An Overview of Fuzzy Quantifiers, Part 1: Interpretations

An Overview of Fuzzy Quanti�ers� Part ��

Interpretations

Liu� Yaxin�

Laboratory of Arti�cial Intelligence

Department of Computer Science and Technology

Peking University

Beijing �� P�R�China

Etienne E� Kerre

Fuzziness and Uncertainty Modelling Research Unit

Department of Applied Mathematics and Computer Science

University Gent� Krijgslaan �� S

B�� Gent� Belgium

January � ��

Abstract

Quanti�cation is an important topic in fuzzy theory and its applica�

tions� 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��

�

a brief review of quanti�ers in �rst order logic� two approaches of gen�

eralizing quantifers are given� the algebraic method and the substitution

method� By distinguishing the fuzziness of predicates and quanti�ers� var�

ious approaches to quanti�cation in fuzzy logic can be organized� Quanti�

�ers in �rst order logic can be generalized in crisp sense� and these gener�

alized 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 di�erent 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� Di�erent to classical logic� var�

ious semantics of propositions in fuzzy logic fall into di�erent frameworks

which are known as the possibility distribution based reasoning system

and the many�valued fuzzy logics� Accordingly� numerical and possibilis�

tic interpretation explored in literature are reviewed conforming to these

two frameworks�

Keywords� fuzzy logic� cardinality of fuzzy sets� fuzzy quanti�er� many�

valued logic� possibility distribution� numerical quanti�er� OWA operator�

� Introduction

The expressive ability of �rst order logic bene�ts a lot from the universal quan�

ti�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 re�

�

searches are mainly undertaken within the framework outlined by Mostowski ��

early in the year of �� Since then a large number of mathematically inter�

esting quanti�ers� known as generalized quanti�ers� are discovered and studied

in two�valued logic and many�valued logics �� Barwise and Cooper ��

motivated the study of generalized quanti�ers in linguistics �� the concept

of which is di�erent to that of logicians and mathematicians�

Since the mid�seventies� Zadeh developed his theory of approximate rea�

soning �� together with PRUF �� based on fuzzy set theory and possibility

theory� and discussed at large the importance of fuzzy quanti�ers in natural

language �� In contrast to linguists and logicians� Zadeh identi�es the

quanti�ers in natural language� for example many� most� etc�� as fuzzy quan�

ti�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 distin�

guishing 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 �� a great number of� close to �� 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 quan�

ti�cation is over x� With the usual contrast of fuzzy versus crisp� we have the

following table�

�

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� �� 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

quanti�cation� Reasoning with fuzzy quanti�ers and the applications of fuzzy

quanti�ers are discussed in Part � of this paper�

� Two�Valued Quanti�cations

�� Quanti�cations 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 �� here we only describe those

parts related to quanti�ers�

The underlying language for �rst order logic consists of the following sym�

bols�

� 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 ��

� and auxiliary symbols of parentheses and comma�

De�nition �� A formulaA is a string of the above symbols de�ned recursively�

� If t�� t�� tn are either constants or free variables and P is an n�ary

predicate symbol� P �t�� t�� 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 �xA�x�u� and �xA�x�u� are formulas� where

A�a�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 IC�a� � U to each constant a�

� assignment of a unique n�ary relation IP �Pn� Un to each n�ary predi�

cate P �

� and the truth evaluation mapping vI which maps each formula to the

truth�value set f��g�

De�nition �� The truth evaluation mapping vI is de�ned recursively as�

� vI�P �t�� t�� tn�� i

IT �t�� IT �t�� IT �tn� IP �P ��

�

where

IT �ti� �

��fIC�ti�g� if ti is a constant�

U� if ti is a free variable

for � 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��xA� � � i vI�A�u�x�� where u is a new free variable�

� vI��xA� � � i there exists a new constant a such that vI �A�a�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�x�� x�� xn�� A� So for convenience� we can write

A�u� if u is a free variable occuring in formulaA� 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 ��

