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On the capability of fuzzy set theory to represent conceptsRadim Bělohlávek a , George J. Klir b , Harold W. Lewis b & Eileen Way b

a Department of Computer Science, Palacký University, Olomouc, Czech Republicb Center for Intelligent Systems and Department of Systems Science and Industrial Engineering,Binghamton University--SUNY, Binghamton, NY, 13902, USA

Version of record first published: 24 Sep 2010

To cite this article: Radim Bělohlávek, George J. Klir, Harold W. Lewis & Eileen Way (2002): On the capability of fuzzy set theory torepresent concepts, International Journal of General Systems, 31:6, 569-585

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ON THE CAPABILITY OF FUZZY SET THEORY TOREPRESENT CONCEPTS

RADIM BELOHLAVEKa, GEORGE J. KLIRb,*, HAROLD W. LEWIS IIIb and

EILEEN WAYb

aDepartment of Computer Science, Palacky University, Olomouc, Czech Republic; bCenter forIntelligent Systems and Department of Systems Science and Industrial Engineering, Binghamton

University—SUNY, Binghamton, NY 13902, USA

(Received 19 March 2002; In final form 28 June 2002)

The aim of this paper is to examine the conclusions drawn by Osherson and Smith [“On the adequacy of prototypetheory as a theory of concepts”, Cognition 9 (1981), pp. 35–58] concerning the inadequacy of the apparatus of fuzzyset theory to represent concepts. Since Osherson and Smith derive their conclusions from specific examples, we showfor each of these examples that the respective conclusion they arrive at is not warranted. That is, we demonstrate thatfuzzy set theory is sufficiently expressive to represent the various strong intuitions and experimental evidenceregarding the relation between simple and combined concepts that are described by Osherson and Smith. To pursueour arguments, we introduce a few relevant notions of fuzzy set theory.

Keywords: Cognitive science; Fuzzy set theory; Theory of concepts; Prototype theory

1. INTRODUCTION

Sometimes, a particular publication can have an extraordinary influence on an entire

discipline with respect to a particular method. We feel that the frequently cited article by

Osherson and Smith (1981) is such a publication, and that it has had an extraordinarily

negative influence on the use of fuzzy logic and fuzzy set theory in psychology. We argue in

this paper that all conclusions drawn by Osherson and Smith (abbreviated as O & S in the

following text) concerning the inadequacy of the apparatus of fuzzy set theory to represent

concepts are refutable. Yet, their almost universal acceptance have caused the psychological

research community to virtually abandon this powerful representational tool.

The amount of influence that a critical publication by reputable scholars has on a field of

research should not be underestimated. A quintessential case from a different field is that of

Minsky and Papert’s book Perceptrons, published in 1969 by the MIT Press. Minsky and

Papert were interested in exploring the potential and limitations of early neural networks,

principally known at that time through Frank Rosenblatt’s work on the perceptron.

ISSN 0308-1079 print/ISSN 1563-5104 online q 2002 Taylor & Francis Ltd

DOI: 10.1080/0308107021000061894

*Corresponding author.

International Journal of General Systems, 2002 Vol. 31 (6), pp. 569–585

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Their book is a thorough mathematical analysis of the kinds of functions a single-layer

perceptron is capable of computing. The well-known and devastating result of this analysis

was that there are important functions that the perceptron could not perform.

Minsky and Papert’s work resulted in the virtual demise of research in neural nets until the

mid 1980’s. In fairness to Minsky and Papert, the conclusion of their book does recommend

that further research be done to determine the possibilities of multi-layer neural networks.

Had this suggestion been taken seriously, the development of neural networks might be much

farther along today. However, because of the excellent reputation of Minsky and Papert and

the rigor of their analysis, most researchers relied on Minsky and Papert’s “intuitive

judgment” that further work on multi-layer networks would prove to be “sterile”. The

artificial intelligence research community concluded for many years that Minsky and

Papert’s book had shown neural networks to be worthless as a research method for intelligent

systems. As we all now know, this was an inaccurate assumption that delayed the

development of what has become an important research area. Surely there is a lesson to be

learned here. We feel that Osherson and Smith’s seminal paper on prototype theory has

incorrectly convinced the psychological research community that fuzzy set theory is a

worthless tool for representing concepts.†

Since the publication of “The Adequacy of Prototype Theory as a Theory of Concepts,” in

1981, in Cognition, there is scarcely any article or book on concepts that does not refer to

O & S and does not summarize their criticism of fuzzy set theory. According to the Web of

Science citation database, this article has been cited 134 times to date since its publication.

This shows its extraordinary influence when compared with 38 citations for Osherson and

Smith (1982) or 65 citations for Smith et al. (1988).

Although O & S’s primary purpose in this article was to criticize prototype theory, they use

fuzzy set theory to explicate the theory of prototypes. They state (p. 41):

Since, in addition, fuzzy-set theory is a natural complement to prototype theory. . . we shall evaluate prototypetheory exclusively in the context of fuzzy-set theory.

They reiterate the same a few pages later (p. 49):

Once again, it is fuzzy-set theory that saves prototype theory from inexplicitness.

Because of this tight connection between the two, O & S’s negative conclusions reflect

strongly on the value of fuzzy set theory in psychology. In fact, they make several explicit

claims about the value of fuzzy set theory. For example, they state (p. 55):

One thing is clear. Amalgamation of any number of current versions of prototype theory with Zadeh’s (1965)rendition of fuzzy-set theory will not handle strong intuitions about the way concepts combine to formcomplex concepts and propositions.

While O & S are correct that Zadeh’s (1965) fuzzy-set operators do not have the properties

they are looking for, this does not mean that fuzzy-set theory as a whole cannot capture

O & S’s intuitions about the way concepts combine. The influence this article has had on

the way in which the psychological community views fuzzy-set theory and fuzzy logic can be

seen by the following sampling of quotes from the literature. Note how quickly the

conclusions go from Zadeh’s operations to conclusions about fuzzy-set theory as a whole.

This quote is from a paper by Roth and Mervis (1983, p. 522):

Recent work of Osherson and Smith (1982) suggests additional problems in applying fuzzy set theory tosemantic categories. Osherson and Smith point out a number of logical inconsistencies that arise when fuzzyset theory is used to form complex concepts from simpler ones. These results taken in combination with our

†It should be noted that Dominic Massaro and his research group provide an exception by using fuzzy set theory torepresent human perception of speech; see, for example, Massaro (1989) and Massaro and Cohen (1993).

R. BELOHLAVEK et al.570

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own results suggest that fuzzy set theory is limited in its usefulness as a formal model of the semantics ofnatural categories.

