Models of concepts

COGNITIVE SCIENCE 8, 27-58 (1984)

Models of Concepts*

B E N J A M I N C O H E N

Department of Computer Science Boston College

G R E G O R Y L . M U R P H Y

Department of Psychology Brown University

Over the lost 10 years research on concepts has produced an important new theory known as prototype theory. Despite its empirical successes, prototype theory has been challenged by various arguments purporting to show its descriptive inadequacy for a variety of phenomena, including complex concepts and quantif ication. These arguments are pr imari ly based on a set- theoretic model of concepts. We consider the advantages and disadvantages of the set-theoretic approach and argue that if we instead model concepts as knowledge representations of a certain kind, it is possible not only to answer prototype theory's critics, but to address more fundamental issues in the theory of concepts. We also consider the implications of these different approaches for psychology, linguistics, and artif icial intell igence (AI). To substan- t iate our claims, a knowJedge representation model of prototype theory is outl ined, based an work in schema theory and AI knowJedge representation.

1. INTRODUCTION

What is a concept? The very term "concept" has so man) common and technical meanings that even to raise the question of how to model concepts, much less provide the answer, poses difficult conceputal issues. If one were to ask a sample of cognitive scientists what a theory of concepts should explain, one could get a different answer from each person, depending on his or her intended use for the theory. Somewhat different notions of a concept may lurk in AI knowledge representation systems, theories of natural

*The order of authorship is alphabetical. The authors would like to thank Herbert Clark, Raymond Gibbs, Frank Halasz, James Hampton, Brian Ross, and Thomas Wasow for comments on an earlier draft. Correspondence and requests for reprints should be addressed to Benjamin Cohen, Department of Computer Science, Boston College, Chestnut Hill, MA 02167 or to Gregory Murphy, Department of Psychology, Brown University, Providence RI 02912.

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28 COHEN AND MURPHY

language understanding, perceptual processors, theories of logic and semantics, and psychological accounts of semantic memory. In some of these enterprises, a concept is regarded as a definition of a term, or a statement of the defining conditions for membership in the class designated by the term. In others, it is regarded as a representation of a chunk of knowledge that can be used to categorize and understand a domain of objects, events, or processes. Suppose we narrow the question and ask: How are concepts mentally represented? Even here, where there has been much recent psychological research (see Mervis & Rosch, 1981; Smith & Medin, 1981), there seems to be confusion over what a theory of concepts is about.

To gain some perspective on the problem of how to model concepts, it is natural to appeal to the logical and semantic tradition which goes back to Frege (1892). Two of its recent products are semantic theories of natural language as developed by linguists and philosophers (see Bierwisch, 1970; Dowty, Wall, & Peters, 1981; Katz, 1972; Partee, 1976) and fuzzy set theory as developed by Zadeh (1965, 1978) and his coworkers. These and related theories attempt to describe and explain such semantic phenomena as logical entailment and quantification. Although the theories proposed shed considerable light on people 's logical and linguistic competence, they tend to take a formal approach and deal with only a small fragment of the range of phenomena that psychological theories of concepts aim to explain. More- over, their connection to psychological representations and processes is tenuous. The difference in approach between psychologicN theories and formal semantic theories is reflected in different approaches to the notion of a concept itself. Because of its mathematical foundations, logical and semantic research has often adopted an extensional or set model approach: a concept is modelled as a set, rather than as some form of mental representation. Thus, for example, the concept furniture might be modelled as the class of objects falling into the category furniture, rather than as a representation of that class. Psychological research is generally ambiguous between this extensional notion and the more recent model of a concept as a certain kind of mentally represented knowledge structure known as a prototype.

In this article we show how this ambiguity has engendered considerable conceptual confusion and the neglect of real issues of representation and process. In our view, the path to conceptual clarity lies in basing a theory of concepts on computat ional theories of knowledge representation, rather than on formal semantic or set models. The latter continue to play an indispensable role, but as a part of the metatheory, not directly as part o f the theory of concepts. Furthermore, we argue that the knowledge representation approach provides a common ground for psychologists, linguists, and researchers in artificial intelligence, and we consider research on concept formation in all three fields in the course of our discussion.

MODELS OF CONCEPTS 29

1.1. Background

This section reviews the basic positions in the theory of concepts. (Readers already familiar with the issues and literature may wish to skip it.)

