An Overview of Fuzzy Quantifiers, Part 1: Interpretations

An Overview of Fuzzy Quanti ers, Part 1: Interpretations Liu, Yaxin Laboratory of Arti
cial Intelligence Department of Computer Science and Technology Peking University Beijing 100871, P.R.China Etienne E. Kerre Fuzziness and Uncertainty Modelling Research Unit Department of Applied Mathematics and Computer Science University Gent, Krijgslaan 281 (S9) B-9000 Gent, Belgium January 9, 1997 Abstract Quanti cation is an important topic in fuzzy theory and its applications. An overview is presented for quanti cation in fuzzy theory. After This work has been supported by the International Projects of the Flemish Community Cooperation with P.R.China (No.9604)

1

a brief review of quanti ers in rst order logic, two approaches of generalizing quantifers are given, the algebraic method and the substitution method. By distinguishing the fuzziness of predicates and quanti ers, various approaches to quanti cation in fuzzy logic can be organized. Quantiers in rst order logic can be generalized in crisp sense, and these generalized quanti ers can also be applied to fuzzy sets. Moreover, quanti ers themselves can be fuzzy, i.e., they can only be represented by a fuzzy set. These dierent kinds of quanti cations are identi ed. Quanti ers relate close to the concept of the cardinality of a fuzzy set, which is summarized before investigating fuzzy quanti cations. Dierent to classical logic, various semantics of propositions in fuzzy logic fall into dierent 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.

Keywords: fuzzy logic, cardinality of fuzzy sets, fuzzy quanti er, manyvalued logic, possibility distribution, numerical quanti er, OWA operator.

1 Introduction The expressive ability of
rst order logic bene
ts a lot from the universal quanti
er and the existential quanti
er, 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 quanti
ers are still not powerful enough to grasp all the quanti
cations in natural language and in logic as well. Intuitionally, quanti
ers relate to the concept of cardinality of sets, which indicates the quantity or counting number of a given set. Logical re2

searches are mainly undertaken within the framework outlined by Mostowski 16] early in the year of 1955. Since then a large number of mathematically interesting quanti
ers, known as generalized quanti
ers, are discovered and studied in two-valued logic and many-valued logics 15, 11]. Barwise and Cooper 2] motivated the study of generalized quanti
ers in linguistics 12, 23], the concept of which is di erent to that of logicians and mathematicians. Since the mid-seventies, 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 quanti
ers in natural language 29, 33, 30]. In contrast to linguists and logicians, Zadeh identi
es the quanti
ers in natural language, for example many, most, etc., as fuzzy quanti
ers with the insight that such quanti
cations are fuzzily de
ned in nature.

The examples of fuzzy quanti
cation are: \There are a lot of skyscrapers in New York", \Most students are single", \Few young men are fat". Also noticing the

relation between quanti
ers and cardinalities, Zadeh treats fuzzy quanti
ers, which relate close to the cardinality of fuzzy sets, as fuzzy numbers while distinguishing quanti
ers of the
rst kind or absolute quanti
ers from quanti
ers of the second kind or relative quanti
ers. 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 quanti
ers such as many and few can be

used in either sense, depending on the context. Generally speaking, quanti
ers in logic take the generic form of QxA(x), where Q is the quanti
er, A(x) is a predicate with a variable x, and the quanti
cation is over x. With the usual contrast of fuzzy versus crisp, we have the following table: 3

A crisp A fuzzy Q

crisp

I

II

Q

fuzzy

III

IV

Type I quanti
cations, such as \All natural numbers are real", \More than half of the countries in the world competed in the Centennial Olympic Games",

are generalized quanti
cations in the viewpoint of classical two-valued logic, usually involving cardinal numbers. Type II quanti
cations can be seen as the extensions of Type I in many-valued fuzzy logic, and the only di erence is that the quanti
ers are applied to fuzzy sets. This type of quanti
cation 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 quanti
cations involve quanti
ers which are represented by fuzzy sets, or more precisely, possibility distributions. Such quanti
cations are exempli
ed by \Almost all birds can y", \Few new pop songs can live long", etc.

As an extension of traditional logical quanti
ers, fuzzy quanti
ers are studied by various authors with similar classi
cation and assumptions to Zadeh's, but vary greatly in the interpretation and reasoning schemas. Corresponding to the classi
cation of the role of fuzzy sets in approximate reasoning made by Dubois et al. 9, 7], the studies of quanti
ers 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 quanti
ers in
rst order logic and looks into two-valued extensions. After that, before diving into quanti
cations involving fuzzy sets, we
rst examine the concept of cardinality of a fuzzy set. The subsequent sections discuss the non-fuzzy quanti
cation of fuzzy sets and fuzzy 4

quanti
cation. Reasoning with fuzzy quanti
ers and the applications of fuzzy quanti
ers are discussed in Part 2 of this paper.

