Annals of the Japan Association for Philosophy of Science 59
A Natural Basis of Fuzzy Set Theory-an Overview
Mamoru SHIMODA*
Abstract
The aim of this paper is to overview our recent studies on mathematical foundation of fuzzy set theory, and to show that large parts of fuzzy set theory can be explained by our natural interpretation. With the natural interpretation of fuzzy set theory in a cumulative Heyting valued model for intuitionistic set theory, various notions and properties of fuzzy sets, fuzzy relations and fuzzy mappings can be acquired consistently. As far as fuzzy set theory is considered as an extension of usual set theory, this interpretation seems to be most natural.
1. Introduction
In ordinary fuzzy set theory, fuzzy sets are 'characterized by' or 'identified with' mappings to the unit interval [0, 1] of real numbers, and the properties or operations of fuzzy sets and relations are defined by equations or inequalities. Various definitions have been proposed on the basic operations such as intersection, union, complement of fuzzy sets, and composition and inverse of fuzzy relations etc. (e.g., [33, 34, 4, 18, 9, 3, 5] ). Occasionally the defining equations are presented without explanation. The original concept of fuzzy sets in the pioneering paper of L. Zadeh [33] was introduced as an extention of characteristic functions of crisp (usual) sets, by enlarging the truth value set of 'grade of membership' from the two value set {0, 1} to the unit interval [0, 1] of real numbers. Accordingly a fuzzy set is said to be ' characterized by' a mapping called 'a membership (characteristic) function' from a crisp set into [0, 1]. Since an explicit, mathematically satisfactory definition of a fuzzy set was not given in the first paper [33], in many of the following papers fuzzy sets are defined just as membership functions (sometimes the word 'identified' is used). Hence the basic relations between two fuzzy sets, such as equality and inclusion as well as the basic fuzzy set operations such as intersection, union, and complement, are defined with some equations or inequalities of membership functions, which also hold for the corresponding characteristic functions of the crisp sets. As for fuzzy relations, the
* Shimonoseki City University , Daigaku-cho, Shimonoseki 751-8510, Japan.
-59- 60 Mamoru SHIMODA Vol. 13 basic operations such as composition and inverse as well as the basic properties such as reflexive, symmetric, transitive etc. are defined with some equations or inequal ities of membership functions as extensions of those in crisp relations. As [0, 1] has much more complex structure than the two element set {0, 1}, many kinds of operations and relations are definable on the unit interval. Accord ingly, various definitions have been proposed for operations and relations of fuzzy sets and fuzzy relations. Sometimes they are evaluated from the viewpoint of effectiveness for applications. Some general notions such as triangular norms and conorms for intersections and uniosns are proposed and have been investigated. However, the primary intention of Zadeh is considered to distinguish the notion of fuzzy set from its membership function and to develop a theory of fuzzy sets as an extention of usual set theory. Our natural interpretation is to give a satisfactory explanation on the problem and give a model of fuzzy set theory on which ordinary set theory can be naturally extended. We interpret fuzzy sets and fuzzy relations in the model VH introduced by the author [20], a cumulative Heyting valued model for intuitionistic set thoery, where H is a complete Heyting algebra and is considered as the set of truth values in the model. The model VH is a kind of so-called sheaf model, constructed by transfinite iteration of power sheaf construction over H. In that model we can easily define basic notions and operations of sets and relations. By the canonical embedding we can obtain most of the standard defining equations of basic notions and operations of fuzzy sets and relations. It is shown that min, max and intuitionistic negation are the most natural basic set operations, and max-min composition is the most natural operation for composition of relations. The extension principle of Zadeh ([35]) is expressed as a theorem on images of mappings for natural extensions of mappings between crisp sets. In this paper only the basic part of the interpretation are summarized. For details, we refer [21, 22, 23, 24, 25]. In the next section the fundamentals of the model is briefly described. After showing the basic construction of the model we deal with basic notions such as sets, relations, mappings etc. in the model. In Section 3 the natural interpretation of fuzzy sets, fuzzy relatioins, fuzzy mappings etc. are dealt with. It is shown that there is a natural correspondence between fuzzy subsets of a crisp set and mappings from the crisp set to H, and that the correspondence preserves inclusion and basic set operations. On fuzzy relations and fuzzy mappings, characterizations of fuzzy mappings and fuzzy equivalence relations are presented. It is also shown that the lifting in the extension principle can be considered as a composition of two steps. In the final section some concludeing statements are provided.
