An overview of fuzzy quantifiers. (I). Interpretations

ELSEVIER Fuzzy Sets and Systems 95 (1998) 1-21

FUZZY sets and systems

Invited Review

An overview of fuzzy quantifiers. (I). Interpretations

Y a x i n L i u a'*' 1, E t i e n n e E. K e r r e b

"Laboratory of Artificial Intelligence, Department of Computer Science and Technology, Peking University, Beijing 100871, People's Republic of China

bFuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, University Gent, Krijgslaan 281 ($9), B-9000 Gent, Belgium

Received July 1997

Abstract

Quantification is an important topic in fuzzy theory and its applications. An overview is presented for quantification in fuzzy theory. After a brief review of quantifiers in first-order logic, two approaches of generalizing quantifiers are given, the algebraic method and the substitution method. By distinguishing the fuzziness of predicates and quantifiers, various approaches to quantification in fuzzy logic can be organized. Quantifiers in first-order logic can be generalized in crisp sense, and these generalized quantifiers can also be applied to fuzzy sets. Moreover, quantifiers themselves can be fuzzy, i.e., they can only be represented by a fuzzy set. These different kinds of quantifications are identified. Quantifiers relate close to the concept of the cardinality of a fuzzy set, which is summarized before investigating fuzzy quantifications. Different to classical logic, various semantics of propositions in fuzzy logic fall into different frameworks which are known as the possibility distribution-based reasoning system and the many-valued fuzzy logics. Accordingly, numerical and possibilistic interpretation explored in literature are reviewed conforming to these two frameworks. © 1998 Published by Elsevier Science B.V.

Keywords: Fuzzy logic; Cardinality of fuzzy sets; Fuzzy quantifier; Many-valued logic; Possibility distribution; Numer- ical quantifier; OWA operator

1. Introduction

The expressive ability of first-order logic benefits a lot from the universal quantifier and the existential quantifier, which enable us to make statements about properties of a class of objects without enumerating them. But it is well known to linguists and logicians that the universal and existential quantifiers are still not

*Corresponding author. Present address: College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA. 1This work has been supported by the International Projects of the Flemish Community Cooperation with People's Republic of

China (No. 9604).

0165-0114/98/$19.00 © 1998 Published by Elsevier Science B.V. All rights reserved PII S0165-01 14 (97)00254-6

2 Y. Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1-21

powerful enough to grasp all the quantifications in natural language and in logic as well. Intuitionally, quantifiers relate to the concept of cardinality of sets, which indicates the quantity or counting number of a given set. Logical researches are mainly undertaken within the framework outlined by Mostowski [16] early in the year of 1955. Since then a large number of mathematically interesting quantifiers, known as generalized quantifiers, are discovered and studied in two-valued logic and many-valued logics [15, 11]. Barwise and Cooper [2] motivated the study of generalized quantifiers in linguistics [12, 23], the concept of which is different to that of logicians and mathematicians.

Since the mid-1970s, Zadeh developed his theory of approximate reasoning [32] together with PRUF [34] based on fuzzy set theory and possibility theory, and discussed at large the importance of fuzzy quantifiers in natural language [29, 33, 30]. In contrast to linguists and logicians, Zadeh identifies the quantifiers in natural language, for example many, most, etc., as fuzzy quantifiers with the insight that such quantifications are fuzzily defined in nature. The examples of fuzzy quantification are: "There are a lot of skyscrapers in New York", "Most students are single", "Few young men are fat". Also noticing the relation between quantifiers and cardinalities, Zadeh treats fuzzy quantifiers, which relate close to the cardinality of fuzzy sets, as fuzzy numbers while distinguishing quantifiers of thefirst kind or absolute quantifiers from quantifiers of the second kind or relative quantifiers. Examples of the former ones are much more than 10, a great number of close to 100, etc., while those of the latter are most, little of about half of etc. Some quantifiers such as many and few can be used in either sense, depending on the context.

Generally speaking, quantifiers in logic take the generic form of.~xA(x), where .~ is the quantifier, A(x) is a predicate with a variable x, and the quantification is over x. With the usual contrast of fuzzy versus crisp, we have the following table:

A crisp A fuzzy

.~ crisp I II

.~ fuzzy III IV

Type I quantifications, such as "All natural numbers are real", "More than half of the countries in the world competed in the Centennial Olympic Games", are generalized quantifications in the viewpoint of classical two-valued logic, usually involving cardinal numbers. Type II quantifications can be seen as the extensions of Type I in many-valued fuzzy logic, and the only difference is that the quantifiers are applied to fuzzy sets. This type of quantification also relates to other many-valued logics, some examples are "Some professors are young", "No more than half of students are tall". Type III and IV quantifications involve quantifiers which are represented by fuzzy sets, or more precisely, possibility distributions. Such quantifications are exemplified by "Almost all birds can fly", "Few new pop songs can live long", etc.

As an extension of traditional logical quantifiers, fuzzy quantifiers are studied by various authors with similar classification and assumptions to Zadeh's, but vary greatly in the interpretation and reasoning schemas. Corresponding to the classification of the role of fuzzy sets in approximate reasoning made by Dubois et al. [9, 7], the studies of quantifiers under the topic of fuzzy sets are also undertaken within two frameworks: one follows the tradition of many-valued logics, while the other is based on possibility distribution.

The following section reviews quantifiers in first-order logic and looks into two-valued extensions. After that, before diving into quantifications involving fuzzy sets, we first examine the concept of cardinality of a fuzzy set. The subsequent sections discuss the non-fuzzy quantification of fuzzy sets and fuzzy quantification. Reasoning with fuzzy quantifiers and the applications of fuzzy quntifiers are discussed in Part 2 of this paper.

Y. Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1 21 3

2. Two-valued quantifications

2.1. Quantifications in first-order logic

First, let us recall the semantics and proper t ies of quantifiers in first-order logic. As an extension of p ropos i t iona l logic, f irst-order logic is enriched mainly by predicates and quantifications. In fact, quantif ica- t ions become natura l when predicates are introduced. Predicates enable us to describe if objects in a given set have a c o m m o n proper ty , while quantif iers enable us to make summar ies on the set wi thout enumera t ing it. The comple te s tudy of f irst-order logic can be found in any tex tbook on mathemat ica l logic [1, 5], here we only describe those parts related to quantifiers.