2 Two-Valued Quantications

2.1 Quantications in First Order Logic First, let us recall the semantics and properties of quanti
ers in
rst-order logic. As an extension of propositional logic,
rst order logic is enriched mainly by predicates and quanti
cations. In fact, quanti
cations become natural when

predicates are introduced. Predicates enable us to describe if objects in a given set have a common property, while quanti
ers enable us to make summaries on the set without enumerate it. The complete study of
rst order logic can be found in any textbook on mathematical logic 1, 5], here we only describe those parts related to quanti
ers. The underlying language for
rst 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 :

_ ^ ! and $

quanti
ers 8 9 and auxiliary symbols of parentheses and comma. 5

De nition 2.1 A formula A is a string of the above symbols de
ned recursively: If t1 t2 : : : tn are either constants or free variables and P is an n-ary predicate symbol, P(t1 t2 : : : tn) is a formula if A ia a formula, :A is a formula if A B are formulas, A _ B A ^ B A ! B and A $ B are formulas if A is a formula, u is a free variable and x is a bounded variable which does not occur in A, then 8xAx=u] and 9xAx=u] are formulas, where

Aa=b] means the formula obtained when all occurrences of a in A are substituted by b. A sub-formula of formula A is a substring of A and itself also a formula. If a formula contains no free variables, it is also called a sentence.

Semantics for
rst order logic, or an interpretation

I

of a formula A will

consist of: a universe U of individuals assignment of a unique individual I C (a) 2 U to each constant a assignment of a unique n-ary relation I P (P n) U n to each n-ary predicate P and the truth evaluation mapping vI which maps each formula to the truth-value set f?

>g.

De nition 2.2 The truth evaluation mapping vI is de
ned recursively as: vI (P (t1 t2 : : : tn)) = > i I T (t1 ) I T (t2 ) : : : I T (tn) I P (P)

6

where

8 > < fI C (ti)g I T (ti ) = > :U

if ti is a constant if ti is a free variable

for 1 i n

vI (:A) = > i vI (A) = ? vI (A _ B) = ? i vI (A) = vI (B) = ? vI (A ^ B) = > i vI (A) = vI (B) = > vI (A ! B) = ? i vI (A) = > vI (B) = ? vI (A $ B) = > i vI (A) = vI (B) vI (8xA) = > i vI (Au=x]) = >, where u is a new free variable vI (9xA) = > i there exists a new constant a such that vI (Aa=x]) = >. In the following sections, we also denote the truth-value of a proposition P as (P) for convenience. Obviously, once an interpretation is determined, a relation on U can be derived from a formula. Furthermore, 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 de
ned by PA (x1 x2 : : : 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 8xA(x) , 8yA(y)

(1)

9xA(x) , 9yA(y)

(2)

7

8xA(x) ,

A(u) where u is a variable

(3)

9xA(x) ,

A(a) where a is a constant

(4)

8x8yA(x

y)

, 8y8xA(x

y)

(5)

9x9yA(x

y)

, 9y9xA(x

y)

(6)

:8xA(x) , 9x:A(x)

(7)

8x(A(x) ^ B(x))

, 8xA(x) ^ 8xB(x)

(8)

9x(A(x) _ B(x))

, 9xA(x) _ 9xB(x)

(9)

It is worth noticing that if the universe U is
nite, the universal and existential quanti
ers have equivalent forms in terms of logical connectives. If we enumerate the element in U as u1 u2 : : : un, and introduce new constants u1 u2 : : : un, which are always assigned as the corresponding elements in U, then we can write P(ui). Such an interpretation is slightly di erent to the above de
nition, but it is easy to verify that the underlying mapping is the same. Now, from the semantics de
ned above, we have: 8xA(x) , A(u1 ) ^ A(u2 ) ^    ^ A(un)

(10)

9xA(x) , A(u1 ) _ A(u2 ) _    _ A(un ):

(11)

and Usually, it is convenient to interpret the truth-values ? and > as real numbers 0 and 1, respectively. We will adopt this numerical interpretation in the consequent sections.