�xA�x� � �yA�y��

�xA�x� � �yA�y��

�

�xA�x� � A�u� where u is a variable� ��

�xA�x� � A�a� where a is a constant� � �

�x�yA�x� y� � �y�xA�x� y��

�x�yA�x� y� � �y�xA�x� y��

��xA�x� � �x�A�x��

�x�A�x� �B�x�� xA�x� � �xB�x��

�x�A�x� �B�x�� xA�x� � �xB�x� ��

It is worth noticing that if the universe U is �nite� the universal and ex�

istential quanti�ers have equivalent forms in terms of logical connectives� If

we enumerate the element in U as u�� u�� un� and introduce new constants

u�� u�� 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�

�xA�x�� A�u�� A�u�� A�un� ��

and

�xA�x�� A�u�� A�u�� A�un��

Usually� it is convenient to interpret the truth�values and � as real num�

bers � and �� respectively� We will adopt this numerical interpretation in the

consequent sections�

�� Generalization of First Order Logic Quanti�ers

The �rst attempts to generalize quanti�ers in classical logic are the de�nitions

of �� and �� which are read as �there exists exactly one� and �there exists at

�

most one� respectively� after the equality predicate � is introduced� They are

de�ned as

��xP �x�� x�P �x� � �y�P �y� � x � y��

��xP �x�� x�y�P �x� � P �y�� x � y� ��

The essence in the above de�nition is the distinction between individuals� Obvi�

ously� more complex extensions will involve the concept of cardinality or cardinal

numbers�

According to Yager �� 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 � 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 � and � de�ned in �� and

��

An alternative approach to investigate quanti�ed propositions is the alge�

braic approach� With the same assumptions as above� we interpret the quanti�

�ed proposition QxP �x� by associating with Q

�� a subset SQ R�

�� a function FQ� FQ�� P �u�� P �u�� P �un�� R such that

� �QxP �x�� if FQ � SQ

For universal quanti�er� Q � �� the association would be�

SQ � fng� FQ �nXi��

� �P �ui��

For existential quanti�er� Q � �� the association is�

SQ � f�� ng� FQ �nXi��

� �P �ui��

This approach is comparable to Mostowski�s method �� in which a quan�

ti�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�

�xP �x�� T��jP�j� jP�j�

�� jP�j � ��

�xP �x�� T��jP�j� jP�j�

�� jP�j � ��

where P� � fu ju � U�P �u� � �g and P� � fu ju � 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 � fdn� �

�e� � � � � ng� FM �

nXi��

� �P �ui��

where dxe indicates the ceiling of x� The de�nition given by the substitution

approach obviously is more complicated�

��

� 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 ��

In classical set theory� cardinalities are de�ned based on equipotency� Equipo�

tent 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 ��

De�nition �� Two fuzzy sets A� B on U are said to be equipotent i for each

natural number i�

infft j jAtj � ig � infft j jBtj � ig� ��

supft j jAtj ig � supft j jBtj ig� ��

or equivalently

De�nition �� If the i�support suppi�F � of a fuzzy set F is de�ned as

suppi�F � � ft j jFtj � ig� i � N � ��

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

suppi�A� � suppi�B�� i � N �

��

�� Scalar Cardinalities

De Luca and Termini �� proposed the following de�nition for a scalar cardinality�

named the power of a fuzzy set�

De�nition �� De Luca and Termini� Let U be the universe� A be a fuzzy

set de�ned on U with the membership function A � U �� and the support

of A� supp�A� � fu ju � U�A�u� � �g� is assumed to be �nite� then the power

of A is de�ned as�

jAj �Xu�U

A�u��

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

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

A �

�a b c d e f g h

��

�

B �

�a b c d e f g h

��

�

C �

�a b c d e f g h

��

�

The cardinals of the sets are

jAj � ��

jBj � ��

jCj � ��

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 properties for cardinalities of crisp sets still hold for this

de�nition� with new interpretations of set operations for fuzzy sets�

Proposition �� Dubois and Prade �� Let A�B be fuzzy sets on a uni�

verse U � then

�� A B � jAj jBj monotonicity�� where A B is de�ned as

�x � U� A�x� B�x��

�� jAj � jU j � jAj when U �nite� coverage property�� where

�x � U� A�x� � �� A�x��

� jA �Bj� jA �Bj � jAj� jBj additivity�� where

�A �B��x� � T �A�x�� B�x�� A �B��x� � S�A�x�� B�x�� x � 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��

S�a� b� � a� b� ab� T �a� b� � ab� ��

S�a� b� � min�� a� b�� T �a� b� � max�� a� b� ��

If we choose S and T as ��

A �B �

�a b c d e f g h

��

�

A �B �

�a b c d e f g h

��

�

��

Thus�

jA �Bj � ��

jA �Bj � ��

jAj� jBj � jA �Bj� jA�Bj � ��

Proposition �� Zadeh �� Let F � G be fuzzy sets on U � then�

max�jF j� jGj� jF �Gj jF j� jGj� ��

max�� jF j� jGj � jU j� jF �Gj min�jF j� jGj��

The following extension of scalar cardinality� referred to as p�power� at�

tributes to Kaufmann ��

jAjp �Xu�U

�A�u��p� ��

where p is a natural number� It is easy to �nd out that

jAj� � jsupp�A�j� jAj� � 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 � ��

Gottwald �� has de�ned the p�power in terms of ��sections� The ��section of

a fuzzy set A is de�ned by

s��A� � fu � U jA�u� � �g� � � � ��

Then the following property holds�

jAjp �X

��

�p � a��

where a� � js��A�j�

�

In practice� a threshold can be applied to a fuzzy set to eliminate the ac�

cumulative e�ects of low membership values� For example� in Example �� if

�� is used as a threshold� the results should be jAj � jBj � �� jCj � ��

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 jAj as �Count�A� following Zadeh

when � and � are de�ned as max and min� respectively�

�� 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 di�erent kinds can be de�ned�

The �rst de�nition of fuzzy cardinality of a �nite fuzzy set A is due to

Zadeh �� based on the ��cut of A� A� � fu jA�u� � �g� for � � ��

De�nition �� 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� n � N � ��

here we de�ne sup � � ��

Example �� For A� B� C and U are same as de�ned in Example �� we have

jAjF �

��

��

�

jBjF �

��

��

�

�

jCjF �

��

��

�

We have jAjF � jBjF �� jCjF� 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 re�ects the

property of the cardinalities of the ��cuts of a fuzzy set�

Proposition �� Dubois and Prade �� A is a fuzzy set on a universe U �

suppose

f� j �u � U�A�u� � �� g � f�� mg�

where � � �� m � �m�� Then for � i m � ��

� �� if �i�� i� then

jAjF�jA�j� � �i�

It is easy to verify this property for the fuzzy sets de�ned in Example ��

The additivity in the case of fuzzy cardinalities should be

jAjF � jBjF � jA�BjF � jA �BjF �

where � is de�ned by the extension principle� But the above de�nition does

not pertain this property�

Example �� For simplicity� we observe fuzzy sets F and G de�ned on the

universe fa� b� cg�

F �

�a b c

��

�� G �

�a b c

��

�

��

then

F �G �

�a b c

��

�� F �G �

�a b c

��

��

We have

jF jF �

��

��

�� jGjF �

��

��

�

and

jF�GjF �

��

��

�� jF�GjF �

��

��

��

But

jF jF � jGjF �

��

��

��

jF �GjF � jF �GjF �

��

��

��

i�e��

jF jF � jGjF �� jF �GjF � jF �GjF �

The reason is that there are �holes� in the cardinal jAjF of the fuzzy set

A if there exists u�� u� � U� u� �� u�� A�u�� A�u�� The following de�nition

recovers the additivity�

De�nition �� Zadeh �� The fuzzy cardinality FGCount�A� of A is given

by�

FGCount�A��n� � supf� j jA�j � ng� n � N ��

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

FGCount�A� �

��

��

�

��

FGCount�C� �

��

��

��

But the de�nition still does not match exactly the idea of a 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 f�� jAjg� A