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2. The Heyting valued model
2.1. Basic set constructions
Let H=<|H|, •È, •É, •È, •É, •¨, •Ê, 0, 1, _??_> be a complete Heyting algebra
with the usual operations and constants. We construct an H-valued model for the
extended intuitionistic set theory, which is a first order intuitionistic logic with
predicates •¸, = and E (called an existence predicate) together with axioms of intuitionistic set theory. Let V be the class of all crisp sets and On be the class of
all ordinals. |H| is identified with H.
Definition 2.1. The H-valued model VH is constructed as follows.
For every ordinal ƒ¿, VHƒ¿ is defined by induction:
VH0=ƒÓ, VHƒ¿=•¾ƒÀ<ƒ¿VHƒÀ(if ƒ¿ is a limit ordinal),
VHƒ¿+1={u=<|u|,Eu>; |u|:Du•¨H, Du•ºVHƒ¿, Eu•¸H,
|u|(x)_??_Eu•ÈEx(•Íx•¸Du)}.
Then VH=•¾ƒ¿•¸ OnVHƒ¿.
We identify |u| with u. An element of the model is called a set in VH. For
a sentence ƒÓ of VH, the Heyting value •aƒÓ•a•¸H is defined as follows.
Definition 2.2. Let u, v•¸VH and ƒÓ, ƒÕ, ƒÓ(a) be formulas of VH. For atomic
formulas,
•a Eu•a=Eu, and by simultaneous induction, •au•¸v•a=•Éy•¸Dv(v(y)•È•au=y•a),
•a u=v•a=•È x•¸Du(u(x)•¨•ax•¸v•a)ƒ©ƒ©y•¸Dv(v(y)•¨•ay•¸u•a)•ÈEu•ÈEv. For compound formulas, by induction (on the number of logical symbols), =•aƒÓ•a•È•aƒÕ•a, •aƒÓ•ÉƒÕ•a=•aƒÓ•a•É•aƒÕ•a, =•aƒÓ•a•¨•aƒÕ•a, •a•ÊƒÓ•a=•Ê•aƒÓ•a,•a•Í xƒÓ(x)•a=•Èu•¸VH(Eu•¨•aƒÓ(u)•a), •a•ÎxƒÓ(x)•a=•Éu•¸VH(Eu•È•aƒÓ(u)•a).
For a sentence ƒÓ of VH, ƒÓ is valid in VH if •aƒÓ•a=1. We say u is a subset of
v in VH and write u_??_v if •au•ºv•a=•Èx•¸Du(u(x)•¨•ax•¸v•a)=1. If u_??_v and v_??_
u, we say u and v are similar (u•`v). If u•`v and Eu=Ev, u and v are said to
be equivalent (u_??_v).
For every crisp set x in V, the check set x•¸VH is defined recursively by:
D(x)={y;y•¸x}, Ex=1, x:y_??_1. The assignment from each crisp set to its
check set is called the canonical embedding.
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Proposition 2. 1. Let ƒÓ(a1, ... , an) be a bounded formula of VH and x1, ..., xn•¸V. Then
ƒÓ(x1, ..., xn) holds iff •aƒÓ(x1, ..., xn)•a=1, and
•ÊƒÓ(x1, ..., xn) holds iff •aƒÓ(x1, ..., xn)•a=0.
Basic sets constructions such as pairs, couples (ordered pairs), (cartesian) products, unions, power sets, as well as basic set operations such as meet, join and difference can be naturally defined in the model. All of these sets have the desirable properties in the model, quite similar to those in usual sets. Then we can prove that all axioms of intuitionistic set theory are valid in VH, that is, VH is a model for intuitionistic set theory. The proof is almost the same as in [32].
2.2. Relations in the model
A relation in VH is a subset of a cartesian product in VH. For R, u, v•¸VH,
R is a relation from u to v in VH if R is a subset of u•~v in VH. We often write
xRy instead of
For each set u in VH, the identity relation Iu on u is defined by:
D(Iu) ={
Definition 2.3. Let R, S•¸VH.