2.2 Generalization of First Order Logic Quantiers The
rst attempts to generalize quanti
ers in classical logic are the de
nitions of 9! and 9!!, which are read as \there exists exactly one" and \there exists at 8

most one" respectively, after the equality predicate  is introduced. They are

de
ned as 4 = 4 9!!xP (x) = 9!xP (x)

9x(P (x) ^ 8y(P(y) ! x  y))

(12)

8x8y(P(x) ^ P (y) ! x  y)

(13)

The essence in the above de
nition 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 quanti
ers can be classi
ed into two approaches: the substitution approach and the algebraic approach. In the substitution approach, a quanti
ed 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
nite set of n individuals and P is a predicate which has truth-values (P(ui)) for each ui 2 U. The truth-value of the quanti
ed proposition QxP (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 8 and 9 de
ned in (10) and (11). An alternative approach to investigate quanti
ed propositions is the algebraic approach. With the same assumptions as above, we interpret the quanti
ed proposition QxP (x) by associating with Q 1. a subset SQ R, 2. a function FQ FQ((P (u1)) (P(u2)) : : : (P(un))) 2 R such that (QxP (x)) = 1 9

if FQ 2 SQ

For universal quanti
er, Q = 8, the association would be: SQ = fng

FQ =

n X i=1

(P (ui)):

For existential quanti
er, Q = 9, the association is: SQ = f1 2 : : : ng FQ =

n X i=1

(P(ui)):

This approach is comparable to Mostowski's method 16], in which a quanti
er Q is equivalent to a second order binary predicate TQ , whose arguments are the cardinalities of each of the two parts of a bi-partition of the universe U given by the quanti
ed predicate P according to its truth-values at the elements of U. In this approach, the universal and existential quanti
ers are represented as: 4 T (jP >j jP ?j) = 4 jP ?j = 0 = 8 4 4 9xP (x) = T9 (jP >j jP ?j) = jP >j > 0

8xP (x)

(14) (15)

where P > = fu j u 2 U P(u) = >g and P ? = fu j u 2 U P(u) = ?g, and jAj indicates the cardinality of A. Obviously the relation between FQ and SQ is equivalent to the second order predicate TQ . In the case of two-valued logic, the de
nitions are equivalent. However in many cases, the algebraic approach gives us a briefer and easier de
nition as in the following example. For the quanti
er \a majority of", M, the de
nition given by the latter is: SM = fd n +2 1 e : : : ng FM =

n X i=1

(P (ui))

where dxe indicates the ceiling of x. The de
nition given by the substitution approach obviously is more complicated. 10

3 Cardinalities of Fuzzy Sets In this section, we mainly concentrate on
nite fuzzy sets. A fuzzy set F is
nite i the support of F is
nite. In logicians' point of view, natural numbers are
rst recognized as the cardinalities of
nite 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 de
ned based on equipotency. Equipotent fuzzy sets should have the same cardinal. The equipotency of fuzzy sets can be de
ned as follows, a special case of Wygralak's de
nition 24]:

De nition 3.1 Two fuzzy sets A, B on U are said to be equipotent i for each natural number i,

inf ft j jAtj  ig = inf ft j jBtj  ig

(16)

supft j jAtj ig = supft j jBtj ig:

(17)

or equivalently

De nition 3.2 If the i-support suppi(F ) of a fuzzy set F is de
ned as suppi (F) = ft j jFtj = ig

i2N

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

suppi (A) = suppi (B) 11

i 2 N:

(18)

3.1 Scalar Cardinalities De Luca and Termini 6] proposed the following de
nition for a scalar cardinality, named the power of a fuzzy set.

De nition 3.3 (De Luca and Termini) Let U be the universe, A be a fuzzy set de
ned on U with the membership function A : U

7! 0

1], and the support

of A, supp(A) = fu j u 2 U A(u) > 0g, is assumed to be
nite, then the power of A is de
ned as: jAj =

X u 2U

A(u):

(19)

It is easy to verify that the de
nition agrees to the equipotency standard.

Example 3.1 Fuzzy set A, B and C are de
ned on U = fa b c d e f g hg as A = B = C =

 a

b

c

d

e

f

g

h

0:1 0:3 0:6 0:6 0:9 0:7 1:0 0:2

 a

b

c

d

e

f

g

h

0:2 0:6 0:7 1:0 0:9 0:6 0:3 0:1

 a

b

c

d

e

f

g

h

! ! !

0:1 0:1 0:2 1:0 1:0 1:0 1:0 0:0

The cardinals of the sets are jAj = 0:1 + 0:3 + 0:6 + 0:6 + 0:9 + 0:7 + 1:0 + 0:2 = 4:4 jB j = 0:2 + 0:6 + 0:7 + 1:0 + 0:9 + 0:6 + 0:3 + 0:1 = 4:4 jC j = 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.