According to the classical theory of concepts, each concept corresponds to a set or collection of entities, in which membership is all-or-none. This tradition may be traced to the Aristotelian view that each concept has a definition characterizing its "essence" and providing necessary and sufficient conditions for concept membership. Membership in a concept is considered to be all-or-none: either an object fulfills all of the conditions in the definition, in which case it is a member, or else it fails some condition(s), in which case it is a nonmember.

This notion leads to certain obvious questions about the psychology of concepts: How do people learn the concept definitions? How do the definitions operate in language comprehension? How do the definitions orga- nize representations of concepts? These questions are addressed in early studies of concept acquisition (e.g., Bourne, 1966; Brunet, Goodnow, & Austin, 1956) and in initial work on semantic memory (e.g., Collins & Quil- lian, 1969). Smith and Medin (1981) provide a more complete account of this tradition.

In linguistics, semantic theorists attempted to specify certain features or components of words that were believed to be necessary and sufficient conditions (e.g., Bierwisch, 1970; Katz, 1972). For example, the word bachelor might contain the semantic components [unmarried], [adult], and [male]. Objects that did not match all of the features could not accurately (and literally) be named by that word. An object that lacked the [unmarried] feature might be called a man, or, if the [married] feature were present, husband.

This elegant and widely held theory of concepts did not stand unchal- lenged, however. In psychology, researchers began to emphasize the importance of nondefining features in people's concepts (e.g., Rosch, 1975; Smith, Rips, & Shoben, 1974). For example, one's concept of a bird seems to fit robins better than chickens, even though both appear to satisfy bird's definition. One's concept of a bachelor does not match the Pope very well, although he is certainly an unmarried adult male. Thus, it seemed that people knew more about concepts than the bare bones of their definitions, and this knowledge was shown to have consequences for their use of these concepts in various tasks (Rosch, 1975; Smith, et al., 1974). Furthermore, some objects did not seem to be clearly in or out of a concept. For example, people are unsure whether a boxing glove is an example of clothing, or whether a tomato is a fruit. These facts challenge the all-or-none character of concepts proposed by the classical model.

Perhaps the greatest problem for the classical view is its inability to specify the defining features for natural kind concepts such as water, horse,

30 COHEN AND MURPHY

or table. In contrast to members of an arbitrary set, members of a natural kind share a relation of similarity that partly explains their belonging to the same kind (Cohen, 1982; Quine, 1969). wittgenstein (1953) first noted the difficulty of finding either sufficient or necessary conditions for something being a game, and Putnam (1975) extended this argument for other natural kind terms, such as lemon, tiger, and gold.

Given these problems, psychologists began to develop other theories of concept representation. Eleanor Rosch (1978; Rosch & Mervis, 1975) proposed that concepts are organized around a best example or prototype. The prototype is usually conceived of as an "average m e m b e r " of the category. That is, it contains the most frequent attributes of the category members. There may be no real object in the world that corresponds to the prototype, as it is an idealized abstraction of the individuals in the category. People are assumed to use the prototype rather than a category definition in identifying members and in reasoning about the category.

This view explains why there are no clear boundaries between the members and nonmembers of a concept: objects vary continuously in their similarity to the prototype, so it is difficult to say just where the category begins and where it ends. It also explains why some category members (e.g., robins and swallows) can be identified as members more quickly than others) (e.g., chickens and vultures): the former are more typical of the concept bird--they resemble the prototype more closely. There is an impressive amount of evidence for this view (summarized in Mervis & Rosch, 1981). Furthermore, this theory of concepts has been incorporated in more general theories of knowledge representation, such as Rumelhar t ' s (1980) version of schema theory and Fahlman 's (1979) NETL.

Some linguists also came to believe that the purely componential view of meaning was greatly oversimplified. Labov ' s (1973) demonstrat ion of fuzzy category boundaries and Lakof f ' s (1973) study of linguistic hedges were particularly influential.