The underlying language for f irst-order logic consists of the following symbols: • constants a, b, c, ... ; • free variables u, v, w . . . . ; • bounded variables x, y, z . . . . ; • n-ary predicate symbols P, Q, R . . . . ; • logical connectives --7, V, A, --, and ~--~; • quantifiers V, 3; • and auxil iary symbols of parentheses and comma.

Definit ion 2.1. A formula A is a string of the above symbols defined recursively: • If t l , t2, . . . , t, are either constants or free variables and P is an n-ary predicate symbol, P(q, t2,..., t,) is

a formula; • if A is a formula, --7 A is a formula; • if A, B are formulas, A V B, A A B, A ~ B and A ~ B are formulas; • if A is a formula, u is a free var iable and x is a bound variable which does not occur in A, then VxA [x/u]

and 3xA [x/u] are formulas, where A [a/b] means the formula obta ined when all occurrences of a in A are subst i tuted by b.

A sub-formula of formula A is a substr ing of A and itself also a formula. If a formula contains no free variables, it is also called a sentence.

Semantics for f irst-order logic, or an in terpre ta t ion ~¢ of a formula A will consist of: • a universe U of individuals; • ass ignment of a unique individual j C ( a ) e U to each cons tant a; • ass ignment of a unique n-ary relat ion J P ( P " ) _c U" to each n-ary predicate P; • and the t ruth evaluat ion m a p p i n g v J which maps each formula to the t ru th-value set { ±, 3-}.

Definit ion 2.2. The t ruth evaluat ion m a p p i n g v ~' is defined recursively as:

• vJ(P(tl,t2, . . . , t , ) ) = y iff

J ( t O x JT(t~) x ... S ( t , ) _= ~P(P),

where

jr(t~) = { {uJC(t~)} if ti is a constant ,

if ti is a free variable

for 1 ~i<~n; • v~(-q A) = T iff v~'(A) = 1 ; • vJ(A V B) = I iff vJ(A) = v~(B) = ±;

4 E Liu, E.E. Kerre / Fuzzy Sets and Systems 95 H998) 1-21

• vJ(A A B) = T i f f vJ(A) = vJ(B) = T; • vJ(A ~ B ) = 3_ iff v J ( A ) = T, vJ (B)= 3_; • vJ(A ~ B) = T i f f v~(A) = vJ(B); • vJ(VxA) = Y iff v~(A[u/x]) = T, where u is a new free variable; • vJ(3xA) = Y iff there exists a new constant a such that w'(A [a/x]) = Y.

In the following sections, we also denote the truth-value of a proposit ion P as z(P) for convenience. Obviously, once an interpretation is determined, a relation on U can be derived from a formula. Further- more, note that the interpretation of a predicate is a relation, we also can claim a formula is equivalent to a predicate, i.e., for any formula A containing n free variables, an n-ary predicate Pa can be defined by PA(Xl, X 2 , . . . , Xn) ~ A. So for convenience, we can write A(u) if u is a free variable occuring in formula A. Moreover, we use the notation A(t) instead of A(u)[t/u] if A(u) is a formula containing a free variable u.

Proposition 2.1.

VxA(x) ~ . VyA(y),

~xA(x) <=> 3yA(y),

VxA(x) ~ . A(u) where u is a variable,

3xA(x) <* A(a) where a is a constant,

VxVyA(x , y) ~ VyVxA(x , y),

3x3yA(x, y) ~ 3y3xA(x, y),

--7 V x A ( x ) ~ 3x -n a(x) ,

Vx(A(x) A B(x)) <:> VxA(x) A VxB(x),

3x(A(x) V B(x)) .¢*, 3xA(x) V 3xB(x).

(i)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

It is worth noticing that if the universe U is finite, the universal and existential quantifiers have equivalent forms in terms of logical connectives. If we enumerate the element in U as u j, u2, . . . , u,, and introduce new constants ul, u2 . . . . , u, which are always assigned as the corresponding elements in U, then we can write P(ui). Such an interpretation is slightly different to the above definition, but it is easy to verify that the underlying mapping is the same. Now, from the semantics defined above, we have

VxA(x) ~ A(ul) A A(u2) A ... A A(u,,) (10)

and

3xA(x) .~ A(ul) V A(u2) V ... V A(u,). (11)

Usually, it is convenient to interpet the truth-values 3_ and T as real numbers 0 and 1, respectively. We will adopt this numerical interpretation in the consequent sections.

2.2. Generalization of first-order logic quantifiers

The first attempts to generalize quantifiers in classical logic are the definitions of 3! and 3!!, which are read as "there exists exactly one" and "there exists at most one", respectively, after the equality predicate ~ is

Y. Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1-21 5

introduced. They are defined as

3!xP(x)A3x(P(x) A V y ( P ( y ) ~ x ~ y)), (12)

?!!xP(x)~=VxVy(P(x) A P(y) ~ x ~ y). (13)

The essence in the above definition is the distinction between individuals. Obviously, more complex extensions will involve the concept of cardinality or cardinal numbers.

According to Yager [25], the extensions of quantifiers can be classified into two approaches: the substitution approach and the algebraic approach.

In the substitution approach, a quantified proposition is represented by an equivalent logical sentence. The sentence involves atoms which are instances of the predicate evaluated at the individuals in a universe U. Assume U is a finite set of n individuals and P is a predicate which has truth-values z(P(ui) ) for each ui ~ U. The truth-value of the quantified proposition ~xP(x) is the truth-value of a logical sentence only involving P(ui), thus only decided by P(ui). Examples of this kind of interpretation are V and 3 defined in (10) and (11).

An alternative approach to investigate quantified propositions is the algebraic approach. With the same assumptions as above, we interpret the quantified proposition ~xP(x) by associating with

1. a subset S~ _ N, 2. a function F~, F~(z(P(ul)), r(P(u2)) . . . . . r(P(u,))) ~ ~ such that

z(~xP(x)) = 1 if Foa~S~.