more reasonable de�nition of a fuzzy cardinality of a fuzzy set is to use the

convex hull of jAjf � or the smallest convex fuzzy set which includes jAjf ��

Wygralak �� introduced a similar de�nition in a di�erent approach� Noted as

jj�jjf � the cardinalities of A and C for Example �� are

jjAjjf �

��

��

�

jjCjjf �

��

��

��

Another kind of fuzzy cardinality FECount was de�ned by Zadeh ��

De�nition �� Zadeh� The fuzzy cardinality FECount�A� of A is given by�

FECount�A��k� � min�supf� j jA�j � kg� supf� j jA��j kg��

where n � jsupp�A�j and � k n�

or equivalently �we keep Zadeh�s notation FECount as above�

De�nition �� Ralescu �� The fuzzy cardinality FECount�A� of A is given

by�

FECount�A��k� � min�A�k�� A�k�� k n� ��

where n � jsupp�A�j� and A�� A�� A�n� are the membership degrees of

elements in supp�A� arranged in non�increasing order� with A�� A�n��

��

��

Following the above de�nition� cardinalities of A and C from Example ��

are

FECount�A� �

��

��

�

FECount�C� �

��

��

��

Proposition �� Ralescu �� Let A a fuzzy set on a universe U � then we

have

�� FECount�A��k� � � i A is a crisp set and jAj � k�

�� FECount�A� is a convex fuzzy set�

� FECount�A��k� � FECount�A��n � k�� n � jsupp�A�j� k � �� n�

Ralescu �� proposed another de�nition of scalar cardinality based on his

de�nition of FECount� The motive behind is to keep the cardinals as natural

numbers�

De�nition �� Ralescu� The numerical cardinality jAjN of a fuzzy set A is

de�ned as follows� For any k � N

jAjN �k� �

��

�� A � �

m� A �� A�m� � ��

m � �� A �� A�m� � ��

� ��

where m � maxfi jA�i�� A�i� � �� i ng�

�� 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 re�ects 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 ��

�Count�BjA� ��Count�A �B�

�Count�A��

here we use the same symbol as in the case of absolute cardinalities to de�

note relative cardinalities� This de�nition is widely adopted in the literature of

fuzzy quanti�ers because of its simplicity� The following property holds for this

de�nition�

Proposition �� Zadeh �� Assume A and B are fuzzy sets on a universe

U � Then we have�

�Count�BjA� � �Count�BjA� � ��

When � is de�ned as a t�norm T � Yager �� adopt the notation �CountT �AjB�

as an extension of the above de�nition� The following property shows the spe�

cialty of the probabilistic product�

Proposition �� Yager �� Assume A and B are fuzzy sets on a universe

U � Then we have�

�� if T is de�ned as T �a� b� � a � b

�CountT �BjA� � �CountT �BjA� � ��

�� for any t�norm T such that �a� b � �� T �a� b�� a � b

�CountT �BjA� � �CountT �BjA� � ��

��

� for any t�norm T such that �a� b � �� T �a� b� a � b

�CountT �BjA� � �CountT �BjA� ��

The fuzzy relative measures are much harder to de�ne� Dubois and Prade ��

provided a proposal to de�ne such a measure on the set of rationals� Some

de�nitions refer to the so�called multi�fuzzy sets �� Since such measures are

scarcely in use� we ignore them in this paper�

� Non�Fuzzy Quanti�cation of Fuzzy Predicates

In this section� we mainly discuss truth�values of propositions involving gener�

alized crisp quanti�cations of fuzzy predicates� while fuzzy quanti�cations will

be discussed in the next section since fuzzily de�ned quanti�ers more or less can