Finally, some logicians also saw the need for changing their logical systems, which always depended on the law of excluded middle (in this context, an object either is in some set or is out of it, with no gradation). Such systems began to seem far removed from human thought and reasoning given the empirical findings discussed above. In 1965, Zadeh proposed a system of logic based on fuzzy sets to more realistically capture people 's judgments of logical relations. Membership in fuzzy sets is graded, thus fuzzy sets are compatible with the basic facts of prototype theory. For each object x and concept C in question, Zadeh 's system assigns a value in [0,1], called a characteristic value, or a degree of typicality, TYPe(x), depending on x 's typicality as a member of the concept C. For example, a garden variety robin named Fred might receive a characteristic value TYPaIRo (Fred) close to 1 for the category BIRD, reflecting its high typicality, whereas a chicken might receive a value closer to .5, and a clear nonmember (e.g., a


rock) would receive a value close to 0. Zedeh then devised logical rules to extend the rules of conjunction, disjunction, and quantification used in classical logic. Zadeh's rules (including recent developments, 1978, 1981) seem to be the only systematic attempt to describe how prototype-based concepts should be combined and evaluated.

1.2. Theories of Representation

Formal mathematical theories of representation, as described by Krantz, Luce, Suppes, and Tversky (1971; or see a less forbidding description in Coombs, Dawes, & Tversky, 1970) distinguish the domain being represented, D, from a representational model, R of D. Often, D is a part of the actual world, and R is some scientific theory about D. As Krantz et al. describe (Chapter 1; or Coombs et al., Chapter 2), R represents D, if every object in D has a corresponding name in R, and every relation in D has a corresponding relation symbol in R. Normally, R uses numeric relations (greater than, equal to, etc.), whereas D's relations are based on actual properties of the objects (mass, velocity, intelligence).

People's mental representations undoubtedly include mental "objects" corresponding to real objects (or events) and mental relations that correspond to or capture the relations between real objects. For example, the spatial relation that obtains between a cat and a mat when Morris the cat is on the mat may be mentally represented by the proposition "Morris is on the ma t . " However, even when people's representations are veridical, they seldom provide a complete picture of the environment. People's mental representations certainly do not include information about the spatial location of all cats, or even about all the cats they know. Palmer (1978) has pointed out that psychology studies mental representations, in the course of which it constructs its own representations (flowcharts, traces, scripts, schemata, etc.). Therefore, psychology's representations are twice removed from the environment: they are (theoretical) representations, R~, of (mental) representations, R2, that in turn represent the external environment~D. Thus, it may be neither necessary nor desirable for psychological representations in R~ to completely represent D. When we offer a theory of concepts, we are not proposing a theory about how the external environment is structured, but rather about how people conceive of that environment. We may well need an independent account of the environment in order to compare people's conceptual structures to the environment 's structure, but our account of concepts themselves need not be a complete, veridical description of the environment.

Set theory is a powerful representation device that can be used to model virtually any domain by assigning an element to each object in the domain and then choosing appropriate sets and relations (sets of ordered pairs, triples, etc.) to correspond to sets and relations in the domain. A

32 COHEN AND MURPHY

small number of operations on sets (intersection, union, complementation) can be used to capture logical operations and relations. Moreover, set theory can be used to model representation itself. Tarski's celebrated theory of truth (1956) uses set theory to define truth as a correspondence between symbolic representations in a formally specified language R (first-order logic) and a set-theoretically represented domain D. Theories of natural language semantics based on Tarski's work (e.g., Montague, 1970) have em- ployed set representations extensively in order to compare sentence meanings to a set-theoretic model of the world.

The importance of set theory to the theory of concepts is twofold: first, it offers a way of modelling virtually any domain; and second, it is able to model formal representations of domain knowledge. In both cases, the role played by set theory is metatheoretic. In the first case, it serves more as an abstract theory of the structure of the environment than as a theory of concepts," and in the second case, its role depends on the extent to which people's representations are set-like. Below, we shall argue that people's representations differ fundamentally from set theory's representations. Therefore, we conclude that set theory is not a proper part of the theory of concepts, but of the metatheory.

Of course, we could use the same (meta-) representational system to describe people's representations and the environment. But doing so would presume that the formal features of a logical model are equally appropriate or useful for a psychological model. If we assume that a theory of concepts should describe mental representations, then the choice of that representation should be based on psychological evidence. We argue below that psychological evidence is quite good for schematic representations, but is not consistent with simple set representations (Johnson-Laird, 1975, 1980). It is perhaps inconvenient that it is so, but people's concepts and their conceptual operations are not designed for formal tractability, nor to embody a formal theory of truth, but rather have evolved to represent many different forms of knowledge within a highly constrained system. The mistake of proposing set models for concepts, we believe, is that it confuses a powerful mathematical model of the world with a model of mental representation.