For universal quantifier, ~ = V, the association would be

S~ -- {n}, F~ = ~ z(P(ui)). i = 1

For existential quantifier, .~ = 3, the association is

Sa-= {1,2, ... ,n}, F~= ~ z(P(u,)). i = 1

This approach is comparable to Mostowski's method [16], in which a quantifier ~ is equivalent to a second-order binary predicate T~, whose arguments are the cardinalities of each of the two parts of a bi-partition of the universe U given by the quantified predicate P according to its truth-values at the elements of U. In this approach, the universal and existential quantifiers are represented as

VxP(x) ~= Tv(IP v [, I P" I) ~ I P ± I -- 0, (14)

3xP(x) & T~ (I pT l, I PZ l) A I pv I > O, (15)

where pv = {u luEU, P(u)= T} and P± = {u]ueU, P(u)= _1_}, and ]AI indicates the cardinality of A. Obviously, the relation between F~ and S~ is equivalent to the second-order predicate T~.

In the case of two-valued logic, the definitions are equivalent. However, in many cases, the algebraic approach gives us a briefer and easier definition as in the following example. For the quantifier "a majority of", M, the definition given by the latter is

where Ix] indicates the ceiling of x. The definition given by the substitution approach obviously is more complicated.


3. Cardinalities of fuzzy sets

In this section, we mainly concentrate on finite fuzzy sets. A fuzzy set F is finite iff the support of F is finite. In logicians' point of view, natural numbers are first recognized as the cardinalities of finite crisp sets. Therefore, as extensions of cardinality from crisp sets to fuzzy sets, two possibilities can be considered. One approach is to extend the set of cardinalities from natural numbers to non-negative reals, while the other extends a natural number to a fuzzy set whose universe is the set of natural numbers. These kinds of cardinalities are referred to as scalar and fuzzy cardinalities, respectively, according to Dubois and Prade [8].

In classical set theory, cardinalities are defined based on equipotency. Equipotent fuzzy sets should have the same cardinal. The equipotency of fuzzy sets can be defined as follows, a special case of Wygralak's definition [24]:

Definition 3.1. Two fuzzy sets A, B on U are said to be equipotent iff for each natural number i,

inf{t[ [At[ ~> i} = inf{t[ ]Bt[ >~ i}, (16)

sup{tllAt[ ~< i} = sup{t[[Bt[ <~ i}. (17)

or equivalently

Definition 3.2. If the / -suppor t suppi(F) of a fuzzy set F is defined as

suppi(F) = {t[[gtl = i}, i ~ Y ,

two fuzzy sets A, B on U are said to be equipotent iff

suppi(A) = suppi(B), i ~ Jff.

(18)

3.1. Scalar cardinalities

De Luca and Termini [6] proposed the following definition for a scalar cardinality, named the power of a fuzzy set.

Definition 3.3 (De Luca and Termini). Let U be the universe, A be a fuzzy set defined on U with the membership function A: U ~ [0, 1], and the support of A, supp(A) = {u]ue U, A(u) > 0}, is assumed to be finite, then the power of A is defined as

I A] = ~ A(u). (19) uEU

It is easy to verify that the definition agrees to the equipotency standard.

Example 3.1. Fuzzy set A, B and C are defined on U = {a, b, c, d, e, f, g, h} as

c e

010 0 6 0 6 0 9 0 7 1 0 k

B = 0.2 0.6 0.7 1.0 0.9 0.6 0.3 0.1 '

{ a b c d e f g h , C = ~0.1 0.1 0.2 1.0 1.0 1.0 1.0 0.0)


The cardinals of the sets are

IAI = 0.1 + 0.3 + 0.6 + 0.6 + 0.9 + 0.7 + 1.0 + 0.2 = 4.4,

IBI = 0.2 + 0.6 + 0.7 + 1.0 + 0.9 + 0.6 + 0.3 + 0.1 = 4.4,

1 C 1 = 0 . 1 + 0 . 1 + 0 . 2 + 1 . 0 + 1 . 0 + 1 . 0 + 1 . 0 + 0 . 0 = 4 . 4 .

Obviously , A and B are equipotent , and each of them is not equipotent to C, but all of them are of the same cardinality.

Obviously , some proper t ies for cardinalit ies of crisp sets still hold for this definition, with new interpretations of set opera t ions for fuzzy sets:

Proposi t ion 3.1 (Dubois and P rade [8]). Let A, B be fuzzy sets on a universe U, then 1. A ~_ B ~ I A I ~< I BI (monotonici ty) , where A ~_ B is defined as

Vx~ U, A(x) <~ B(x);

2. I AI = I U I -- I A I (when U finite) (coverage property) , where

Vx ~ U, A(x) = 1 - A(x);

3. I A u B I + I A n B I = [A[ + IBI (additivity), where

(AnB)(x) = T(A(x), B(x)), (AuB)(x) = S(A(x), B(x)), V x 6 U,

and this only holds for proper choices of definitions of T (t-norm) and S (t-conorm).

Some suitable definitions for T and S are the following:

S(a, b) = max(a , b), T(a, b) = min(a, b),

S(a, b) = a + b - ab, T(a, b) = ab,

S(a, b) = min(1, a + b), T(a, b) = max(0, a + b - 1).

If we choose S and T as (20),

( a b c d e f g h ) A u B = 0.2 0.6 0.7 1.0 0.9 0.7 1.0 0.2 '

( a b c d e f g h ) A n B = 0.1 0.3 0.6 0.6 0.9 0.6 0.3 0.1 "

Thus,

[AwB

[AnB

I Z l +

[ = 0.2 + 0.6 + 0.7 + 1.0 + 0.9 + 0.7 + 1.0 + 0.2 = 5.3,

I = 0.1 + 0.3 + 0.6 + 0.6 + 0.9 + 0.6 + 0.3 + 0.1 = 3.5,

Inl = Ihun l + IZnnl = 8 . 8 .