(1) The composition S_??_R is defined by:
D(S_??_R)={
S_??_R:
(2) The inverse relation R-1 is defined by:
D(R-1)={
The definitions are not affected by the choice of the ordinal ƒ¿, in the sense that the composition (resp. the inverse) is determined "up to equivalence". If R and S
are relations from u to v and from v to w respectively, then S_??_R is a relation from
u to w and R-1 is a relation from v to u.
Definition 2.4. Let R be a relation from u to v in VH.
(1) For A•¸VH, the image R(A) is defined by:
D(R(A))=Dv, E(R(A))=ER•ÈEA, R(A):y_??_•a(•Îx•¸A)xRy•a
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(2) For B•¸VH, the inverse image R-1(B) is defined by: D(R-1(B))=Du, E(R-1(B))=ER•ÈEB,
R-1(B):x_??_•a(•Îy•¸B)xRy•a.
A relation R from u to v in VH is called total if (•Íx•¸u) (•Îy•¸v) (xRy) is
valid, and is called surjective if (•Íy•¸v) (•Îx•¸u) (xRy) is valid. R is called
injective if (•Íx, y•¸u) (•Íz•¸v) (xRz•ÈyRz•¨x=y) is valid, and is called univalent
if (•Íx•¸u) (•Íy, z•¸v) (xRy•ÈxRz•¨y=z) is valid. Then R is total iff Iu_??_
R-1_??_R, R is surjective iff Iv_??_R_??_R-1, R is injective iff R-1_??_R_??_Iu, and R is
univalent iff R_??_R-1_??_Iv.
A relation R on u is called reflexive if (•Íx•¸u) (xRx) is valid, is symmetric if
•Íx•Íy (xRy•¨yRx) is valid, and is transitive if •Íx•Íy•Íz (xRy•ÈyRz•¨xRz) is
valid. R is called antisymmetric if •Íx•Íy (xRy•ÈyRx•¨x=y) is valid, and is
connected if (•Íx•¸u) (•Íy•¸u) (xRy•ÉyRx) is valid. Then R is reflexive iff
Iu_??_ R, is symmetric iff R-1_??_R iff R-1_??_R, is transitive iff R_??_R_??_R, is antisym
metric iff R•¿R-1_??_Iu, and is connected iff u•~u•`R•¾R-1. An equivalence relation
is a reflexive, symmetric, and transitive relation, an order relation is a reflexive,
antisymmetric, and transitive relation, and a linear order is a connected order
relation.
2.3. Mappings in the model
A mapping in VH is a total and univalent relation in VH. For u, v, f•¸VH,
f is a mapping from u to v in VH(f: u•¨v in VH) if f is a total and univalent relation from u to v in VH. For every f in VH, f is a mapping from u to v iff f
_??_u•~v, Iu_??_f-1_??_f, and f_??_f-1_??_Iv. We often write f(x)=y instead of or xfy.
An injection (resp. a surjection) is an injective (resp. surjective) mapping, and
a bisection is an injective and surjective mapping. For a mapping f from u to v, f
is injective iff Iu_??_f-1_??_f and is surjective iff f_??_f-1_??_Iv. For f: u•¨v in VH,
f-1_??_f is always an equivalence relation on u. We can define images and inverse images of mappings similarly to those of
relations (Cf. Definition 2.4).
Most of the basic properties of relations and mappings of usual set theory also
hold in the model.
3. The natural interpretation
3.1. Fuzzy sets We distinguish a fuzzy set, a fuzzy subset of a crisp set (which is called a universe for the fuzzy subset), and a membership function of a fuzzy set on a crisp set (universe). Let X be a crisp set.
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Every set in VH is an H-fuzzy set . For each H-fuzzy set A, the mapping
ƒÊA=ƒÊXA:X•¨H; x_??_•ax•¸A•a is called the membership function of A on X.
An H-fuzzy subset of X is a subset of X in VH . Obviously A is an H-fuzzy subset
of X iff A_??_X for every A•¸VH. It is easily proved that every mapping from X to H is the membership function of some H-fuzzy subset of X. Hence there is a natural correspondence between H-fuzzy subsets of X and mappings from X to H. The order relation and basic set operations of mapppings into H are defined pointwise as following.