2. SET MODELS

There are two basic distinctions to be drawn between different models of concepts. First, there is the distinction between classical (all-or-none) and fuzzy models. Second, there is the distinction between extensional (based on set theory) and representational models. These distinctions are independent, in that all four possibilities can be entertained as models of concepts. Table 1 summarizes the four possibilities. We consider each of the possibilities be-


low. In Section 1. l, we reviewed the shift in psychology from a classical to a fuzzy model. What are the issues in the extensional-representational distinction?

TABLE I

Fuzzy Classical

Representational prototype theory componential semantics

Extensional fuzzy set theory set theory 096s)

In extensional models, concepts are regarded as equivalent if their corresponding sets have the same members, and concept formation is based on operations that combine extensions, e.g., set union. In representational models, concepts are modelled as complex structured descriptions. They are regarded as equivalent if they have the same components structured in the same way, and concept formation is based on structural rules for forming complex representations. A paradigm case of an extensional model is the fuzzy set model (Zadeh, 1965); a paradigm case of a representational model is a knowledge representation language (e.g., Bobrow & Winograd, 1977).

Extensional models reduce all questions of conceptual equivalence and concept formation to questions of class membership. No further descriptive knowledge of instances of a concept, of its associated features, or its relations to other concepts needs to be taken into account. Instead, classes are identified by their members and combined by Boolean operations. The simplicity and transparency of extensional models accounts for a large part of their continuing appeal. A further advantage much prized by semantic theorists (e.g., Dowty et al., 1981) is that they have been invalu- able in constructing a theory of truth for various fragments of natural language.

Unfortunately, extensional models also have certain fundamental disadvantages. To use a well known example, on an extensional model, the complex concepts underlying the descriptions "creature with a hear t" and "creature with kidneys" are equivalent if the set of all creatures with a heart has the same extension as the set of all those with kidneys. Yet the meanings of these descriptions surely differ. The inability of extensional models to capture such differences is one of their main limitations.

2.1. Classical Models

In this section, we formulate representational and extensional versions of the classical theory of concepts. The extensional version is simply standard

34 COHEN AND MURPHY

set theory, in which a concept is modelled by all the objects that exemplify it. Different versions of this theory can be created with different processes to operate on these sets.

In the representational version of the classical model, a concept is modelled by:

1. a finite list of primitive (undefined) names or primitive descriptions of individual instances; or

2. a definition based on a (finite) list of defining conditions or features that are shared by all instances.

There are many possible representation schemes that can satisfy (1) and (2). The particular representations given below are merely intended to illustrate the classical character of the model and make no attempt at empirical adequacy or processing efficiency. An example of (1) is the finite enumeration (in LISP-like prefix notation):

3. (IMPLIES (BIRD X) (OR EQUAL X b,) (EQUAL X b2).. . (EQUAL X b.) ) )

In this example, b, are primitive names or exemplars and the list is represented as a disjunction of equality statements (EQUAL X b,). In the simplest case, to check whether something is a bird, one might sequentially search (3) (or some equivalent knowledge structure) attempting to match the possible instance to the stored descriptions b,. An example of (2) is the following pair of rules.

4. (IMPLIES (BIRD X) (AND (FLIES X) (HAS-WINGS X) (HAS-BEAK X) ) );

5. ( IMPLIES (AND (FLIES X) (HAS-WINGS X) (HAS-BEEK X) ) (BIRD X) ).

The first rule lends itself to making inferences about necessary properties of BIRDs and has a conjunction of conditions or features on the right hand side. The second lends itself to recognizing sufficient conditions for BIRD- hood, and has the same conjunction on the left hand side.

What makes (4) and (5) "def ini t ional" is an interpreter with a set of recognition procedures that uses rule (5) to recognize instances of the class BIRD by checking to see whether a given object satisfies all of the (sufficient) features, together with a set of inference procedures that use rule (4) to infer that a given instance of BIRD has each (necessary) feature. Thus, it is both the representation scheme and its associated processes that make this a classical model.