Proposition 3.2 (Zadeh [29]). Let F, G be fuzzy sets on U, then

max(IFl, lGI) ~< IF~GI <<. IFI + Ial,

max(0, IFI + I a l - I U I ) ~< I F n G I ~< m i n ( I F I , IGI).

(20)

(21)

(22)

(23)

(24)


The following extension of scalar cardinality, referred to as p-power, attributes to Kaufmann [13]:

IAIp = ~ (A(u)) p, (25) u~U

where p is a natural number. It is easy to find out that

IAIo = [supp(A)[, JAil = IAI.

The property of monotonicity is still valid for [ A[p, and additivity is only valid for T -- rain, S = max, while coverage property is violated except for p = 1. Gottwald [10] has defined the p-power in terms of a-sections. The a-section of a fuzzy set A is defined by

s~(A) = {ue VIA(u) = a}, 0 < a <~ 1.

Then the following property holds:

[Alp= ~ ~"'a~, (26) 0<0t~<l

where a~ = Is~(Z) l. In practice, a threshold can be applied to a fuzzy set to eliminate the accumulative effects of low

membership values. For example, in Example 3.1, if 0.3 is used as a threshold, the results should be I A I = I B I = 4.1, I C I = 4.0. Another consideration is to associate weights to each element in the universe, and calculate the cardinality as a weighted sum.

In the following sections, we will denote l a] as Z, Count(A) following Zadeh when u and n are defined as max and min, respectively.

3.2. Fuzzy cardinalities

A fuzzy cardinality of a fuzzy set is itself also a fuzzy set on the universe of natural numbers. But with the variety of interpreting natural numbers, fuzzy cardinalities of different kinds can be defined.

The first definition of fuzzy cardinality of a finite fuzzy set A is due to Zadeh [32], based on the a-cut of A,A , = {vIA(u) >~ a}, for a > 0.

Definition 3.4 (Zadeh). The fuzzy cardinality of A, such that supp(A) is finite, is denoted as [A I~, whose membership function is as follows:

Ial~(n) = sup{al [Z~] = n}, ne~Ar, (27)

here we defined sup 0 = 0.0.

Example 3.2. For A, B, C and U are same as defined as Example 3.1, we have

(000 1 2 3 4 5 6 7 8 ; i ; ) IAI~= 1.0 0.9 0.7 0.0 0.6 0.3 0.2 0.1 '

I B I s = ( 0 0 0 110 0~9 0~7 040 0~6 063 0~2 081 ; i ; ) '

(000 1 2 3 4 5 6 7 8 ; i ; ) ICl~= 0.0 0.0 0.0 1.0 0.2 0.0 0.1 0.0 "

We have I A Is = [ B Is ~ [ C I~. More generally, for fuzzy cardinalities, two fuzzy sets are equipotent iff they are of the same cardinal. So in the following examples, we omit B.


The following property follows directly from the definition. It reflects the property of the cardinalities of the e-cuts of a fuzzy set.

Proposition 3.3 (Dubois and Prade [8]). A is a fuzzy set on a universe U, suppose

{~13u~g,A(u)-- ~,0 < ~ < 1} -- {cq,~2,...,~,,},

where 0 = ao < ~1 < "'" < ~,, < c~m+l = 1. Then for 1 <. i <. m + 1, Va, O < ~ <. 1, if ai-1 < c~ <. ~, then

IAI~(IAI~) = ~.

It is easy to verify this property for the fuzzy sets defined in Example 3.1. The additivity in the case of fuzzy cardinalities should be

[AI.~®IBI~ = I A u B I ~ @ IAnBI~,

where • is defined by the extension principle. But the above definition does not pertain to this property.

Example 3.3. For simplicity, we observe fuzzy sets F and G defined on the universe {a, b, c}.

(. c) c) F = 0.2 0.8 0.3 ' 0.3 0.9 0.2 '

then

( a b c ) F n G = ( a b c ) F u G = 0.3 0.9 0.3 ' 0.2 0.8 0.2 "

We have

and

But

i.e.,

(100 1 2 3 ; i ; ) [ G l s * = ( 0 1 2 3 ; i ; ) F I~,= 0 . 8 0 . 3 0 . 2 ' 1 . 0 0 . 9 0 . 3 0 . 2

(100 1 2 3 ;i;) IFnG[~-=(0 1 2 3 ;i;) F u G I ~ = 0.9 0.0 0.3 ' 1.0 0.8 0.0 0.2 "

(100 1 2 3 4 5 6 ; i ; t F I~ (~ IGIs ,= 0.9 0.8 0.3 0.3 0.2 0.2 '

( 0 1 2 3 4 5 6 ; i ; ) FuGI.~OIFnGIs~= 1.0 0.9 0.8 0.3 0.3 0.0 0.2

FI~ O IG1~ ~ IFuGls~ G [FnGls~.

The reason is that there are "holes" in the cardinal I A[~ of the fuzzy set A if there exists u l ,u2~ U, Ux ~ u2, A(ul) = A(u2). The following definition recovers the additivity:

Definition 3.5 (Zadeh [29]). The fuzzy cardinality FGCount(A) of A is given by

FGCount(A)(n) = sup{a[ IA~I >/n}, n~A/'. (28)


Now, the cardinalities of the fuzzy sets A and C from Example 3.1 are

1.0 1.0 0.9 0.7 0.6 0.6 0.3 0.2 0.1

1.0 1.0 1.0 1.0 1.0 0.2 0.1 0.1 0.0

But the definition still does not match exactly the idea ofa cardinality. In the case of crisp sets, a cardinality of a set should be the number of elements exactly contained in the set, while it gives a set of integers (0, 1 ... . , I A [}. A more reasonable definition of a fuzzy cardinality of a fuzzy set is to use the convex hull of f A I~, of the smallest convex fuzzy set which includes I A If [8]. Wygralak [24] introduced a similar definition in a different approach. Noted as I1" II f, the cardinalities of A and C for Example 3.1 are

( 0 1 2 3 4 5 6 7 8 010) H A H f = 0.0 1.0 0.9 0.7 0.6 0.6 0.3 0.2 0.1

( 0 1 2 3 4 5 6 7 8 010) I lCl l f - - -0 .0 0.0 0.0 0.0 1.0 0.2 0.1 0.1 0.0

Another kind of fuzzy cardinality FECount was defined by Zadeh [29]:

Definition 3.6 (Zadeh). The fuzzy cardinality FECount(A) of A is given by

FECount(A)(k) = min(sup{a[ Ih, I I> k}, sup{a[ lAx-,[ ~< k}),

where n = I supp(A)l and 1 ~< k ~< n.