Johnson-Laird (1983, p. 199) says:

All the obvious ways of specifying rules for and and not (in fuzzy logic) do violence to our intuitions aboutnatural language. These arguments count decisively against the adoption of a fuzzy-set semantics for naturallanguage (see Osherson and Smith, 1981, for other, though related, shortcomings of fuzzy logic).

More recently, Hampton (1993, pp. 81–82) states:

Fuzzy logic is thus inappropriate for describing the case of conceptual combination of the kind exemplified inpet fish.

Kamp and Partee (1995) also attack fuzzy-set theory, arguing

. . .that many of the problems O & S discovered are due to difficulties that are intrinsic to fuzzy set theory”(p. 129).

Kamp and Partee’s paper is not so concerned with prototype theory per se, but rather with

attacking fuzzy-set theory as a useful methodology. They state that the aim of their paper

is to point out some well-known defects of fuzzy-set theory which render it unsuitable in connection with anytheory of vagueness (p. 130)

The “inadequacies of fuzzy-set theory” according to Kamp and Partee (abbreviated as

K & P in the rest of the text), turn out to hinge, again, on the definitions given in Zadeh’s

(1965) paper. K & P re-iterates arguments that O & S give concerning logically empty and

logically universal concepts. As we discuss in Section 3.2, both O & S and K & P are correct

that the law of contradiction and the law of excluded middle do not hold when using Zadeh’s

standard operation for complement, union, and intersection. However, this does not condemn

fuzzy-set theory since there are many possible functions that can be used for complement,

intersection and union. This is not to say that Zadeh’s operations are flawed or useless. On the

contrary, they possess certain properties that make them suitable, or even essential, in some

applications (control, information retrieval, clustering etc.). It is just that they are not the

correct choice of operators for the purpose of capturing certain conceptual combinations.

In Section 3.2, we show a combination of functions in fuzzy set theory whereby the laws of

excluded middle and contradiction are satisfied.

The fact that there are many possible choices for fuzzy set operations for complement,

intersection (or conjunction) and union (or disjunction) has been well known in fuzzy set

theory for quite some time. The general foundations of fuzzy set theory and fuzzy logic and

its philosophy can be found in Goguen (1967 and 1968–9) as well as Dubois and Prade

(1980). Note that all of these were published before the original O & S article in 1981, and

certainly before the 1995 article by K & P. In fact both these articles refer to Goguen’s (1967)

work, indicating that there was at least some awareness of the developments in the field of

fuzzy-set theory.

K & P are clearly aware of the flexibility of fuzzy set theory as can be seen by their

comments in footnote 11 (p. 148):

It was pointed out to us by Filem Novak‡ (p.c. to B.H.P) that there are some versions of fuzzy-set theory (suchas that presented in Zadeh, 1978) on which there is not just a single formula for conjunction but potentiallyinfinitely many, and one must take into account the content of the conjuncts in deciding which operation to usein a given case.

‡Filem Novak is really Vilem Novak, currently a professor at University of Ostrava and Director of the Institute forResearch and Applications of Fuzzy Modeling (IRAFM). He is one of the principal contributors to fuzzy set theoryand fuzzy logic; see, for example, Novak (1989; 1992) and Novak et al. (1999).

CAPABILITY OF FUZZY SET THEORY 571

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However, K & P seem to be unmoved by this objection, stating, further in this footnote

(p. 148):

We should make it clear that when we refer to fuzzy-set theory, we, like O & S, are referring principally to itsclassic version in Zadeh (1965), the basic principles of which we have heard Zadeh himself continue toarticulate well into the 1980s.

While, as we have stated above, it is true that Zadeh’s original formulation of max and min

for conjunction and disjunction leads to counterintuitive results when modeling conceptual

combination, these operations are not all there is to fuzzy-set theory. The very fact that so

many publications in psychology assume these “well-known defects of fuzzy-set theory”

show that an entire field has been condemned based on a misunderstanding of its capabilities,

a situation that this paper is intended to begin to remedy.

Again we can see the long-range effects of O & S’s article on the perception of fuzzy-set

theory in psychology in (Hampton 1997, pp. 139–140):

Although fuzzy logic had some success in accounting for intuitions about the conjunction of unrelatedstatements (Oden, 1977), it soon became clear, following a key article by Osherson and Smith (1981), that notonly the minimum rule, but in fact any rule that takes as input solely the truth value of the two constituentstatements is doomed to failure. . . .Smith et al. (1988) took the striped apple example from Osherson andSmith (1981) and collected empirical evidence that it is indeed true that a picture of a brown apple (forexample) is considered more typical of the conjunctive concept “brown apple” than it is of the simple concept“apple”. The almost trivial nature of this demonstration highlights the failing of the fuzzy logic approach tocope with predictions of typicality in complex concepts.

We claim that just as the artificial intelligence community was mistaken in accepting

Minsky and Papert’s intuition that perceptrons were a dead-end, so too is the psychological

community mistaken to reject the power of fuzzy set theory as a representational tool. Just as

Minsky and Papert were mistaken in concluding that the limitations of a single-layer

perceptron apply to multi-layer networks, so too, were O & S mistaken in concluding that the

properties of Zadeh’s classic fuzzy set operators capture the representational ability of all of

fuzzy set theory. We feel that Osherson and Smith’s (1981) article has gone unanswered in

specific terms too long and that the negative attitude it has generated in psychology towards

fuzzy set theory has impeded possible progress in representing concepts.

In this paper, we address the supposed inadequacies and shortcomings of fuzzy set theory

and show that in fact, it does have sufficient power and flexibility to capture O & S’s or

anyone else’s “strong intuitions” about concepts. In fact, fuzzy set theory has an expressive

power that is far beyond that of classical set theory. For instance, the counterparts to the

classical set-theoretic operations of complement, intersection and union in fuzzy set theory

are not unique. There are, in fact, an infinite set of each of these operations from which to

choose. Moreover, fuzzy set theory has aggregation operations and modifying operations that

have no counterparts in classical set theory. Given this richness of representation, fuzzy sets

can capture any relationships among natural categories revealed by empirical data or

required by intuition.

Some may feel that this multitude of possible operations makes fuzzy set theory too

flexible: it can fit any kind of data. However, this is precisely what we want in a powerful

representational formalism. There seems to be an unspoken expectation in O & S (and others)

that somehow fuzzy set theory will automatically give the right answers to such difficult

questions as conceptual combination. They state in another paper (Smith and Osherson,

1984, p. 339):

Fuzzy-set theory (e.g. Zadeh, 1965) is of interest to cognitive scientists because it offers a calculus forcombining prototype concepts.