12

Obviously some properties for cardinalities of crisp sets still hold for this de
nition, with new interpretations of set operations for fuzzy sets:

Proposition 3.1 (Dubois and Prade 8]) Let A B be fuzzy sets on a universe U , then 1. A B ) jAj jB j (monotonicity), where A B is de
ned as 8x 2 U

A(x) B(x)

2. jAj = jU j ; jAj (when U
nite) (coverage property), where 8x 2 U

A(x) = 1 ; A(x)

3. jA  B j + jA \ B j = jAj + jB j (additivity), where

(A \ B)(x) = T (A(x) B(x))

(A  B)(x) = S(A(x) B(x)) 8x 2 U

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

Some suitable de
nitions for T and S are the following: S(a b) = max(a b)

T(a b) = min(a b)

(20)

S(a b) = a + b ; ab

T(a b) = ab

(21)

T(a b) = max(0 a + b ; 1):

(22)

S(a b) = min(1 a + b) If we choose S and T as (20), AB = A\B =

 a

b

c

d

e

f

g

h

0:2 0:6 0:7 1:0 0:9 0:7 1:0 0:2

 a

b

c

d

e

f

g

h

0:1 0:3 0:6 0:6 0:9 0:6 0:3 0:1 13

! !

Thus, jA  B j = 0:2 + 0:6 + 0:7 + 1:0 + 0:9 + 0:7 + 1:0 + 0:2 = 5:3 jA \ B j = 0:1 + 0:3 + 0:6 + 0:6 + 0:9 + 0:6 + 0:3 + 0:1 = 3:5 jAj + jB j = jA  B j + jA \ B j = 8:8

Proposition 3.2 (Zadeh 29]) Let F , G be fuzzy sets on U , then: max(jF j jGj) max(0 jF j + jGj ; jU j)

jF  Gj jF j + jGj

(23)

jF \ Gj

(24)

min(jF j jGj):

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

=

X u2U

(A(u))p

(25)

where p is a natural number. It is easy to
nd out that jAj0 = jsupp(A)j

jAj1

= jAj:

The property of monotonicity is still valid for jAjp, and additivity is only valid for T = min S = max, while coverage property is violated except for p = 1. Gottwald 10] has de
ned the p-power in terms of -sections. The -section of a fuzzy set A is de
ned by s (A) = fu 2 U jA(u) = g

0 <  1:

Then the following property holds: jAjp

=

X 0
0.

De nition 3.4 (Zadeh) The fuzzy cardinality of A, such that supp(A) is
nite, is denoted as jAjF , whose membership function is as follows: jAjF (n) = supf j jA j = ng

n2N

(27)

here we de
ne sup  = 0:0.

Example 3.2 For A, B, C and U are same as de
ned in Example 3.1, we have jAjF

jB jF

= =

 0 1

2

3

4

5

6

7

8



0:0 1:0 0:9 0:7 0:0 0:6 0:3 0:2 0:1 0:0

 0 1

2

3

4

5

6

7

8



0:0 1:0 0:9 0:7 0:0 0:6 0:3 0:2 0:1 0:0 15

!

!

jC jF

=

 0 1

2

3

4

5

6

7

8



!

0:0 0:0 0:0 0:0 1:0 0:2 0:0 0:1 0:0 0:0

We have jAjF = jB jF = 6 jC jF . More generally, for fuzzy cardinalities, two fuzzy sets are equipotent i they are of the same cardinal. So in the following examples, we omit B. The following property follows directly from the de
nition. It reects the property of the cardinalities of the -cuts of a fuzzy set.

Proposition 3.3 (Dubois and Prade 8]) A is a fuzzy set on a universe U , suppose f j 9u 2 U

A(u) =  0 <  < 1g = f1 2 : : : m g

where 0 = 0 < 1 <    < m < m+1 = 1. Then for 1 i m + 1 8 0
< QII (x) x a II Q; (x) = >: 1 x > a:

8> < QII (x) x  b II Q+ (x) = >: 1 x < b:

(64)

Then we can have QII

= QII; and QII+ :

where QII (x) = T (QII ; (x) QII+ (x))

with T any t-norm operator. Therefore, the overall truth-value is calculated as  = T (; + )

(65)

where ; and + are the truth-values calculated with the quanti
ers QII; and QII+ ,

respectively. 35

Besides the above interpretations, the substitution approach can also be applied 25]. The substitution approach involves the rewriting of an equivalent formula of the quanti
ed proposition in terms of a set of predicates connected by logical connectives. Suppose we investigate the proposition P:\QX's are F", where X is the universe, F is a fuzzy set on X, acting as a predicate. Let VF be the set of all logical sentences whose atomic propositions consist of the predicate F applied to an element in X. We represent a quanti
er Q by a fuzzy set SFQ on the universe VF , such that for each v 2 VF , SFQ (v) indicates the degree of possibility of v as a meaning for Q. Furthermore, for each v 2 VF , let TF (v) indicate the truth-value of v resulting from its logical structure and the predicate F . From them we can calculate the truth of the quanti
ed proposition P as: (P ) = vmax 2V min(SFQ (v) TF (v)): F

(66)

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

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