(29)

or equivalently (we keep Zadeh's notation FECount as above)

Definition 3.7 (Ralescu [19]). The fuzzy cardinality FECount(A) of A is given by

FECount(A)(k) = min(A(k), 1 -- A(k+ l)), 1 <~ k <~ n, (30)

where n = [ supp(A)I, and A(1), A(2), ..., A(,) are the membership degrees of elements in supp(A) arranged in non-increasing order, with A(o) = i, A(,+ 1) = 0.

Following the above definition, cardinalities of A and C from Example 3.1 are

(000 1 2 3 4 5 6 7 8 010) FECount (A)= 0.1 0.3 0.4 0.4 0.6 0.3 0.2 0.l '

(000 1 2 3 4 5 6 7 8 010) FECount(C)= 0 . 0 0 . 0 0.0 0.8 0.2 0.1 0.1 0.0 "

Proposition 3.4 (Ralescu [19]). Let A be a fuzzy set on a universe U, then we have 1. FECount(A)(k) = 1 iff A is a crisp set and I Ar = k. 2. FECount(A) is a convex fuzzy set. 3. FECount(A)(k) = FECount(A)(n -- k), n = [ supp(A)[, k = 0, 1, . . . , n.

E Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1-21 11

Ralescu [19] proposed another definition of scalar cardinality based on his definition of FECount. The motive behind is to keep the cardinals as natural numbers:

Definition 3.8 (Ralescu). The numerical cardinality [ A L~ of a fuzzy set A is defined as follows. For any k e ,At {0! I A Ix(k) = ' - 1,

A = 0 ,

A ¢ 0, A(,,) ) 0 . 5 ,

A ¢ O, A(m) < 0.5,

(31)

w h e r e m = m a x { i l A ( i 1) + A(1) > 1,1<~ i <~ n}.

3.3. Relative measures of cardinalities

In the discussions of the following sections, the relative measures of cardinalities play an important role. The relative measure of cardinalities of two fuzzy sets A and B reflects the proportion of A in B. Since we often use the ratio of the numbers of two sets to indicate such a quantity, the natural extension is to use the ratio of cardinalities of these two sets if the cardinalities are represented as scalars [29]:

XCount(Ac~B) ZCount(BIA) = , (32)

XCount(A)

here we use the same symbol as in the case of absolute cardinalities to denote relative cardinalities. This definition is widely adopted in the literature of fuzzy quantifiers because of its simplicity. The following property holds for this definition:

Proposition 3.5 (Zadeh [29]). Assume A and B are fuzzy sets on a universe U. Then, we have

XCount(BIA) + XCount(B[A) ) 1. (33)

When c~ is defined as a t-norm T, Yager [26] adopt the notation XCountr(AlB) as an extension of the above definition. The following property shows the speciality of the probabilistic product:

Proposition 3.6 (Yager [26]). Assume A and B are fuzzy sets on a universe U. Then we have 1. I f T is defined as T(a,b) = a 'b

XCountT(BIA) + XCountr(B[A) = 1;

2. For any t-norm T such that Va, b ~ [0, 1], T(a, b) ) a" b

XCountT(BIA) + XCountT(BIA) ) 1;

3. ,for any t-norm T such that Va, b ~ [0, t], T(a, b) <. a" b

XCountT(BlA) + XCountT(BIA) <<, 1.

The fuzzy relative measures are much harder to define. Dubois and Prade [8] provided a proposal to define such a measure on the set of rationals. Some definitions refer to the so-called multi-fuzzy sets [31]. Since such measures are scarcely in use, we ignore them in this paper.


4. Non-fuzzy quantification of fuzzy predicates

In this section, we mainly discuss truth-values of propositions involving generalized crisp quantifications of fuzzy predicates, while fuzzy quantifications will be discussed in the next section since fuzzily defined quantifiers more or less can be linked to possibility distributions.

4.1. Quantification of fuzzy predicates

This kind of quantifications is mainly discussed in many-valued logics tradition, and follows Mostowski's definition of quantifiers as second-order predicates of two-valued logic.

Definition 4.1 (Thiele [22]). A general fuzzy quantifier on a universe U is defined as

:o~(u) ~ [0, 11 (34)

More restrictions should be applied to this definition to obtain useful fuzzy quantifiers both in theory and in applications. In order to define these restrictions, Thiele [22] introduced some equivalence relations between arbitrary fuzzy subsets F and G on U.

Definition 4.2 (Thiele [22]). Let F, G be fuzzy sets on a universe U. Then 1. F and G are isomorphic (F-iso G) iff there exists a bijection f on U such that f (F) = G, where

f (F)(x) = V(f(x)), Vx ~ U. 2. F and G are cardinality equivalent (F~-c,ra G) iff for every real number re [0, 1], the equation

Card{x IF(x) -- r, x ~ U } = Card{xlG(x) = r, x ~ U} (35)

holds, where Card indicates the cardinality. 3. F and G are value equivalent (F --w, G) iff the equality

{v(x)lxe F} = {6(x)lxe v} (36)

holds.

Using the above definition, the following restrictions of the concept on a general fuzzy quantifier can be introduced:

Definition 4.3 (Thiele [22]). Let .~ be a fuzzy quantifier on a universe U, 1..~ is a cardinal quantifier iff for any fuzzy sets F and G on U, F -c,ra G implies .~(F) = -~(G). 2. .~ is a extensional quantifier iff for any fuzzy sets F and G on U, F -=v,t G implies .~(F) = .~(G).