be linked to possibility distributions�

�� Quanti�cation of Fuzzy Predicates

This kind of quanti�cations is mainly discussed in many�valued logics tradition�

and follows Mostowski�s de�nition of quanti�ers as second order predicates of

two�valued logic�

De�nition �� Thiele �� A general fuzzy quanti�er on a universe U is

de�ned as�

Q � F�U � ��

More restrictions should be applied to this de�nition to obtain useful fuzzy

quanti�ers both in theory and in applications� In order to de�ne these restric�

tions� Thiele �� introduced some equivalence relations between arbitrary fuzzy

subsets F and G on U �

��

De�nition �� Thiele �� Let F�G be fuzzy sets on a universe U � Then

�� F and G are isomorphic F�isoG� i there exists a bijection f on U such

that f�F � � G� where f�F ��x� � F �f�x�� x � U �

�� F and G are cardinality equivalent F�cardG� i for every real number

r � �� the equation

Cardfx jF �x� � r� x � Ug � Cardfx jG�x� � r� x � Ug ��

holds� where Card indicates the cardinality�

� F and G are value equivalent F�valG� i the equality

fF �x�jx � Ug � fG�x�jx � Ug ��

holds�

Using the above de�nition� the following restrictions of the concept on a

general fuzzy quanti�er can be introduced�

De�nition �� Thiele �� Let Q be a fuzzy quanti�er on a universe U �

�� Q is a cardinal quanti�er i for any fuzzy sets F and G on U � F�cardG

implies Q�F � � Q�G��

�� Q is an extensional quanti�er i for any fuzzy sets F and G on U � F�valG


Proposition �� Thiele �� Let Q be a fuzzy quanti�er on a universe U �

�� Q is a cardinal quanti�er i for any fuzzy sets F and G on U � F�isoG


�� If Q is an extensional quanti�er� then Q is a cardinal quanti�er� but not

vice versa�

��

�� Interpretation of Various Quanti�ers

�� as min and max

First� we invest the truth�values of � and � quanti�ed propositions� Recall in

the case of a �nite universe� �xA�x� is equivalent to A�x��A�x�� A�xn��

and �xA�x� is equivalent to A�x��A�x�� A�xn�� If we adopt the numer�

ical representation of truth�values� we have another form of the interpretation

mapping of a � or � quanti�ed proposition�

� ��xA� � min��i�n

� �A�xi��

� ��xA� � max��i�n

� �A�xi��

The de�nition can be easily extended into in�nite universes using inf and sup�

If we keep this form of de�nition when truth�values are not limited to � and

�� a natural extension in many�valued fuzzy logic is obtained� The de�nition

is widely accepted in literature of many�valued logics �� Some extension of

other quanti�ers based on �� and other modi�ers or connectives can be de�ned

similarly�

�� t� and s�Quanti�ers

It is well known that min and max are special cases of more general t�norms and

t�conorms� respectively� Thiele �� de�ned his t�quanti�ers and s�quanti�ers

based on this analogy� he replaced min and max by a t�norm and a t�conorm

to de�ne these new quanti�ers� Common t�norms �t�conorms� have only two

arguments� we need �rst to generalize a t�norm �t�conorm� to take an arbitrary

number of elements as arguments�

Let f be any binary function �� and

��

�� f��r�� r��

�� fn��r�� r�� rn� rn�� f�fn�r�� r�� rn�� rn��

Thus� we can have the following de�nition�

De�nition �� Let F be a fuzzy set on U � for a given t�norm T �

�T �F � � inffTn�F �x�� F �xn�� jn � � � x�� x�� xn � Ug� ��

similarly� for a t�conorm S�

�S�F � � supfSn�F �x�� F �xn�� jn � � � x�� x�� xn � Ug� � ��

Obviously� � � �min� � � �max�

It can be proven that �T satis�es the following properties�

Proposition �� Thiele �� Let T be a t�norm�

� TQ�� T is a cardinal quanti�er�

� TQ�� For any fuzzy set F on U and every x � U � if F �y� � � for every

y � U with y �� x� then �T �F � � F �x��

� TQ�� For any fuzzy set F on U � if there exists an x � U with F �x� � ��

then �T �F � � ��

� TQ � For any fuzzy sets F and G on U � if F G� then �T �F � �T �G��

� TQ�� For any fuzzy set F on U and every mapping f � U �� U � if f is a

bijection on U � then �T �f�F �� T �F �� where f�F ��x� � F �f�x��

� TQ�� For any fuzzy sets F and G on U and for every x� y � U � the equality

�T �F�T �G�jx� � �T �G

�T �FG�y�jx�jy�

�

holds� where

F cjx�y� �

��

c� if x � y�

F �y�� else�

Thiele �� also de�ned the quanti�ers which satisfy TQ�� as t�quanti�ers�

He proved the existance of a �� correspondence between such�de�ned t�quanti�ers

and t�norms� Similar results are got for t�conorms and s�quanti�ers� Obviously�

t� and s�quanti�ers are extensional quanti�ers�

The t� and s�quanti�ers have the following property�

Proposition ��

�T �F � ��F ��

�S �F � � ��F ��

�� Nov�ak�s Generalized Quanti�ers

Nov�ak �� de�ned his generalized quanti�ers in the context of a special case of

L�fuzzy sets� Here� we rewrite the de�nition for fuzzy subsets whose membership

functions take values in the unit interval �� and the de�nition is also modi�ed

a little so as not to mention too much of Nov�ak�s system and deviate from the

main purpose of this paper�

De�nition �� Nov�ak �� A generalized quanti�er is a mapping Q � P��

�� which ful�lls here� we omit the parentheses to make an analogy to � and

��

Qfag � a� for all a � ��

Qfa� b j a � Kg QK � b� � �

�Qf��a� b� j a � Kg ��Qf�a j a � Kg�� b� � �

�

�K QK �K� � ��

where a� b � max�� a� b� �� and �a � �� a� K ��

De�nition �� Let Q be a generalized quanti�er� the adjoint quanti�er �Q of Q

is de�ned as�

�QK � �Qf�a j a � Kg� for any K �� K ��

It is easy to verify that the de�nition is meaningful� According to the de�nitions�

� and � are generalized quanti�ers� and are adjoint to each other�

�� Aggregation Operators

Aggregation operators can also be regarded as a generalized operator according

to Thiele�s de�nition�

De�nition �� An aggregation operator is a mapping� h � �� n ��

satisfying at least the �rst three of the following conditions�

�� Boundary conditions�

h�� h��

�� Monotonicity� for any a�� a�� an and b�� b�� bn� ai� bi � ��

i n�� if ai bi�� i n�� then�

h�a�� a�� an� h�b�� b�� bn��

� h is a continuous function�

�� Commutativity� for any permutation p of f�� ng�

h�a�� a�� an� � h�ap�� ap�� ap�n��

��

�� Idempotency� for all a � ��

h�a� a� � � � � a� � a�

For any fuzzy predicate on a �nite universe� the truth�value of its quanti�ed

form can be obtained from a class of aggregation operators which only vary in

n� But for an in�nite fuzzy set� the de�nition should be extended using a limit

process�

The uni�norm aggregation operators de�ned by Yager and Rybalovi �� are

prospective members of the family of general fuzzy quanti�ers in Thiele�s sense�

De�nition �� Yager and Rybalovi �� A uni�norm R is a mapping R �

�� 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 � �� such that �a � �� R�a� e� � a� Iden�

tity�

Also we can extend the de�nition to n�ary uni�norms� From the following prop�

erty� we can prove that the extension to countable in�nite arguments is mean�

ingful applying a limit process�

Proposition �� Yager and Rybalovi �� Assume R is a uni�norm with

identity e� then we have

R�a�� a�� an� � R�a�� a�� an� an�� if an�� e� � ��

R�a�� a�� an� R�a�� a�� an� an�� if an�� e� � �

��

�R� the dual of R� can be de�ned by

�R�a� b� � ��R� a� b� ��

with identity e� where e � �� e�

� Fuzzy Quanti�cations

�� Possibility Distributions as Quanti�ers

Using the term fuzzy quanti�cations� we mean that the quanti�ers themselves

are fuzzy� in another word� the quanti�ers are represented as fuzzy sets� Dif�

ferent to propositions in traditional meaning and notation� the objects mainly

investigated here are so�called canonical forms of �There are Q A�s� and �Q

A�s are B�s�� In the following sections� the cardinalities involved are �nite ones�

These kinds of quanti�ers are usually represented as possibility distributions�

In daily life� we use both natural numbers and percentages to refer to the quan�

tity of a given set� Analogically� there are two kinds of quanti�ers related to

possibility distributions de�ned on di�erent universes�

Obviously� propositions of the form �There are Q A�s� relate to the so�called

absolute or �rst�kind quanti�ers� noted as QI � which are looked as possibility

distributions of cardinalities of fuzzy sets� while propositions of the form �QA�s

are B�s� relate to relative or second�kind quanti�ers� noted as QII � 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 quanti�ers 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 universe of natural numbers� In his approach� Zadeh �� represents

the desired distribution as on the non�negative real universe� for he mainly

adopts the corresponding de�nition of cardinalities �De�nition �� for fuzzy

sets� We also follow this approach� Thus� in the below� both kinds of quanti�ers

are represented by normal convex closed fuzzy sets� but they are identi�ed by

the underlying universe� a quanti�er of the �rst kind is on the universe of the

non�negative reals R� � f�g� while a quanti�er of the second kind is on the

universe of unit interval �� 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 de�ned as normal convex closed fuzzy sets onR� Furthermore� the extension

principle can be used to de�ne their arithmetic�

In this kind of interpretation� the classical quanti�ers � and � are degener�

ated fuzzy sets f�g and fxj� � x �g respectively�

The quanti�ers of the second kind are discussed more thoroughly in the

literature� here we de�ne some special sub�categories of these quanti�ers�

De�nition �� Yager �� Zadeh �� Let QII be a fuzzy quanti�er of the

second kind� Then

�� QII is regular non�decreasing if

� QII��

� QII��

� if x� � x�� then QII�x�� QII�x��

�� QII is regular non�increasing if

�

�

��

��

��

�

��

��

��

�

��

��

��

�

��

��

��

a� Regular non�decreasing quanti�er b� Regular non�increasing quanti�er

c� Regular unimodal quanti�er d� A quanti�er and its antonym

QII ant QII

Figure �� Special cases of relative quanti�ers�

� QII��

� QII��

� if x� � x�� then QII�x�� QII�x��

� QII is regular unimodal if for some � a b ��

� QII�� QII��

� QII�x� � �� for a x b�

� if x� � x� a� then QII�x�� QII�x��

� if b x� � x�� then QII�x�� QII�x��

�� ant QII � the antonym of QII� is de�ned by�

�ant QII��x� � QII�� x�� x ��

The de�nitions are illustrated in Figure ��

��

The interpretation of propositions of such forms falls into two distinct cate�

gories� possibilistic interpretation 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�

�� Possibilistic Interpretation

This approach is mainly developed by Zadeh in his fuzzy linguistic logic or theory

of approximate reasoning with the aid of PRUF �� The following quotation

re�ects 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 de�ning the meaning�M �P ��

of p� with the possibility distribution computed by P constituting

the information� I�P �� conveyed by p� �The procedure de�ned 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��

Propositions with canonical forms are represented in the following way�

There are QIA�s � Card�A� is QI � ��

QIIA�s are B�s � Prop�BjA� is QII � ��

where Prop�BjA� indicates the relative measure of cardinality of B in A� In the

following passages� we assume the cardinalities are de�ned as �Count�

��

The right hand side of the above two equivalences can be translated into

possibility distributions as common propositions�

Card�A� is QI � �Card�A� � QI � � �

Prop�BjA� is QII � �Prop�BjA� � QII � ��

Therefore� the resulted possibility distribution shows the consistency of the

available data �Card�A� or Prop�BjA�� with the quanti�ers �QI or QII�� So

this kind of interpretation is completely di�erent 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