There may be a hope that if fuzzy logic based upon fuzzy set theory truly represents human

reasoning, that somehow the underlying logic of concepts might be revealed through its

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mechanisms. In this way, all we would need to do is to tap into the right logic and the

problems of conceptual combination will be solved—the correct answers will fall out from

fuzzy set theory automatically. And if this fails to occur, then fuzzy logic is just not the

correct model for conceptual structure. However, we argue that it is a mistake to view fuzzy

set theory as a complete context-independent model of concepts and reasoning rather than a

representational language. Fuzzy set theory is a formal system, albeit one that has a great deal

of expressive power, but how to use this expressive power must ultimately depend upon

empirical data—it will not in itself answer the hard questions of cognition or any other

domain.

It is important to emphasize here that this paper has a narrow focus to specifically address

the criticism of O & S. We are planning future papers that will consider the efficacy of fuzzy

set theory in concepts and other areas of psychology.

In Section 2, we introduce those notions of fuzzy set theory that are relevant to our

discussion in this paper. We describe six different types of operations on fuzzy sets and show

how they work. In Section 3, we respond in detail to each of O & S’s arguments against the

utility of fuzzy set theory for representing concepts. We show in each case that the

conclusions stated by O & S are not warranted. Our overall conclusions are then presented in

Section 4.

2. FUZZY SET THEORY: RELEVANT NOTIONS AND NOTATION

Let X denote a conceptual domain in the sense used by O & S. To denote fuzzy sets defined

on X, we adopt a notation that is currently predominant in the literature on fuzzy set theory.

According to this notation, symbols of fuzzy sets, A, B,. . ., are not distinguished from

symbols of their membership functions.{ Since each fuzzy set is uniquely defined by one

particular membership function, no ambiguity results from this double use of the same

symbol. Following O & S, we consider only standard fuzzy sets, in which degrees of

membership are expressed by real numbers in the unit interval [0,1].

Contrary to the symbolic role of numbers 1 and 0 in characteristic functions of classical

(crisp) sets, numbers assigned to objects in X by membership functions of fuzzy sets have

clearly a numerical significance. This significance, which is preserved when crisp sets are

viewed (from the standpoint of fuzzy set theory) as special fuzzy sets, allows us to

manipulate fuzzy sets in numerous ways, some of which have no counterparts in classical set

theory.

Since most of the arguments made by O & S in their paper involve operations on fuzzy

sets, we need to introduce all relevant operations on fuzzy sets to facilitate our discussion.

For our purpose in this paper, we present only a digest of the meaning and basic properties of

the various operations. The notation is adopted from Klir and Yuan (1995), where further

details (including relevant proofs) are covered.

Each of the following six types of operations on fuzzy sets are relevant to our discussion:

. modifiers

. complements

. intersections

. unions

{Observe that O & S use the term “characteristic function” (adopted from classical set theory) rather than thecommonly used term “membership function”.

CAPABILITY OF FUZZY SET THEORY 573

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. implications

. averaging operations

Modifiers and complements are unitary operations; intersections and unions are defined as

binary operations, but their application can be extended to any number of fuzzy sets via their

property of associativity; implications are binary operations that satisfy certain monotonicity

conditions; averaging operations, which are not associative, are defined, in general, as n-ary

operations ðn $ 2Þ:All of these operations on fuzzy sets are induced by corresponding operations on [0,1] in

the following way: an n-ary operation s on [0,1] extends to an n-ary operation on fuzzy sets§

by assigning to fuzzy sets A1, A2, . . ., An a fuzzy set sðA1;A2; . . .;AnÞ whose membership

function is given by

½sðA1;A2; . . .;AnÞ�ðxÞ ¼ s½A1ðxÞ;A2ðxÞ; . . .;AnðxÞ�:

The operation on fuzzy sets defined in this way is said to be based on the corresponding

operation on [0,1].

The purpose of modifiers is to modify fuzzy sets representing, in each given context, some

linguistic terms to account for added linguistic hedges, such as very, somewhat, barely, etc.

in the same context. Each modifier, m, is based on a one-to-one (and usually continuous)

function of the form

m : ½0; 1�! ½0; 1�;

which is order preserving. Sometimes, it is also required that mð0Þ ¼ 0 and mð1Þ ¼ 1:The most common modifiers either increase or decrease all values of a given membership

function. A convenient class of functions, ml; that qualify as increasing or decreasing

modifiers is defined by the simple formula

mlðaÞ ¼ al; ð1Þ

where l [ ð0;1Þ is a parameter whose value determines which way and how strongly ml

modifies a given membership function. For all a [ ½0; 1�; mlðaÞ . a when l [ ð0; 1Þ;mlðaÞ , a when l [ ð1;1Þ; and mlðaÞ ¼ a when l ¼ 1: The farther the value of l from 1,

the stronger the modifier ml.

In addition to the most common class of modifiers defined by Eq. (1), numerous other

classes of modifiers were proposed and studied (Kerre and De Cock, 1999), some of which

involve more than one parameter. Several modifiers may also be composed to form a more

descriptive modifier. Which modifier to choose is basically an experimental question. Given

any data regarding the meaning of a linguistic hedge in a given context, we need to choose a

class of modifiers that qualitatively conforms to the data and, then, to select the modifier from

the class (characterized by specific values of the parameters involved) that best fits the data.

Complements of fuzzy sets are based on functions, c, of the same form as modifiers, but

they are order reversing and such that cð0Þ ¼ 1 and cð1Þ ¼ 0: Moreover, they are usually

required to possess for all a [ ½0; 1� the property

c½cðaÞ� ¼ a;

which is referred to as involution.

§For simplicity, this extension of s is denoted also by s; moreover, we use a common name for both of thecorresponding operations, i.e. the operation on [0,1] and the operation on fuzzy sets.

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A practical class of involutive complements, cl, is defined for each a [ ½0; 1� by the

formula

clðaÞ ¼ ð1 2 alÞ1=l; ð2Þ

where l [ ð0;1Þ is a parameter whose values specify individual complements in the class.

When l ¼ 1; the complement is usually referred to as the standard complement. Various

other classes of complements are known and can be used when desirable. Some of them

involve more than one parameter. As with modifiers, the choice of a particular complement

in a given context is an experimental issue.