Proposition 4.1 (Thiele [22]). Let .~ be a fuzzy quantifier on a universe U, 1. ~ is a cardinal quantifier iff for any fuzzy sets F and G on U, F -iso G implies .~(F) = .~(G). 2. I f ~ is an extensional quantifier, then .~ is a cardinal quantifier, but not vice versa.

4.2. Interpretation o f various quantifiers

4.2.1. V, 3 as min and max First, we invest the truth-values of V and 3 quantified propositions. Recall in the case of a finite universe,

VxA(x) is equivalent to A(xl) A A(xz) A ... A A(x,), and 3xA(x) is equivalent to A(xl) V A(x2) V ... V A(x,). If we adopt the numerical representation of truth-values, we have another form of the interpretation


mapping of a V or 3 quantified proposition:

z(VxA) = rain z(A(xi)), (37) l<.i<~n

z(3xA) = m a x z(A(xi) ). (38) l<~i<~n

The definition can be easily extended into infinite universes using inf and sup. If we keep this form of definition when truth-values are not limited to 0 and 1, a natural extension in many-valued fuzzy logic is obtained. The definition is widely accepted in literature of many-valued logics [20]. Some extension of other quantifiers based on V, 3 and other modifiers or connectives can be defined similarly.

4.2.2. t- and s-Quantifiers It is well known that min and max are special cases of more general t-norms and t-conorms, respectively.

Thiele [21-] defined his t-quantifiers and s-quantifiers based on this analogy, he replaced min and max by a t-norm and a t-conorm to define these new quantifiers. Common t-norms (t-conorms) have only two arguments, we need first to generalize a t-norm (t-conorm) to take an arbitrary number of elements as arguments.

L e t f b e any binary function [0, 132 ~--~ [0, 13 and 1. f l ( r l ) = rl; 2. f " + l ( r l , r2, ... ,r , , r,+l) = f ( f " ( r l , r2, ... ,r,), r,+l).

Thus, we can have the following definition.

Definition 4.4. Let F be a fuzzy set of U, for a given t-norm T,

V T ( F ) = in f {T"(F(x l ) , ... ,F (x , ) ) l n >>. 1 A X l , X 2 . . . . , X n ~ U } ,

similarly, for a t-conorm S,

3s(F) = sup{S"(F(xl) , . . . , F(x,)) I n >t 1 A x l , x2 . . . . , x , ~ U }.

(39)

(40)

Obviously, V = Vmin, 3 = 3ma x. It can be proved that •T satisfies the following properties:

Proposition 4.2 (Thiele [-21-]). Let T be a t-norm, • TQ0: VT is a cardinal quantifier. • TQI: For any f u z z y set F on U and every x ~ U, i f F ( y ) = l for every y ~ U with y ~ x, then VT(F) = F(x). • TQ2: For any f u z z y set F on U, if there exists an x ~ U with F(x) = O, then VT(F) = 0. • TQ3: For any f u z z y sets F and G on U, i f F ~_ G, then Vr(F) ~< Vr(G). • TQ4: For any f u z z y set F on U and every mapping f : U ~-~ U, i f f i s a bijection on U, then VT(f(F)) = VT(F),

where f (F ) ( x ) = F( f (x ) ) . • TQ5: For any f u z z y sets F and G on U and for every x, y ~ U, the equality

VT(FVT~c~I x) = VT(GVT~F°'~"x)IY)

holds, where

c, i f x = y; FClX(Y)= F(y), else.


Thiele [21] also defined the quantifiers which satisfy TQ1-5 as t-quantifiers. He proved the existence of a 1-1 correspondence between such-defined t-quantifiers and t-norms. Similar results are got for t-conorms and s-quantifiers. Obviously, t- and s-quantifiers are extensional quantifiers.

The t- and s-quantifiers have the following property:

Proposition 4.3.

VT(F) ~< V(F); (41)

3s(F) >>, 3(F). (42)

4.2.3. Novhk's generalized quantifiers Novfik [17] defined his generalized quantifiers in the context of a special case of L-fuzzy sets. Here, we

rewrite the definition for fuzzy subsets whose membership functions take values in the unit interval [0, 1], and the definition is also modified a little so as not to mention too much of Nov~ik's system and deviate from the main purpose of this paper.

Definition 4.5 (Novfik [17]). A generalized quantifier is a mapping ~ : ~([0, 1]) ~ [0, 1], which fulfills (here, we omit the parentheses to make an analogy to V and 3):

.~{a} = a , foral l ae[O, 1],

. ~ { a ® b l a e K } < . ~ K ® b ,

- n ~ { - q ( a ® b ) laeK} <~ (-q.~{--qalaeK})Q b,

VK <~ ~ K <~ 9K,

where a ® b = max(O, a + b - 1) and -qa = 1 - a, K __ [0, 1].

(43)

(44)

(45)

(46)

Definition 4.6. Let ~ be a generalized quantifier, the adjoint quantifier ~ of ~ is defined as

~ K = ~ . ~ { - n a l a e K } , for any K _ [0, 1], K # 0. (47)

It is easy to verify that the definition is meaningful. According to the definitions, V and 3 are generalized quantifiers, and are adjoint to each other.

4.2.4. Aggregation operators Aggregation operators can also be regarded as a generalized operator according to Thiele's definition.

Definition 4.7 (Klir and Yuan [14]). An aggregation operator is a mapping: h: [0, 1]" ~-~ [0, 1] satisfying at least the first three of the following conditions: 1. Boundary conditions:

h(0, ... ,0) = 0, h(1, .. . , 1) = 1;

2. Monotonicity: for any al , a 2 . . . . . an and bl, b2 . . . . . b,, al, bie[O, 1] (1 <~ i <~ n), ifai <~ bi (1 ~< i ~< n) then:

h(al, a2 . . . . ,a,) <~ h(bl,b2, ... ,b,);

3. h is a continuous function; 4. Commutativity: for any permutation p of {1, 2 , . . . , n},

h(aa, a2 ... . a,) = h(aptl), ap(2) . . . . , a~,(,));

Y. Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1 21 15

5. Idempotency: for all a e [0, 1],

h(a,a, ... ,a) = a.