Smith and Medin (1981) review the evidence demonstrating that such simple classical models and other more sophisticated versions cannot account for the psychological data on concepts. Fuzzy models of concepts seem to have greater empirical adequacy.

2.2. Fuzzy Set Models

The standard extensional model of fuzzy concept theory is the original fuzzy set model proposed by Zadeh (1965) and formalized by Osherson and Smith (1981). This extensional model represents the concept bird by a quadruple < B, TYPB,Ro(X), b, d>consis t ing of a .class B of birds, a characteristic function TYPB,Ro that measures degrees of "birdiness" by mapping all birds in B into the continuum [0,1], a prototypical bird b, and a distance metric d between members of B. Here, a concept is primarily represented by assigning a typicality value TYP,,~o(x) to each object x. Although the central typicality facts are readily captured by this model, it says little about how the concept bird is mentally represented. The function TYP~,Ro is given ex- tionsionally, that is, there are no rules specified for actually computing TYP~,~o(x) for any x. There is no syntactic or lexical information in the model - -no t even the name " b i r d " appears--and nothing is said about typical properties of birds, e.g., having wings, or flying, or about bird's relations to other concepts, e.g., the superordinate animal. Although the quadruple < B, TYPs,~o (x), b, d > includes the prototype b, none of the rules of concept formation actually use b, nor do they depend on domain knowledge. This strict separation of concepts from domain knowledge is characteristic of an extensional approach.

One justification of the fuzzy extensional approach is offered by Osherson and Smith who say that they think it wiser "in the present era of psychological understanding" to "settle for a specification, not of the representation (i.e., not of [the concept] C itself), but of the associated quadruple (i.e., of what C represents)" (1981, p. 40). That is, they wish their model to describe classes of objects ("what C represents") rather than mental representations ( "C itself"). It is not immediately obvious how to derive a psychological model from accounts of this form. A concept in this extensional model is not a representation of a category or set, but the set itself. Yet, everyone will agree that people who recognize birds do not have the set of birds stored in their memories. And it remains unclear how merely giving extensional descriptions of classes of objects in a given domain will tell us how people conceive of those objects, or how they represent the concepts that pick out those objects. (In fact, standard textboks of set-theoretic semantics often bypass the question of psychological representation com- p le te ly-see , for example, Dowty et al., p. 13. Again, these theorists are interested in formal correctness and compositionality rather than the psy-

36 COHEN AND MURPHY

chological issues.) Of course, characterizing classes and their combinations might lead to a theory of representations, but we have seldom seen this argument spelled out (see Goguen, 1981).

At this point, we should present a fuzzy representational model. How- ever, there are too many possible models to be able to consider them all (see, for example, those discussed by Smith & Medin, 1981). In general, this variety of concept theory says that concepts are mental descriptions (or images) which objects fit to a greater or lesser degree, thereby making the concept fuzzy. Many of these models suggest that concepts are representations of a best example or prototype. We propose a specific representational model after considering the objections to fuzzy theories in general.

3. OBJECTIONS TO FUZZY MODELS

Understanding how concepts are combined to form complex concepts provides considerable insight into the structure of concepts and domain knowledge in general. In this section, we discuss data on conjunctive, disjunctive, and contradictory complex concepts, and also on quantification, in order to review certain objections to the fuzzy models of each phenomenon. In our view, when these objections are valid, they arise from the lack of domain knowledge in standard extensional models rather than from fuzzy models per se. That is, these objections are based on fuzzy set models, but have been used as evidence against all fuzzy models.

3.1. Evaluation Criteria

How does one evaluate models of concepts? Above, we argued that fuzzy models (both extensional and representational) improved on classical models by allowing degrees of membership, thus rendering them compatible with psychological studies of concepts. Such empirical constraints, however, are not the only way to evaluate a model. In a recent paper, Osherson and Smith (1981) suggest two further criteria of adequacy. The first is the compositionality constraint: any account of concept formation should provide rules that construct complex concepts using only the representations for simple concepts. Thus, for example, if A and B correspond to natural language concepts, one should provide rules for constructing the complex concepts A-and-B, A-or-B, and A-B (where appropriate). The second constraint is to give truth conditions for quantificational statements such as "All A's and B's." They argue that prototype theory satisfies neither of these criteria, and is therefore an inadequate theory of concepts.