Intersections and unions of fuzzy sets, denoted in this paper by i and u, respectively, are

generalizations of the operations of intersection and union in classical set theory. They are

based on functions from [0,1]2 to [0,1] that are commutative, associative, and monotone

nondecreasing. The only property in which they differ is a boundary condition: for all

a [ ½0; 1�; iða; 1Þ ¼ a while uða; 0Þ ¼ a:Contrary to their classical counterparts, intersections and unions of fuzzy sets are not

unique. However, they are bounded for all a; b [ ½0; 1� by the inequalities

iminða; bÞ # iða; bÞ # minða; bÞ;

maxða; bÞ # uða; bÞ # umaxða; bÞ;

where

iminða; bÞ ¼minða; bÞ when maxða; bÞ ¼ 1;

0 otherwise;

(ð3Þ

umaxða; bÞ ¼maxða; bÞ when minða; bÞ ¼ 0;

1 otherwise:

(ð4Þ

Operations min and max are usually called standard operations, while operations imin and

umax are often referred to as drastic operations. Numerous classes of functions are now

available that fully cover these ranges of intersections and unions. Examples are classes iland ul that are defined for all a; b [ ½0; 1� by the formulas

ilða; bÞ ¼ 1 2 min{1; ½ð1 2 aÞl þ ð1 2 bÞl�1=l}; ð5Þ

ulða; bÞ ¼ min½1; ðal þ blÞ1=l�; ð6Þ

where l [ ½0;1Þ is a parameter whose values specify individual intersections or unions in

these classes. It is easy to show that the drastic operations are obtained for l ¼ 0; while the

standard operations are obtained in the limit for l!1:Functions i and u, which represent possible intersections and unions of fuzzy sets,

respectively, are usually referred to in the literature as triangular norms (or t-norms) and

triangular conorms (or t-conorms), respectively (Klir and Yuan, 1995; Klement, et al.,

2000).

The fact that the primitive operations of complementation, intersection and union are not

unique in fuzzy set theory means that the whole family of theories is actually subsumed under

the term “fuzzy set theory”. Individual theories in the family are distinguished from one

another by their mathematical structures, which are induced by the primitive operations

employed. Although each fuzzy set theory violates some properties of Boolean lattice or

algebra, the underlying mathematical structure of classical set theory, different theories

CAPABILITY OF FUZZY SET THEORY 575

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violate different properties. This feature of fuzzy set theory is essential for pursuing some of

our argument in Section 3.

Another important class of operations are fuzzy implications. These are binary operations

) on [0,1] that are usually required to satisfy ð0 ) aÞ ¼ 1; ð1 ) aÞ ¼ a; a # a0 implies

ða0 ) bÞ # ða ) bÞ; b $ b0 implies ða ) bÞ # ða ) b0Þ; and possibly also other conditions

(see Klir and Yuan, 1995; Hajek, 1998). Fuzzy implications model the semantics of the

implication connective in fuzzy setting and generalize in an obvious way the classical case.

The most common example is the Lukasiewicz implication defined by ða ) bÞ ¼

minð1; 1 2 a þ bÞ:The last type of operations on fuzzy sets are averaging operations. These operations have

no counterparts in classical set theory. Since they are not associative, averaging operations

must be defined as functions of n arguments for any n $ 2: That is, they are based on

functions h of the form

h : ½0; 1�n ! ½0; 1�;

which are monotone nondecreasing and symmetric in all arguments, continuous, and

idempotent. It is significant that

minða1; a2; · · ·; anÞ # hða1; a2; · · ·; anÞ # maxða1; a2; · · ·; anÞ

for any n-tuple ða1; a2; · · ·; anÞ [ ½0; 1�n: As noted above, in fuzzy set theory, unions and

intersections each have well-defined continuous ranges. The averaging operations have their

own range that happens to fill the gap between the largest intersection (the minimum

operator) and the smallest union (the maximum operator).

One class of averaging operations, hl, where l is a parameter whose range is the set of all

real numbers is called generalized means. This class, which covers the entire interval

between min and max operations, is defined for each n-tuple ða1; a2; · · ·; anÞ in the Cartesian

product [0,1]n as follows:

hlða1; a2; · · ·; anÞ ¼al

1 þ al2 þ · · · þ al

n

n

� �1=l

ð7Þ

when l – 0, 21, 1 and, as is well known (Klir and Yuan, 1995),

h0ða1; a2; . . .; anÞ ¼l!0lim

al1 þ al

2 þ · · · þ aln

n

� �1=l

¼ ða1·a2· · ·anÞ1=n; ð7aÞ

h21ða1; a2; . . .; anÞ ¼l!21lim

al1 þ al

2 þ · · · þ aln

n

� �1=l

¼ minða1; a2; . . .; anÞ; ð7bÞ

h1ða1; a2; . . .; anÞ ¼l!1lim

al1 þ al

2 þ · · · þ aln

n

� �1=l

¼ maxða1; a2; . . .; anÞ: ð7cÞ

Other classes of averaging operations are now available, some of which use weighting

factors to express relative importance of the individual fuzzy sets involved. For example, the

function

hðai;wiji ¼ 1; 2; . . .; nÞ ¼Xn

i¼1

wiai; ð8Þ

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where the weighting factors wi usually take values in the unit interval [0,1] and

Xn

i¼1

wi ¼ 1;

expresses for each choice of values wi the corresponding weighted average of values

aiði ¼ 1; 2; . . .; nÞ: Again, the choice is an experimental issue.

3. RESPONSE TO OSHERSON AND SMITH

In this section, we respond in specific ways to all arguments regarding the utility of fuzzy

set theory in representing and combining concepts that are pursued by Osherson and

Smith (1981). These arguments led O & S to the conclusion that “fuzzy-set theory will

not handle strong intuitions about the way concepts combine to form complex concepts

and propositions” (p. 55).

Our response is based on the following position. We do not contest the soundness of

“strong intuitions” (the term used by O & S), presumably supported by outcomes of relevant

experiments, regarding the relation between simple and combined concepts, as described

by O & S. Our only aim is to answer the question: is the apparatus of fuzzy set theory capable

of representing the various strong intuitions and/or experimental evidence described

by O & S. Moreover, our answers are restricted to the specific examples employed

by O & S. In each of the following subsections, we deal with one of the examples. For

convenience, we use the simplified notation introduced in Section 2.

3.1. Striped Apples

In this example, the domain, F, is the set of all fruits, and three relevant concepts

are considered: apple, striped, and striped apple. In their discussion, O & S use a drawing of

a normal apple with stripes on it (p. 44) and refer to this striped apple as a. They argue that

the mem-bership degree of a in the set of striped apples, SA, should be, from the

psychological point of view, greater than the membership degree of a in the set of apples, A.

That is, they argue that

SAðaÞ . AðaÞ ð9Þ

[inequality (2.11) in O & S]. So far so good. However, prior to stating this reasonable

requirement, they assume, without any justification, that SA is an intersection of A and the set

of all striped fruits, S. Using this assumption, they then argue that the intersection of A and S

violates Eq. (9). That is, using (according to O & S) the standard intersection,

SAðaÞ ¼ min½ðSðaÞ;AðaÞ�;

we obtain

SAðaÞ # AðaÞ

[inequality (2.12) in O & S], which contradicts (9) [or (2.11) in O & S]. Although this

argument is correct, the assumption is wanting, as also recognized by Lakoff (1987, p. 142):

“The assumption that noun modifiers work by conjunction is grossly incorrect.”