For any fuzzy predicate on a finite universe, the truth-value of its quantified form can be obtained from a class of aggregation operators which only vary in n. But for an infinite fuzzy set, the definition should be extended using a limit process.

The uni-norm aggregation operators defined by Yag, er and Rybalovi [28] ae prospective members of the family of general fuzzy quantifiers in Thiele's sense.

Definition 4.8 (Yager and Rybalovi [-28]). A uni-norm R is a mapping R:[,0, 112w -* [-0, 1] having the following properties: • R(a, b) = R(b, a) (Commutativity), • if a ~< c, b ~ d, then R(a, b) <~ R(c, d) (Monotonicity), • R(a, R(b, c)) = R(R(a, b), c) (Associativity), • there exists an identity e e [0, 1], such that Vae [-0, 1], R(a, e) = a (Identity).

Also, we can extend the definition to n-ary uni-norms. From the following property, we can prove that the extension to countable infinite arguments is meaningful applying a limit process:

Proposit ion 4.4 (Yager and Rybalovi [-28]). Assume R is a uni-norm with identity e, then we have

R(a~, a2 . . . . ,a.) >~ R ( a l , a 2 . . . . . a , ,a .+l) , i f a ,+l < e, (48)

R(a l , a2 . . . . ,a,) >~ R ( a l , a 2 . . . . . a , ,a ,+l) , / f a ,+ l > e. (49)

/~, the dual of R, can be defined by

/~(a, b) = 1 - R(ci,/~) (50)

with identity & where ~ -- 1 - e.

5. Fuzzy quantifications

5.1. Possibility distributions as quantifiers

Using the term f u z z y quantifications, we mean that the quantifiers themselves are fuzzy, in another word, the quantifiers are represented as fuzzy sets. Different to propositions in traditional meaning and notation, the objects mainly investigated here are so-called canonical forms of"There are ~ A's" and "~ A's are B's". In the following sections, the cardinalities involved are finite ones.

These kinds of quantifiers are usually represented as possibility distributions. In daily life, we use both natural numbers and percentages to refer to the quantity of a given set. Analogically, there are two kinds of quantifiers related to possibility distributions defined on different universes.

Obviously, propositions of the form "There are ~ A's" relate to the so-called absolute or first-kind quantifiers, noted as 2 I, which are looked as possibility distributions of cardinalities of fuzzy sets, while propositions of the form "~A's are B's" relate to relative or second-kind quantifiers, noted as ~" , which are interpreted as possibility distributions of the relative measure or proportion of cardinalities of fuzzy sets.

Since the cardinality of a fuzzy set can be a non-negative real number or a fuzzy set on the universe of natural numbers, the possibility distribution related to absolute quantifiers can be either a distribution over the universe of non-negative real numbers or a high-order distribution over the universe of fuzzy sets on the

16 E Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1-21

universe of natural numbers. In his approach, Zadeh [32, 29] represents the desired distribution as on the non-negative real universe, for he mainly adopts the corresponding definition of cardinalities (Definition 3.3) for fuzzy sets. We also follow this approach. Thus, in the below, both kinds of quantifiers are represented by normal convex closed fuzzy sets, but they are identified by the underlying universe: a quantifier of the first kind is on the universe of the non-negative reals ~ + w {0}, while a quantifier of the second kind is on the universe of unit interval [0, 1]. For convenience, we would refer to both of these fuzzy sets as fuzzy numbers, as long as they are subsets of fuzzy numbers which are defined as normal convex closed fuzzy sets on ~ . Furthermore, the extension principle can be used to define their arithmetic.

In this kind of interpretation, the classical quantifiers V and 3 are degenerated fuzzy sets {1} and {x]0 < x ~< 1}, respectively.

The quantifiers of the second kind are discussed more thoroughly in the literature, here we define some special sub-categories of these quantifiers:

Definition 5.1 (Yager [27] and Zadeh [29]). Let .~H be a fuzzy quantifier of the second kind. Then 1. ~ is regular non-decreasing if

• . ~ " ( 0 ) = 0, • . ~ n ( 1 ) = 1, • if X 1 > X2, then ,,~II(xl) ~ ..~II(x2).

2. ~H is regular non-increasing if • ~ n ( O ) = 1,

• ~ H ( 1 ) = 0, • if xl > X2, then .~n(xa) ~< ~II(x2).

3..~*~ is regular unimodal if for some 0 ~< a ~< b ~< 1, • . ~ " ( 0 ) = ~ n ( 1 ) = 0, • ~ n ( x ) = l , fora~<x~<b, • if xl < x2 ~< a, then .~n(xl) <<. ~n(x2), • if b ~< xl < x2, then ~n(Xx) ~> .~II(x2).

4. ant .~H, the antonym of ~x~, is defined by (ant ~n)(x) = ~ H ( 1 - x), 0~<x~<l . (51)

The definitions are illustrated in Fig. 1. The interpretation of propositions of such forms falls into two distinct categories: possibilistic interpreta-

tion interprets the extension of a proposition as a possibility distribution, while numerical interpretation looks at the extensions of a proposition as a real number indicating truth value, possibility or certainty of the proposition.

5.2. Possibilistic interpretation

This approach is mainly developed by Zadeh in his fuzzy linguistic logic or theory of approximate reasoning with the aid of PRUF [34]. The following quotation reflects his idea:

If p is an expression in a natural language and P is its translation in PRUF, that is,

p ~ P ,

then the procedure P may be viewed as defining the meaning, M(P), of p, with the possibility distribution computed by P constituting the information, I(P), conveyed by p. (The procedure defined by an expression in PRUF and the possibility distribution which it yields are analogous to the intension and extension of a predicate in two-valued logic.) [34]


1.0

I t

0.5 1.0

1.0

(a) (b) 0.5 1.0

1.0

m-

0.5 1.0

1.0

(c) (d)

. . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . , . ° . , , , , ° 0 ° ~ . . . . . . .