�� denoted as u�true ��

u�true�t� � t� t � ��

while the propositions can be quali�ed by another truth�value � � in which case

the truth�value of the proposition is � �

�� Numerical Interpretations

More easy�to�use interpretations are numerical interpretations� This class of

interpretations can be recognized as truth�value interpretations and possibil�

ity�certainty interpretations �� Here we mainly discuss the truth�value in�

terpretations� Other interpretations �� relate to the possibility!certainty mea�

sure� and more generally� fuzzy measure theory� Some recent developments ��

involve Sugeno and Choquet fuzzy integrals�

Truth�value interpretation keeps the tradition that the extension of a propo�

sition 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 calcu�

lated based on the membership function of the quanti�er�

The simplest way to calculate the truth�value is to use cardinalities� that

is ��

� �There are QIA�s� � QI��Count�A��

� �QIIA�s are B�s� � QII��Count�BjA��

Recall Yager�s classi�cation of de�ning semantics� this approach is an algebraic

one�

Yager �� proposed to use OWA operators to calculate the truth values when

the proposition takes the form of �QIIU �s are A�s�� where U is the universe

concerned� OWA operators� or Ordered Weighted Averaging operators� are

de�ned as follows�

De�nition �� An OWA operator of dimension n is a mapping

f � Rn �� R

which has an associated n�vector W

W � �w�� w�� wn�

such thatnXi��

wi � �� i n�wi � ��

Furthermore�

f�a�� a�� an� �nXj��

wj � bj� ��

where bj is the j�th largest of a�� a�� an�

��

For a regular non�decreasing quanti�er QII� an associated OWA operator

fQII is de�ned with

wi � QII�i

n� �QII�

i� �

n�� i n� ��

The de�nition is meaningful because of the boundary conditions of a regular

quanti�er� Thus� the truth�value is

� � fQII �a�� a�� an��

The truth�value of a proposition related to a regular non�increasing quanti�er

is de�ned with the aid of its antonym� Instead of considering the proposition

QIIU �s are A�s�

we investigate the equivalent proposition

�ant QII�U �s are A�s�

Thus following the above trick� we obtain

wi � �ant QII��i

n�� ant QII��

i� �

n��

Since �ant QII��r� � QII�� r�� that is

wi � QII��i

n�� QII��

i� �

n��

Recall A�t� � ��A�t�� we have

� � fQII �� a�� a�� an��

Decomposition is applied to a unimodal quanti�er to get an overall truth�

value� Notice that a regular unimodal quanti�er QII is non�decreasing at the

left hand side of a� and non�increasing at the right hand side of b� therefore�

�

�

�

��

��

��

QII

QII� QII

�

Figure �� Decomposition of unimodal quanti�er

it can be represented by a regular non�decreasing quanti�er QII� and a regular

non�increasing quanti�er QII� �� These two quanti�ers are de�ned as follows�

QII��x� �

��

QII�x�� x a�

�� x � a�

QII��x� �

��

QII�x�� x � b�

�� x � b�

��

Then we can have

QII � QII� and QII

��

where

QII�x� � T �QII��x��Q

II��x��

with T any t�norm operator� Therefore� the overall truth�value is calculated as

� � T ��

where �� and �� are the truth�values calculated with the quanti�ers QII� and

QII�� respectively�

�

Besides the above interpretations� the substitution approach can also be

applied �� 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 SF�Q on the universe VF � such that for each v � VF � SF�Q�v� indicates the

degree of possibility of v as a meaning for Q� Furthermore� for each v � 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 � � maxv�VF

min�SF�Q�v�� TF �v��

And for practical use� we should point out that the maximumcan be equivalently

obtained over the support of SF�Q�

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An Overview of Fuzzy Quantifiers, Part 1: Interpretations

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