The issue of adjective-noun combinations (such as striped apples) is thoroughly discussed

by Kamp and Partee (1995). They argue that the assumption that adjectives in these

combinations are intersective is not warranted. The fact that the inequality (9) is required,

on whatever grounds by O & S, for the adjective–noun combination “striped apple” is itself

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an indicator that this conceptual combination is not intersective. In the following, we discuss

how this combination can be represented by the language of fuzzy set theory.

First, we recognize that to determine SA( f ) as a function of S( f ) and A( f ) for any f [ F; it

is essential to require that SAð f Þ ¼ 0 whenever Sð f Þ ¼ 0 or Að f Þ ¼ 0: Consider, for example,

O & S’s drawing (b) in Fig. 1. Clearly, SðbÞ ¼ 0; AðbÞ . 0 (and fairly large), and certainly

SAðbÞ ¼ 0: O & S do not explicitly state this requirement, but it is implicitly covered by their

assumption that SA is an intersection of S and A. This assumption seems to be inspired by

classical set theory, where the set intersection is the only operation that satisfies the

requirement. In fuzzy set theory, contrary to classical set theory, the requirement is satisfied

not only by the intersection operations, but also by some averaging operations. Using, for

example, the class of generalized means, defined by Eq. (7), the requirement is satisfied by

any generalized mean for which l # 0: The largest value of SA, which is the most desirable

choice in our case, is obtained for l ¼ 0: This happens to be the geometric average, as

indicated in Eq. (7a). Hence,

SAðaÞ ¼ ½SðaÞ·AðaÞ�1=2

Assuming now that SðaÞ . AðaÞ; which is reasonable for the given drawing, we get the

desired inequality SAðaÞ . AðaÞ: Moreover, we get SAðbÞ ¼ 0 for drawing (b), as required.

Let us remark that the example of striped apples, as presented by O & S, has also a more

fundamental deficiency: values compared in O & S’s inequality (2.11) are degrees of

prototypicality of an object (drawing (a) in Fig. 1) pertaining to the respective concepts,

while values compared in their inequality (2.12) are degrees of membership of the object in

fuzzy sets representing the concepts. For any given object and a considered concept, these

two kinds of degrees are generally distinct, which implies that the inequalities (2.11) and

(2.12) are not necessarily inconsistent. However, this issue is beyond the intended scope of

this paper. It will be thoroughly discussed in our second paper regarding the role of fuzzy set

theory in cognitive science.

3.2. Logically Empty and Logically Universal Concepts

In Section 2.3.2 (pp. 45 and 46) of their paper, O & S argue that the concept apple that is not

an apple is logically empty and, similarly, that the concept fruit that either is or is not an

apple is logically universal. As a consequence, the sets representing these concepts are,

respectively, the empty set and the universal set. These properties are of course correct in

terms of classical logic and classical set theory, where they are called the law of

contradiction and the law of excluded middle, respectively. However, O & S require that

these laws hold in fuzzy set theory as well. They correctly show, that they, in fact, do not hold

when standard operations of complement, intersection, and union are employed.

Requiring that the laws of contradiction and excluded middle must hold in fuzzy set theory

to correctly represent concepts in natural language needs justification, which O & S do not

provide. The requirement has already been disputed in numerous ways; see, e.g. Zadeh

(1982, p. 588); Lakoff (1987, p. 141), and Fuhrmann (1988a, pp. 323 and 324). Instead of

presenting here the various arguments against the requirement, we prefer to show that the

conclusions drawn by O & S regarding the two laws are erroneous regardless of whether we

accept or reject the requirement.

Since the counterparts of the three classical set-theoretic operations (complements,

intersections, unions) are not unique in fuzzy sets theory (as described in Section 2), we can

choose different combinations of these operations from the delimited classes of functions as

needed. Whatever combination of these functions we choose, some properties of the classical

operations (properties of the underlying Boolean algebra) are inevitably violated. This is

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a consequence of imprecise boundaries of fuzzy sets. However, different combinations violate

different properties, and this is crucial for our argument. The standard operations, for

example, violate only the law of contradiction and the law of excluded middle, as correctly

demonstrated by O & S. Some other combinations preserve these laws, but violate some other

properties, usually distributivity and idempotence. For example, when the standard

complement is combined with imin and umax, the laws of contradiction and excluded middle

are satisfied, but they are also satisfied when the intersection and union operations are defined

for all a; b [ ½0; 1� by the formulas

iða; bÞ ¼ maxð0; a þ b 2 1Þ;

uða; bÞ ¼ minð1; a þ bÞ;

respectively. Furthermore, a procedure is well established (Klir and Yuan, 1995, Section 3.5)

by which classes of operations can be constructed that satisfy the two laws. Consider, for

example, the classes of complements, cl, and unions, ul, defined for each a; b [ ½0; 1� by the

formulas

clðaÞ ¼1 2 a

1 þ la;

ulða; bÞ ¼ minð1; a þ b þ labÞ;

where l [ ð21;1Þ: Then,

ulða; clðaÞÞ ¼ 1

for all a [ ½0; 1� and each l [ ð21;1Þ: This means that the law of excluded middle is

satisfied for any pair (ul,cl) of operations from these classes.

It is clear that fuzzy set theory can represent cognitive situations in which the laws of

contradiction and/or excluded middle should hold according to experimental evidence.

Again, O & S are wrong in their conclusions.

3.3. Liquidity, Investment, and Wealth

In Section 2.4.3 (pp. 46–48) of their paper, O & S discuss an example involving the concept

financial wealth and its connection to concepts liquidity and investment. They consider three

persons, A, B, C, whose assets are given in the following table.

They describe the concepts liquidity, investment, and wealth, respectively, by fuzzy sets

L, I, and W (our notation), and argue correctly that the following inequalities should be

satisfied on intuitive grounds:

LðAÞ . LðBÞ

IðCÞ . IðBÞ

WðBÞ . WðAÞ

WðBÞ . WðCÞ

Person Liquidity Investment

A $105,000 $5000B $100,000 $100,000C $5000 $105,000

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The basic issue now is how W(x ) is determined for any x [ X in terms of L(x ) and I(x ).