Definition 3.1. Let ƒÊ, v: X•¨H.
(1) ƒÊ_??_v iff ƒÊ(x)_??_v(x) for all x•¸H.
(2) The mappings ƒÊ•Èv, ƒÊ•Év, ƒÊ•_v, •ÊƒÊ: X•¨H are defined by:
ƒÊ•È v: x_??_ƒÊ(x)•Èv(x), ƒÊ•Év: x_??_ƒÊ(x)•Év(x),ƒÊ•_
v: x_??_ƒÊ(x)•È•Êv(x), •ÊƒÊ: x_??_•ÊƒÊ (x).
Theorem 1. Let A, B be H-fuzzy sets and all the membership functions below be on X.
(1) If A and B are H-fuzzy subsets of X, then A_??_B iff ƒÊA_??_ƒÊB, and A•`B iff ƒÊA=ƒÊB
•K(2) ƒÊA•¿B=ƒÊA•ÈƒÊB, ƒÊA•¾B=ƒÊA•ÉƒÊB,
ƒÊA•_B=ƒÊA•_ƒÊB, ƒÊX•_A•Ê=ƒÊA•K
Therefore there is a one-to-one correspondence between the set of all mappings
from X to H and the set of all equivalent classes of H-fuzzy subsets of X with
respect to the equivalence relation •`, and this correspondence preserves inclusion
(order) and the basic set operations.
3.2. Fuzzy relations
Every relation in VH is called an H fuzzy relation. Let X, Y, Z be crisp sets.
For R, u, v•¸VH, R is an H-fuzzy relation from u to v if it is a relation from u
to v in VH. For every R•¸VH, the membership function of R from X to Y is the
membership function of R on X•~Y. An H-fuzzy relation from X to Y is an
H-fuzzy subset of X•~Y. For every R•¸VH, the membership function of R from
X to Y is the mapping ƒÊR: X•~Y•¨H;
Proposition 3.1. Let R be an H-fuzzy relation from X to Y, S be an H-fuzzy
relation from Y to Z, and ƒÊR, ƒÊs, ƒÊS_??_R, ƒÊR-1 be the membership functions on X•~
Y, Y•~Z, X•~Z, Y•~X respectively.
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(1) The composition S_??_R is an H-fuzzy relation from X to Z, and for all x•¸X, z•¸ Z,
S_??_R
(2) The inverse relation R-1 is an H-fuzzy relation from Y to X, and for all x•¸X, y•¸Y,
ƒÊR-1
The equations are extensions of the defining equations of max-min composition and inverse for ordinary fuzzy relations.
Proposition 3.2. Let R be an H-fuzzy relation from X to Y, and ƒÊR be its
membership function on X•~Y.
(1) For every A•¸VH, the image R(A) is an H fuzzy subset of Y, and for all y•¸Y
ƒÊR(A)(y)=•Éx•¸X(ƒÊA(x)•ÈƒÊR
where ƒÊA, ƒÊR(A) are the membership functions on X and on Y respectively.
(2) For every B•¸VH, the inverse image R-1(B) is an H-fuzz y subset of X, and for
all x•¸X
ƒÊ R-1(B)(x)=•Éy•¸Y(ƒÊB(y)•ÈƒÊR
where ƒÊB, ƒÊR-1 (B) are the membership functions on Y and on X respectively.
On H-fuzzy relations between crisp sets, the properties of total, surjective, injective, and univalent are characterized by equations or inequalities of member ship functions.
Proposition 3.3. Let R be an H-fuzzy relation from X to Y, and ƒÊA, ƒÊB, ƒÊR be the
membership functions on X, Y, X•~Y respectively.
(1) R is total iff •Éy•¸YƒÊR
(2) R is surjective iff •Éx•¸XƒÊR
(3) R is injective iff ƒÊR
.(4) R is univalent iff ƒÊR
An H-fuzzy relation from X to X is called an H-fuzzy relation on X. On H-fuzzy relations on a crisp set, the properties of reflexive, symmetric, transitive, antisymmetric, and connected are characterized by equations or inequalities of the membership functions. -65-
ƒÊ 66 Mamoru SHIMODA Vol. 13
Theorem 2. Let R be an H-fuzzy relation on X and ƒÊR be its membership functions on X•~X.
