is independent of the implementation: since we use classes, we don’t need the term “universe of discourse,” an implementation/interpretation dependent term.

‘Actually we introduced an expanded version of the VNB system, a version that includes the axiom of choice. We think that this axiom is an essential part of axiomatic set and class theory. The historical argument concerning the role of the axiom of choice in set theory [2] is beyond the scope of this article.

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ET AL.

A careful formalization of FST using only set theory may be possible, yet this kind of formalism will have to be based and dependent on a universe of discourse, and thus will be less flexible and susceptible to inconsistencies. Our formalism of FST is the minimal one that is free from a universe of discourse. Our proposed system is minimal also in the sense that no other FST proposed in the literature has less axioms.

CONSISTENCY

Not every collection of items is a set; thus treating some collections as sets (for example the set of all sets) is susceptible to contradictions and Russell-like paradoxes. One solution to this problem is the concept of orders of collections. The first order of collections is sets; the second order-collections of sets-is classes. Thus the class of all the sets or the class of all the sets that do not contain themselves is a noncontradictory notion. Note that the term “the class of all classes” has the same problems as the term “the set of all sets.” In our system we have used only collections of sets and not collections of classes. Thus, in the framework of classes there is no danger of Russell-like paradoxes in our system. We raise an objection to the conclusion of Fuhrmann [7] that since classes are defined via natural languages, they may lead to contradictions and cannot serve as a tool for axiomatic FST. Fuhrmann concludes that every system based on classes is susceptible to contradictions. But if this were true, then the widely accepted theory of categories on which Goguen based his axiomatic FST would also be inconsistent.

6.

CONCLUSIONS

In this paper we have attempted to formalize axiomatic FST. The motive underlying our approach is the desire to avoid paradoxes. The difference between the proposed system and other systems are: (1) The proposed system requires only class theory. (2) The system is independent of a universe of discourse. (3) A fuzzy operator between two fuzzy sets df,,dfi provided that fl fl fi # 0.

is well defined,

This work can be extended. In particular, some important terms of FST (for example fuzzy containment) have not been defined in this paper. Nevertheless

AXIOMATIC

APPROACH

it seems that our significant ways:

approach

TO FUZZY

SET THEORY

may contribute

(1) It establishes a more reliable (though theory of fuzzy sets. (2) It generalizes the previous approaches.

to the field

more restricted)

83 of FST

in two

basis for the

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