That is, we want to find a function f such that

WðxÞ ¼ f ½LðxÞ; IðxÞ�

for any x [ X is sensible on intuitive and/or experimental grounds. O & S argue that “if

fuzzy-set theory is to represent the conceptual connection among liquidity, investment, and

wealth, it would seem that the only option is to employ fuzzy union,” and what they mean is

one particular fuzzy union, the one expressed by the max operation. This argument is clearly

unwarranted, since f can be chosen from an infinite set (a continuum) of functions, as

described in Section 2. Moreover, it is easy to see that the right function in this example

should be an averaging function, which can be chosen, for example, from the class of

functions characterized by Eq. (7). If there is no special experimental evidence to do

otherwise, we may as well choose the arithmetic average (e.g. the function for l ¼ 1 in Eq.

(7)), so that

WðxÞ ¼LðxÞ þ IðxÞ

2

for all x [ X: Clearly, this function satisfies the required inequalities. For example, let

LðAÞ ¼ 0:92; LðBÞ ¼ 0:9; LðCÞ ¼ 0:05; IðAÞ ¼ 0:1; IðBÞ ¼ 0:94; and IðCÞ ¼ 0:95: Then,

WðAÞ ¼ 0:51; WðBÞ ¼ 0:92; and WðCÞ ¼ 0:5: We can see that all the required inequalities

are satisfied.

This example is particularly illuminating. It shows that fuzzy set theory has some

capabilities that have no counterparts in classical set theory. Indeed, classical sets cannot be

averaged!

3.4. Truth Conditions of Thoughts

The last example discussed by O & S concerns the so-called “truth conditions of thoughts”.

What O & S mean by this term is basically the issue of determining the truth degree of a

proposition of the form

All A’s are B’s: ð10Þ

They claim that propositions of this form are normally assigned the truth condition

ð;x2DÞðcAðxÞ # cBðxÞÞ; ð11Þ

where D is a universe of discourse and cA and cB are membership functions of fuzzy sets

representing the terms A and B in Eq. (10), respectively. Then, they present a

“counterintuitive result”: let D denote the universe of all animals and let A and B denote,

respectively, the concepts grizzly bear and inhabitant of North America. That is, Eq. (10)

becomes “All grizzly bears are inhabitants of North America”. Then, if there is a squirrel

(call it Sam) who lives on Mars and if cAðSamÞ ¼ a . 0 and cBðSamÞ , a; it follows that

cAðSamÞ # cBðSamÞ is not the case, and thus the existence of a squirrel on Mars makes the

truth value of “All grizzly bears are inhabitants of North America” 0 (false). The authors

conclude that “. . .fuzzy set theory does not render prototype theory compatible with the truth

conditions of inclusion”.

For this example, it is essential to recognize that Eq. (11) is not the definition of the truth

degree of Eq. (10). In fact, Eq. (11) defines a crisp (bivalent) relation Sbiv, between fuzzy sets

in D. Sbiv is verbally described as follows: “fuzzy sets cA and cB are in the relation Sbiv, if and

only if for all x from D, it is true that the degree to which x belongs to cA is less or equal to

the degree to which x belongs to cB”. Then, the existence of Sam as described above makes

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the fact that Sbiv holds between cA and cB false, which is completely in accord with the

meaning of Sbiv. To interpret Eq. (10) in the right way, let us rewrite it as:

For all x of the universe: if x belongs to A then x belongs to B.

Using the basic principles of interpreting logical formulas in fuzzy logic, one obtains that

the truth degree S(cA,cB) of Eq. (10) is

SðcA; cBÞ ¼ infx2DðcAðxÞ ) cBðxÞÞ

where ) is a fuzzy implication, i.e. a binary operation in [0,1] that corresponds to an

implication connective, and inf denotes the infimum (minimum if D is finite) in [0,1].

As an illustration, if Lukasiewicz implication,

ða ) bÞ ¼ minð1; 1 2 a þ bÞ;

is used, then one gets

SðcA; cBÞ ¼ infx[Dminð1; 1 2 cAðxÞ þ cBðxÞÞ:

Therefore, Sam’s existence implies that the truth degree of “All grizzly bears are inhabitants

of North America” is at most minðl; 1 2 cAðSamÞ þ cBðSamÞÞ: If, for instance, cAðSamÞ ¼

0:1 and cBðSamÞ ¼ 0:05 then the truth degree of Eq. (10) is at most 0.95 (the existence of

other animals that count to a certain degree for grizzly bears and live outside North America

can make S(cA,cB) still smaller).

The meaning of Eq. (10), when interpreted correctly, therefore satisfies the basic intuitive

requirements of O & S (note that the correct definition of the “subsethood degree” S(cA,cB)

can be found in Goguen’s (1968–9), a paper to which O & S refer).

3.5. Other Issues Related to Truth Conditions

We now briefly comment on further intuitive requirements and criticism formulated by

O & S that are connected to propositions of the form Eq. (10) and to “truth conditions

of thoughts”. They give three reasons for their “distrust of a fuzzy-truth remedy to the

problems cited with fuzzy inclusion”. First, they claim that a calculus that allows one to

assign partial truth degrees to Eq. (10) requires a nontrivial theory not yet undertaken.

We have demonstrated that the right interpretation of Eq. (10) that was already available

in 1968 (in a paper referred by O & S) leads to results that are in accord with the

intuition demonstrated by O & S’s example. If one feels that the more squirrels on

Mars, the smaller the truth degree of Eq. (10) should be the case, which is an issue

mentioned by O & S, one actually has in mind a quantifier like “many”, not “all”,

i.e. instead of Eq. (10) one deals with a proposition of the form “Most A’s are B’s.”

Quantifiers like “many”, “most” etc. have traditionally been studied in logic under the

name generalized quantifiers. A good and up-to-date overview can be found in Krynicki

et al. (1995); in the context of fuzzy logic, generalized quantifiers are studied in Hajek

(1998), Chapter 8.

Second, O & S express a general distrust of a calculus that allows propositions to have

intermediate truth degrees. They claim, without any serious justification, that “infinite

valued logics violate strong intuitions about truth, validity, and consistency.” Their only

example is the following: O & S claims that in Lukasiewicz infinite valued logic, the

sentence “If John is happy, and if John is happy only if business is good, then business is

good” is not a tautology (i.e. has not always the truth degree 1), contrary to intuition.

However, this is not true: if p denotes “John is happy” and q denotes “business is good”

then the above sentence corresponds to a propositional formula q ¼ ð p&ð p ) qÞÞ ) q:

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If n( p ) and n(q ) are truth degrees of p and q, then the truth degree assigned to q in

Lukasiewicz calculus is

minð1; 1 2 maxð0; nð pÞ þ minð1; 1 2 nð pÞ þ nðqÞÞ2 lÞ þ nðqÞÞ

which is always 1, i.e. q is a tautology. Objections to graded approach to truth and to

fuzzy logic can be confronted with literature; for example, Hajek (1998), Novak et al.