0.5 1.0

Fig. 1. Special cases of relative quantifiers. (a) Regular non-decreasing quantifier; (b) regular non-increasing quantifier; (c) regular unimodal quantifier; (d) a quantifier and its antonym.

Propositions with canonical forms are represented in the following way:

There are ~*A's - Card(A) is ,~I, (52)

~HA's are B's - Prop(BIA) is 2 u, (53)

where Prop(BIA) indicates the relative measure of cardinality of B in A. In the following passages, we assume the cardinalities are defined as ZCount.

The right-hand side of the above two equivalences can be translated into possibility distributions as common propositions:

Card(A) is .~l ~ T[Card(A ) = o,~I, (54)

Prop(BIA) is ,~" ~ ~Prop(BlA ) = A I I . (55)

Therefore, the resulted possibility distribution shows the consistency of the available data (Card(A) or Prop(B[A)) with the quantifiers (~i or .~u). So this kind of interpretation is completely different to the well-established logical understanding of the extension of a proposition as a truth-value. The truth-value of possibilistically interpreted propositions is also a possibility distribution over [0, 1], denoted as u-true 1-34]:

u- t rue( t )= t, t~[0 ,1] ,

while the propositions can be qualified by another truth-value v, in which case the truth-value of the proposition is z.

5.3. Numerical interpretations

More easy-to-use interpretations are numerical interpretations. This class of interpretations can be recognized as truth-value interpretations and possibility/certainty interpretations [19]. Here we mainly discuss the truth-value interpretations. Other interpretations [18] relate to the possibility/certainty measure,

18 Y. Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1~1

and more generally, fuzzy measure theory. Some recent developments [3, 4] involve Sugeno and Choquet fuzzy integrals.

Truth-value interpretation keeps the tradition that the extension of a proposition is a truth-value. Similar to many-valued logic, the truth-values are taken from the unit interval. Usually, the truth-value of a given proposition is calculated based on the membership function of the quantifier.

The simplest way to calculate the truth-value is to use cardinalities, that is [35]

z(There are SA's) = ~1(ZCount(A)), (56)

~(~HA's are B's) = ~H(~Count(BlA)). (57)

Recall Yager's classification of defining segments, this approach is an algebraic one. Yager [27] proposed to use OWA operators to calculate the truth values when the proposition takes the

form of "~HU's are A's", where U is the universe concerned. OWA operators, or Ordered Weighted Averaging operators, are defined as follows:

Definition 5.2. An OWA operator of dimension n is a mapping

f : ~ " ~

which has an associated n-vector W

W = (Wl, w2, . . . , w,)

such that

~ w i = l , Vl<<.i<~.n, wi~[O, 1]. (58) i = 1

Furthermore,

f (a l ,a2 .. . . , a,) = ~ wj. b j, (59) j = l

where bj is the jth largest of al, a2, ..., a,. For a regular non-decreasing quantifier ~n, an associated OWA operator f~,, is defined with

The definition is meaningful because of the boundary conditions of a regular quantifier. Thus, the truth-value is

= f~,,(aa, a2, .. . , a,). (61)

The truth-value of a proposition related to a regular non-increasing quantifier is defined with the aid of its antonym. Instead of considering the proposition

31tU's a r e A's,

we investigate the equivalent proposition

(ant .~H)U's are A's.

E Liu, E.E. Kerre / Fuzzy Sets and Systems 95 (1998) 1-21 19

Thus following the above trick, we obta in

Since (ant ~II)(r) = .~tI(1 - r), that is

Recall A(t) = 1 - A(t), we have

z =f~,,(1 - a l , 1 - az, . . . , 1 -- a,). (63)

Decompos i t i on is appl ied to a un imoda l quantif ier to get an overall t ruth-value. Not ice that a regular un imoda l quantif ier ~ x is non-decreas ing at the left-hand side of a, and non-increasing at the r ight -hand side of b, therefore, it can be represented by a regular non-decreasing quantifier .~ i and a regular non-increasing quantif ier ~ I (2 ) . These two quantif iers (Fig. 2) are defined as follows:

~ ( x ) = ~ ~H(x) ' x ~< a, ~.~(x) = ~ ~H(x) ' x >~ b, (64) ( 1, x > a . ( 1 , x < b .

Then we can have

.~ii = 02H and . ~ ,

where

~H(x) = T ( ~ ( x ) , ~ ( x ) )

with T any t -norm operator . Therefore, the overall t ru th-value is calculated as

z = T ( r _ , z + ) , (65)

where z_ and z + are the t ruth-values calculated with the quantifiers . ~ and ~ , respectively.

1.0-

Q"-

- - - . . . . . . . . . . . ° . . . . . . . . . . . . .

I

0.5 1.0

Fig. 2. Decomposition of unimodal quantifier.


Besides the above interpretations, the substitution approach can also be applied [25]. The substitution approach involves the rewriting of an equivalent formula of the quantified proposition in terms of a set of predicates connected by logical connectives. Suppose we investigate the proposition P: ".~X's are F", where X is the universe, F is a fuzzy set on X, acting as a predicate. Let Vv be the set of all logical sentences whose atomic propositions consist of the predicate F applied to an element in X. We represent a quantifier .~ by a fuzzy set SF, a on the universe Vr, such that for each v ~ VF, SF,~(v) indicates the degree of possibility of v as a meaning for .~. Furthermore, for each v e VF, let Tr(v) indicate the truth-value of v resulting from its local structure and the predicate F. From them we can calculate the truth of the quantified proposition P as:

z(P) = max min(SF,~(v), Te(v)). (66) v~Vr

And for practical use, we should point out that the maximum can be equivalently obtained over the support of SF, ~.

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[28] R.R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996) 111-120. [-29] L.A. Zadeh, A computational approach to fuzzy quantifiers in natural languages, Comput. Math. Appl. 9 (1983) 149-184. 130] L.A. Zadeh, A theory of commonsense knowledge, in: H.J. Skala et al. (Eds.), Aspects of Vagueness, Reidel, Dordrecht, 1984,

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An overview of fuzzy quantifiers. (I). Interpretations

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