Certainly, we can imagine a considerably different set of constraints on a theory of concepts (some of which are probably implicitly assumed by


proponents of these two constraints), such as the requirement that the theory should be compatible with what we know about lexical organization or that it should explain how children acquire concepts, and the like. Al- though we accept (a restricted form of) the truth-condition constraint, we shall argue that people's concepts simply do not obey the compositionality constraint and, therefore, it is not a proper constraint.

3.2. Conjunctive Concepts

In extensional fuzzy models, concepts like A-and-B are formed by conjoin- ing the sets associated with A and B. To form conjunctive concepts, it uses the rain rule:

TYPA N 8(X) = min [TYPa(x), TYPe(x) ],

which assigns typicalities (TYPA AB(X) ) tO objects based on their typicality in A and B. For example, the degree of membership in the concept red apple is determined by:

TYPR~o N appL~(X) = min[ TYPR~o(x), TYPApeLE(X) 1.

So, if an object has a characteristic value of .9 for apple and .8 for red, its value for redapple will be the minimum of the two, .8. To form the complex concepts red apple or pet fish, the min rule requires no knowledge of prototypes or exemplars of the component categories. Thus, it requires no knowledge that apples are fruit, that instances of fruit have a color, that red is a color, or that a goldfish is an exemplar of pet fish.

A harmless looking implication of the min rule is the following:

TYPA f) ~(x) < TYPA(x), TYPB(x).

This inequality is also entailed by classical set theory, in which the characteristic function is restricted to the binary values 0 and 1. It says that the typicality or degree of membership in a conjunction is never greater than in either of the conjuncts; therefore, it ensures that if something is not a " g o o d " A, it cannot be a " g o o d " A B. Osherson and Smith observe that the implication is false for real world categories: for example, the goldfish is a more typical pettish than either a fish or a pet. (Smith & Osherson, 1982, provide extensive evidence for the existence of such cases.) Thus, the min rule is falsified: typicality may be higher in the combination than in the components.

These and related counterexamples may seem to challenge prototype theory. Our view, however, is that it is the intersective combination rule that

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is c h a l l e n g e d , n o t p r o t o t y p e t h e o r y . ' A l t h o u g h the m i n ru le is c l ea r ly i nade -

q u a t e , the m a i n a p p e a l o f p r o t o t y p e t h e o r y c o n t i n u e s to h o l d fo r so -ca l l ed

c o n j u n c t i v e c o n c e p t s as well as fo r s imp le c o n c e p t s : c o n j u n c t i v e c o n c e p t s

h a v e b o t h a g r a d a t i o n o f m e m b e r s h i o p a n d u n c l e a r cases . F o r e x a m p l e , the

pe t g u p p y in o n e ' s a q u a r i u m is sure ly a typ ica l pet f ish , bu t the t r o u t in

o n e ' s p o n d is o n l y a m a r g i n a l m e m b e r . ' We sugges t b e l o w tha t ru les o f c o n -

cep t f o r m a t i o n based on d o m a i n k n o w l e d g e , in p a r t i c u l a r , on k n o w l e d g e o f

p r o t o t y p e s , can be used to a v o i d the p r o b l e m s o f the m i n rule .

3.3. Complex Concepts in Natural Language

In na tu r a l l a n g u a g e , c o m p l e x c o n c e p t s a re exp re s sed by n o u n ph ra se s o f t he

f o r m a d j e c t i v e - n o u n or n o u n - n o u n . F i r s t , we will c o n s i d e r n o u n - n o u n c o m -

b i n a t i o n s . T h e r e is a vas t r ange o f n o m i n a l c o m p o u n d s tha t s i m p l y c a n n o t

be i n t e p r e t e d in t e rms o f c o n j u n c t i o n : " m o r n i n g f l i g h t , " " e n g i n e r e p a i r , "

" U n i t e d S ta tes S e n a t e M i c h i g a n b e a n s o u p , " " o c e a n d r ive . ' '3 In each o f

these c o m p o u n d s , k n o w l e d g e o f a mediat ing relation b e t w e e n c o n c e p t s is

r e q u i r e d . A typ ica l " m o r n i n g f l i g h t " re fe rs to a f l igh t occurring in t he

m o r n i n g . A " m o r n i n g c o a t " re fe rs to a c o a t t yp ica l ly worn in t he m o r n i n g .