(1) R is reflexive iff ƒÊR
(2) R is symmetric iff ƒÊR
(3) R is transitive iff ƒÊR
(4) R is antisymmetric iff ƒÊR
(5) R is connected iff ,ƒÊR
Therefore H-fuzzy equivalence relations, H-fuzzy order relations, and H-fuzzy linear orders on a crisp set are characterized by the corresponding equations or inequalities of the membership functions. If H=[0, 1], the conditions are almost same to the usual definitions in fuzzy literature, but the condition (5) and the definition of linear order is stronger than ordinary definitions.
3.3. Fuzzy mappings and the extension principle
Every mapping in VH is called an H-fuzzy mapping, that is, H-fuzzy mappings
are total and univalent H-fuzzy relations. Let X, Y be crisp sets.
For f, u, v•¸VH, f is an H-fuzzy mapping from u to v if it is a mapping from
u to v in VH. An H-fuzzy mapping from X to Y is a mapping from X to Y in VH.
Obviously an H-fuzzy mapping from u to v is a total and univalent H-fuzzy
relation from u to v.
H-fuzzy mappings are characterized by equations or inequalities of the member
ship functions.
Theorem 3. If f is an H-fuzzy relation from X to Y and ƒÕ=ƒÊf is its member
ship function on X•~Y, then f is an H-fuzzy mapping from X to Y iff ƒÕ satisfies the following two conditions:
(T) •Éy•¸YƒÕ
(U) ƒÕ
Conversely, if a crisp mapping ,fr: X•~Y•¨H satisfies the two conditions, then there is an H-fuzzy mapping f from X to Y whose membership function on X•~Y
is identical with ƒÕ.
As far as we know, the above properties which characterizes fuzzy mappings are different from any other definition or characterization of fuzzy mappings (some times called as fuzzy functions, fuzzy multivalued mappings, fuzzifying functions etc.) in the literature. Since every H-fuzzy mapping is an H-fuzzy relation, Proposition 3.2 also holds for H-fuzzy mappings. We now consider fuzzy mappings induced from crisp -66- No. 1 A Natural Basis of Fuzzy Set Theory-An Overview 67
mappings and their images and inverse images.
Theorem 4. Let ƒÓ: X•¨Y be a crisp mapping. Then the check set ƒÓ is an
H-fuzzy mapping from X to Y, and the followings hold.
(1) For every A•¸VH, ƒÓ(A) is an H-fuzzy subset of Y, and for all y•¸Y
ƒÊƒÓ(A)(y)=•Éx•¸XƒÓ(x)=yƒÊA(x),
where ƒÊA, ƒÊƒÓ(A) are the membership functions on X and on Y respectively.
(2) For every B•¸VH, ƒÓ-1(B) is an H-fuzzy subset of X, and for all x•¸X
ƒÊƒÓ-1(B)(x)=ƒÊB(ƒÓ(x)),
where ƒÊB, ƒÊƒÓ-1(B) are the membership functions on Y and on X respectively.
The extension principle by Zadeh can be expressed as follows:
Every crisp map ƒÓ: X•¨Y induces a map ƒÓ:_??_X•¨_??_Y such that
ƒÓ(A)(y)=•Éx•¸ƒÓ-1(y)ƒÊA(x)(•Íy•¸Y)
for all A•¸_??_X, where _??_X={A; a fuzzy subset of X}.
Sometimes in the literature ƒÓ(A) is simply written as ƒÓ(A). It is obvious that
if A is an H-fuzzy subset of X the equation in Theorem 4 (1) coincides with that
of the extension principle.
The theorem shows that the check set ƒÓ of a crisp mapping ƒÓ is in fact a
mapping in the model (that is, an H-fuzzy mapping) and the image ƒÓ(A) is the
direct image in the model. Therefore the lifting in the extention principle by Zadeh
is considered to be a combination of two steps; the first step is to lift a crisp
mapping between crisp sets to a fuzzy mapping between them, and the second one
is to assign a fuzzy mapping to a crisp mapping which maps each fuzzy (sub) set to
its direct image in the model.