(1999) and Gottwald (2001) are examples of recent monographs where fuzzy logics are

well-covered.

Third, O & S “suspect that the partial falsification proposal results from mistaking degrees

of belief for degrees of truth”. We strongly object to the claim that degrees of truth are, in

fact, degrees of belief in disguise. Calculi for dealing with degrees of truth (fuzzy logic) and

degrees of belief (Dempster-Shafer theory of evidence and related theories) are nowadays

well developed; for Dempster-Shafer theory, see e.g. Shafer (1976), and both epistemological

and formal differences between truth and belief are well-recognized.

4. CONCLUSIONS

Our main points in this paper are that: (i) fuzzy set theory is a formalized language of set

theory of much greater expressive power than classical set theory; (ii) to utilize this

expressive power for representing and combining concepts, we need to construct relevant

membership functions and operations by which they are combined or modified in the context

of each particular application; and (iii) the construction may be guided by our strong

intuitions (as understood by O & S), but it must ultimately be based on experimental

evidence.

One cannot expect the complexity of human cognition to be easily captured by any

formalism. In fact, it may well be that there are aspects of cognition that cannot be

represented in a formal system—we do not know. However, fuzzy set theory extends the

domain of formalizability. For example, as we have shown, there are operators, such as

averaging, that are not available in classical set theory.

The current tools of fuzzy set theory are considerably more sophisticated than those

proposed in Zadeh’s original seminal paper (Zadeh, 1965). They are covered

comprehensively in several volumes of a Handbook of Fuzzy Sets published by Kluwer

(especially Vol. 1, edited by Dubois and Prade, 1999), a large Handbook of Fuzzy

Computation edited by Ruspini et al. (1998), as well as in two volumes of selected papers

by Lotfi A. Zadeh (Yager et al., 1987; Klir and Yuan, 1996). In this paper, we introduce

only those that are relevant to our discussion. Other tools of fuzzy set theory can be found

in most up-to-date textbooks in the field (see e.g. Klir and Yuan, 1995). Furthermore,

fuzzy set theory is being continuously extended by an active and growing community of

researchers.

The focus of this paper is intentionally very narrow: a demonstration that each of the

examples of combining or modifying concepts that are discussed by O & S can adequately be

handled by fuzzy set theory to satisfy the strong intuitions or experimental evidence.

Although some criticism of O & S’s conclusions was expressed by, for example, Zadeh

(1982), Lakoff (1987) and Fuhrmann (1988a,b; 1991), this criticism is not sufficiently

specific. We felt that a specific response to each of the specific conclusions made by O & S is

long overdue.

To compensate for the intentional specificity of this article, we intend to discuss some

broader issues of the role of fuzzy set theory in cognitive science in two companion papers. In

one of them, we plan to present a comprehensive overview of current resources offered

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by fuzzy set theory to cognitive science. The other article will be more oriented to discussing

the various conceptual, philosophical, and methodological issues regarding the use of fuzzy

set theory for representing and manipulating concepts.

In writing these three articles, our intent is solely constructive. We want to show that the

frequent negative statements about fuzzy set theory in cognitive science literature, usually

based on the O & S’s paper, are ill founded, and that shying away from fuzzy set theory is

counterproductive to healthy development of the field. It is our hope that our articles will

help to renew interest in fuzzy set theory by at least some cognitive scientists in the years

ahead.

References

Dubois, D. and Prade, H. (1980) Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York).Dubois, D. and Prade, H. (1999) Fundamentals of Fuzzy Sets (Kluwer Academic Publishers, Boston).Fuhrmann, G.Y. (1988a) “Prototypes and fuzziness in the logic of concepts”, Synthese 75, 317–347.Fuhrmann, G.Y. (1988b) “Fuzziness of concepts and concepts of fuzziness”, Synthese 75, 349–372.Fuhrmann, G.Y. (1991) “Note on the integration of prototype theory and fuzzy-set theory”, Synthese 86(1), 1–27.Goguen, J.A. (1967) “L-fuzzy sets”, J. of Math. Analysis and Applications 18(1), 145–174.Goguen, J.A. (1968–9) “The logic of inexact concepts”, Synthese 19, 325–373.Gottwald, S. (2001) A Treatise on Many-Valued Logic (Research Studies Press, Philadelphia).Hajek, P. (1998) Metamathematics of Fuzzy Logic (Kluwer Academic Publishers, Boston).Hampton, J.A. (1993) “Prototype models of concept representation”, In: Van Mechelen, I., Hampton, J.A.,

Michalski, R.S. and Theuns, P., eds, Categories and Concepts: Theoretical Views and Inductive Data Analysis(Academic Press, London).

Hampton, J.A. (1997) “Conceptual combinations”, In: Lamberts, K. and Shanks, D., eds, Knowledge, Concepts, andCategories (MIT Press, Cambridge, MA).

Johnson-Laird, P.N. (1983) Mental Models (Harvard University Press, Cambridge, MA).Kamp, H. and Partee, B. (1995) “Prototype theory and compositionality”, Cognition 57, 129–191.Kerre, E.E. and De Cock, M. (1999) “Linguistic modifiers: an Overview”, In: Chen, G., Ying, M. and Cai, K.Y., eds,

Fuzzy Logic and Soft Computing (Kluwer Academic Publishers, Boston).Klement, E.P., Mesiar, R. and Pap, E. (2000) Triangular Norms (Kluwer Academic Publishers, Boston).Klir, G.J. and Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic: Theory and Applications (Prentice Hall, PTR, Upper

Saddle River, NJ).Klir, C.J., Yuan, B., eds, (1996) Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh

(World Scientific, Singapore).Krynicki, M., Mostowski, M., Szczerba, L.W., eds, (1995) Quantifiers: Logics, Models, and Computing Surveys

(Kluwer Academic Publishers, Boston) Vol. I.Lakoff, G. (1987) Women, Fire, and Dangerous Things (University of Chicago Press, Chicago).Massaro, D.W. (1989) “Testing between the TRACE model and the fuzzy logical model of speech perception”

Cognitive Psychology 21(3), 398–421.Massaro, D.W. and Cohen, M.M. (1993) “The paradigm and the fuzzy logical model of perception are alive and

well”, J. of Experimental Psychology: General 122(1), 115–124.Minsky, M. and Papert, S. (1969) Perceptrons: An Introduction to Computational Geometry (MIT Press, Cambridge,

MA).Novak, V. (1989) Fuzzy Sets and Their Applications (Adam Hilger, Bristol).Novak, V. (1992) The Alternative Mathematical Model of Linguistic Semantics and Pragmatics (Plenum Press,