" U n i t e d Sta tes S e n a t e M i c h i g a n b e a n s o u p " re fe r s to a s o u p made o f beans

grown in M i c h i g a n a n d served to me mbe r s o f the S e n a t e o f the U n i t e d

Sta tes . O n c e the m e d i a t i n g r e l a t i o n is k n o w n , o n e can g e n e r a l l y r e c o v e r

'Prototype theory may even be tenable within the extensional framework. For example, given two concepts, X and Y, or fish and pet, one might form the conjunctive concept XY (pet fish) by finding a new prototype that is an "average member" of the intersection X Y, using the principles suggested by Reed (1972) or Rosch and Mervis (1975). The distances from each object in the domain to the new prototype determine a characteristic function for the complex concept XY. Using this rule, unlike the min rule, the fact that a goldfish is an atypical fish does not affect its typicality as a pet fish, since it might be an "average exemplar" of the conjunction even though it is not an average member of either concept alone. The problem with this revision, as with all such conjunctive accounts, is that it fails to generalize to the large class of non-intersective concepts that we discuss below.

2Tversky and Kahneman (1982) give some empirical evidence that conjunctive concepts have prototypes and characteristic functions similar to simple concepts. They described to their subjects a woman who was a social activist in college (they also gave other details), and then asked the subjects to rank how likely it was that certain descriptions of her were true. Over 88% of the subjects (including a group familiar with probability theory) said it was more likely that the woman "is a bank teller who is active in the feminist movement" than she "is a bank teller" even though this defies the law of probability that the conjunction of two events is not more likely than either event alone. Tversky and Kahneman make a convincing argument that people judged how representative the woman's description was of each category, which is consistent with our suggestion that people form prototypes for conjunctive concepts like bank teller and feminist and make typicality judgments relative to these concepts. Such a finding is evidence against the fuzzy set min rule as well as against a classical model with no prototype. (These rules preserve the probability law that people apparently violate.)

'We owe some of these examples to Sandra Cook (personal communication) and Terry Winograd (personal communication).


some form of conjunctive analysis--presumably the extension of "morning flight" is something like the intersection of " the class of all events occurring in the morning" and " the class of all fl ights." But this conjunctive analysis is post hoc. It is the mediating relation occurring in the morning that is crucial, not set membership. No information is gained by simply considering the intersection of the class of mornings and the class of flights. Further- more, other complex concepts with the same modifier may invoke different mediating relations: morning coat is not the intersection of the class of events occurring in the morning and the class of coats. Even in paradigm cases of conjunctive concepts such as red apple, we would argue that the concept of color mediates between the two concepts.

The analysis in terms of mediating relations avoids the well-known problem mentioned above of identifying the meaning of a term with its extension: the mental representation of creature with a heart and creature with kidneys will have different mediating relations (based on knowledge about the differing functions of the heart and kidneys), even though the concepts may have the same extension. An extensional account does not appear to be able to capture this difference, since it represents the set of creatures with a heart as opposed to people's knowledge about such creatures.

In a linguistic analysis of compound noun phrases, Kay and Zimmer (1976) point out that there may be an indefinite number of interpretations of a single compound NP, such as "f inger cup . " They argue that there are no uniform linguistic rules (like conjunction) that specify what these phrases mean. Rather, there is some relation between the two nouns that the listener can deduce in context, based on the meaning of the words and the rest of the utterance. (Adams, 1973, Chapter 5, also documents the diversity of relations between nouns in nominal compounds.) If this analysis is even roughly correct, the set model of concept formation would seem to have little relevance to comprehension of compound NPs, since it provides an in- flexible, knowledge-independent rule for combining concepts. We will later discuss a knowledge representation approach that is more consistent with Kay and Zimmer's analysis.

Turning to adjective-noun combinations, phrases such as "large mos- qu i to , " "expensive pencil ," and "fas t snail" are equally unsusceptible to the intersection analysis given by the min rule. ' That is, there is nothing that is both fast and a snail, so there should be no fast snails. So-called syncate- gorematic adjectives such as "a l leged," " u t t e r , " and " m e r e , " which seem to define no sets (and apparently have no prototypes) are worse, yet they combine with nouns in a predictable manner. Even so-called "abso lu te"

' A formal semanticist might suggest that the intersective analysis L AM of large mosquito be rejected in favor of treating a concept such as large as a choice function L that selects a subset L(X) o f any concept X that it modifies. Al though this analysis does not say how the subset is determined, it is compatible with the non-emptiness of the concept large mosquito as ordinarily understood. Unfortunately, this rule also needs mediating relations to account for nominal compounds such as ocean drive.