4. Concluding Remarks
Although there has been much development on fuzzy set theory and fuzzy logic especially in application area, its mathematical foundation seems to have been somewhat vague and not stable in the fuzzy literature. Our interpretation probably gives a new basis for a consistent theory which describes various definitions and properties of most of the standard notions in fuzzy set theory in a uniform way, especially from the viewpoint of extending usual set theory. In the following we deal with the relation of our results to previous studies in some aspects. We give an explicit definition of a fuzzy set, which is different from either a -67- 68 Mamoru SHIMODA Vol. 13 fuzzy subset of a crisp set or the membership function of a fuzzy set on a crisp set. Fuzzy sets and fuzzy subsets can be treated as sets, not a kind of characteristic function, in almost the same way as in usual set theory. As mentioned previously, we seldom find such a distinction in most of the fuzzy literature. Since membership functions on a crisp set X are mappings from X to H, they are a kind of L-fuzzy sets ([7]). The standard relations of equality and inclusion between fuzzy subsets and the operations min and max are also natural in the interpretation. But the complement naturally corresponds to the intuitionistic (Godel) negation, not to the ordinary (Zadeh) negation x_??_1-x. On fuzzy relation, the max-min composition seems to be most natural as an extension of composition of crisp relations. The standard definition of inverses, images, and inverse images are also natural in the interpretation. Most of the defining properties for ordinary fuzzy equivalence and order relation are also natural, but the ordinary definition of fuzzy linear order is weaker than the corre sponding definition in the model. There have been proposed several definitions of fuzzy mappings or fuzzy functions, one of which identifies fuzzy mapping with fuzzy relation (e.g., [1, 4, 18, 9, 2]). Our interpretation and characterization of fuzzy mappings seems to be unique and quite natural, which is probably different from any of preceding definitions or characterizations in the literature. Zadeh's extension principle is said to be most important in fuzzy theory, but it seems to have never been treated relating to fuzzy mappings nor have been considered as a combination of two steps. In this paper we have shown only the most basic parts of the natural interpreta tion, but we can consider natural interpretations of various notions which are not referred to here. For example, in a natural way we can define basic set operations of fuzzy subsets of difierent universes and define the inclusion relation between them, whereas most of the literature on fuzzy sets treat only fuzzy subsets of a certain set called universe. We can also define relations and mappings between fuzzy subsets of crisp sets. In addition, we can consider fuzzy equivalence rela tions, fuzzy partitions, and fuzzy groups etc. The Heyting valued model VH ([20]) is very similar to the sheaf models in [10, 26, 29, 30, 32], which are extensions of (so called Scott-Solovay) Boolean valued models for classical set theory. G. Takeuti and S. Titani [27, 28, 31] also use similar sheaf models for studying fuzzy logic and fuzzy set thoery. But in their papers the internal logic has extra axioms (and logical symbols in [31]) to express properties of the unit interval [0, 1], and the operations and relations such as set inclusion, basic set operations, and compositions of relations etc. are not treated in connection with their membership functions. As far as we know, the only precedent of our interpretation appears in H. Kodera [17], where the interpretation is applied
-68- No. 1 A Natural Basis of Fuzzy Set Theory-An Overview 69 only to elements of a group. The internal logic of our model is the intuitionistic logic with identity and existence, introduced by D. Scott [19]. The semantics for this logic using Heyting valued sets are studied in [6, 11, 8]. Relating to this logic and its extentions, U. Hohle [12, 13, 14] investigates category theoretical properties of M-valued sets and sheaves over M (M is a kind of algebra) in connection with fuzzy set theory. Lately he claims that "large parts of fuzzy set theory are actually subfields of sheaf theory" ([15]). As a category the model VH is equivalent to the category of sheaves over H and the category of H-valued sets ([20]). So our interpretation might be seen as an example of his claim from a category theoretical viewpoint. But our work is one of set-theoretic approaches, not a category theoretic approach of studying fuzzy set theory ([16]). In our model set theory can be developed in almost similar way to usual (naive) set theory. Therefore in light of developing fuzzy set theory as an extension of usual set theory, our interpretation seems to be most natural and useful.
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