New York).Novak, V., Perfilieva, I. and Mockor, J. (1999) Mathematical Principles of Fuzzy Logic (Kluwer Academic

Publishers, Boston).Oden, G.C. (1977) “Fuzziness in semantic memory: choosing exemplars of subjective categories”, Memory &

Cognition 5(2), 198–204.Osherson, D.N. and Smith, E.E. (1981) “On the adequacy of prototype theory as a theory of concepts”, Cognition 9,

35–58.Osherson, D.N. and Smith, E.E. (1982) “Gradeness and conceptual combination”, Cognition 12, 299–318.Roth, E.M. and Mervis, C.B. (1983) “Fuzzy set theory and class inclusion relations in semantic categories”, J. of

Verbal Learning and Verbal Behaviour 22, 509–525.Ruspini, E.H., Bonissone, P.P., Pedrycz, W., eds, (1998) Handbook of Fuzzy Computation (Institute of Physics,

Philadelphia).Shafer, G. (1976) A Mathematical Theory of Evidence (Princeton University Press, Princeton, NJ).Smith, E.E. and Osherson, D.N. (1984) “Conceptual combinations with prototypes concepts”, Cognitive Science 8,

337–361.

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Smith, E.E., Osherson, D.N., Rips, L.J. and Keane, M.T. (1988) “Combining prototypes: a selective modificationmodel”, Cognitive Science 12, 485–527.

Yager, R.R., Ovchinnikov, S., Tong, R.M., Nguyen, H.T., eds, (1987) Fuzzy Sets and Applications—Selected Papersby L. A. Zadeh (Wiley, New York).

Zadeh, L.A. (1965) “Fuzzy sets”, Information and Control 8(3), 338–353.Zadeh, L.A. (1978) “PRUF—a meaning representation language for natural languages”, Intern. J. of Man–Machine

Studies 10(4), 395–460.Zadeh, L.A. (1982) “A note on prototype theory and fuzzy sets”, Cognition 12, 291–297.

EPILOG

The purpose of this epilog is twofold. First, we want to explain what preceded the publication

of this paper in this journal; and, second, we want to announce our plans for further writings

on the subject initiated in this paper.

Since this paper deals with specific claims made by O & S in Cognition, we considered it

appropriate to publish it in the same journal. Hence, we submitted it to Cognition (in August

2001). We received a negative response from the editor, Jacques Mehler, on February 21,

2002. He decided not to publish the present paper, but encouraged us to submit to Cognition

our second, broader paper, which is mentioned in the present one. His decision was based

upon two reviews. One of them is rather short:

The paper is yet another response to Osherson and Smith (1981). This might be sufficient reason to reject it, butsince you sent it to me to review, you apparently are still open to furthering this debate. Personally, I think thatthis is not appropriate 21 years later, so I prefer not to review this paper.

The second review is quite thorough and highly positive. It consists of a general statement

and a series of detailed comments. The following is the general statement:

Overall I think this is a useful paper and deserves to be published, and should indeed be published in the samejournal as the paper it principally addresses and responds to, Osherson and Smith (1981). I would rate it asacceptable without revision, though I have some suggestions for improvements below. I think it is fine on all ofthe six items listed in the instructions to reviewers.

The suggestions made by this reviewer are relatively minor, but useful and constructive,

and we revised the paper according to virtually all of them. We are grateful to the anonymous

reviewer to pointing our attention to the relevant paper by Kamp and Partee (1995), with

which we had not been familiar.

Needless to say, we strongly disagree with the first reviewer that it is not appropriate to

respond to the paper by O & S 21 years later. On the contrary, we deem it even more

important now than it was some two decades ago to show in specific terms that virtually all

conclusions made by O & S about representing concepts by fuzzy logic are mistaken. It is

more important now, because these erroneous conclusions have been uncritically accepted

for all these years by a whole generation of cognitive scientists.

We thus dismiss the argument in the first review (or, rather, nonreview) that it is not

appropriate to publish “another response to O & S”, especially because none of the very few

responses that addressed O & S’s criticism of fuzzy set theory were sufficiently specific and

comprehensive. We publish the paper in this journal on the basis of the second, positive

outside review. The version published here is a revised version in which the various minor

changes suggested by the reviewer are incorporated.

We are starting now to write the follow-up paper tentatively entitled “Concepts and

Fuzzy Set Theory”, which we plan to submit again to Cognition. In this paper, we intend

to discuss issues regarding the use of fuzzy set theory for representing and manipulating

concepts. This paper will be considerably broader than the present paper, not restricted to

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O & S or to the prototype theory of concepts. However, the present paper will be an

important reference in it.

Finally, our third paper in the sequence will be a comprehensive overview of all resources

available now within the areas of fuzzy set theory and fuzzy logic, viewed in the broadest

sense, and a discussion about the utility of these resources in cognitive science.

Radim Belohlavek graduated in 1994 from Palacky University,

Olomouc. In 1998, he received his Ph.D. in Computer Science from

the Technical University of Ostrava, in 2001 he received a Ph.D. in

Algebra from Palacky University, Olomouc. In 1992–93, he obtained

a scholarship and spent 8 months at the University of Bern

(Switzerland). In 1999, he obtained a post-doc position at the State

University of New York at Binghamton, in 2000 he obtained the

NATO Advanced Fellowship. Since 2001, Radim Belohlavek is the

Head of Department of Computer Science, Palacky University,

Olomouc (Czech Republic). His interests are in pure and applied

algebra and logic (fuzzy logic, universal algebra, lattices), processing

of indeterminate information (soft computing, uncertainty calculi,

data analysis), and foundations of mathematics and computing. He published one book and

over 30 journal papers in these areas.

Since 1998 Hal Lewis has been in his current position as associate

professor of systems science at Binghamton University. He previously

taught for 8 years at Fukushima University in Japan and has 13 years

of industrial experience working in the U.S. and Japan. His research

interests include soft computing and other hybrid approaches to

control and to the modeling of human behavior.

Eileen C. Way is currently a professor in the Department of Systems

Science and Industrial Engineering at Binghamton University,

Binghamton, New York. She received her Ph.D. in Advanced

Technology from Binghamton University in 1988. Prior to joining the

faculty in Systems Science and Industrial Engineering in 2000, she

was director of the program in Philosophy, Computers and Cognitive

Science in Harpur College. Her books include Knowledge

Representation and Metaphor and Realism Rescued, co-authored

with J. Aronson and R. Harre. Her areas of research interest include

cognitive science, intelligent systems and systems design.

For biography and photograph of George J. Klir, please see Vol. 30, No. 2, 2001,

pp. 132–132 of this journal.

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