40 COHEN AND MURPHY

concepts (Katz, 1972) such as color terms seem to ascribe different properties depending on what they are conjoined with (e.g., " red ha i r" vs. "red apple" vs. " red cabbage") .

These and similar examples of complex NPs make it doubtful that concepts are represented extensionally and combined independently of domain knowledge. We conclude that complex noun phrases are not intersective.

3.4. Disjunctive Concepts

Disjunctive concepts are composed of concepts connected by or: for example, the explicit logical disjunction "elephants or t igers." It has been argued that natural language superordinate concepts (e.g., vehicle, furniture, tool) are mentally represented as disjuncts of more subordinate concepts (Miller & Johnson-Laird, 1976; Murphy & Smith, 1982; Newport & Bellugi, 1978). Thus, vehicle might be represented as the disjunctive concept car or boat or helicopter; or furniture as chair or table or rug or desk. Any theory of concepts must capture this sort of structure. How does prototype theory explain it? The fuzzy set rule for disjunctive concepts is the max rule, the dual of the min rule, which assigns typicality values to objects in the disjunctive category:

T Y P ~x) = T Y P c U n tO n(x) = max[ TYPc(x), TYPB(x), TYPn(x) ].

This rule says that something can only be as typical a vehicle V as it is typical of one of the disjuncts car, boat or helicopter. Thus, a raft is an atypical boat, which makes it an equally atypical vehicle. The problem with this rule is that any object that is a good example of one of the disjuncts is a good example of the superordinate category, which appears to be false. For example, the most typical helicopter is still an atypical vehicle, yet the max rule will give it a high value as a vehicle. Once again the fuzzy set rule is wrong. Yet here again, concepts fit the description of prototype theory: as it predicts, disjunctive categories (such as natural language superordinates) do have graded membership and unclear members (Rosch & Mervis, 1975). Although some concepts may well be disjunctive, the max rule is not the only way to represent them.

3.5. Contradictory Concepts

In the extensional analysis, a contradictory concept such as apple that is not an apple, has a null extension--nothing can belong both to a classical set A and its complement not-A. Yet membership gradients allow objects to belong (i.e., have non-zero degrees of membership) in both apple and n o t -


apple. These facts led Osherson & Smith (1981) to the conclusion that prototype theory violates strong intuitions about contradictions.

Unlike previous objections, this one does not depend on fuzzy set theory. Any version of prototype theory must claim that objects have graded membership in concepts. The problem with this objection is that it begs the question. If concepts are like classical sets and have well-defined extensions, the "cont rad ic tory" concept apple that is not an apple is indeed empty. But if concepts are ill-defined, then complex concepts of the form " X that is not an X " may simply not conform to classical set theory. Prototype theory does not say that there must be objects that are " g o o d " examples of both apple and not an apple, only that there may be objects that are fair examples of both (or a good example of one, and a poor example of the other, etc.).

The reader is invited to look for the use of "contradictory concepts" in actual discourse. Outside of classrooms in formal logic, people do understand such concepts as apple that is not an apple as true of objects that are like apples in some respects (appropriate size, color, and shape), but unlike in others (made of plastic). Indeed it takes training to learn the logical interpretation of contradictions. Such constructions seem especially common with concepts that do not have well-accepted defining characteristics, such as a solution that is not a solution, a game that is not a game. The "strong intuit ion" against such concepts may be explained by the inculcation of logical conventions that do not always correspond to the natural use of concepts, and also by the particular linguistic form, " X that is not X . " Replac- ing the n o t - X concept with some other concept disjoint with X often makes much more sense, even though from the extensional point of view the two should be the same, since the conjunction of X with any disjoint set is empty. Examples of "empty concepts" that make good sense are f i sh that is a m a m m a l (e.g., the whale), f ru i t that is a vegetable (e.g., the tomato; see Osherson & Smith, p. 45), robot cat, a bizarre feline from some science fiction movie, or bachelorette, a female bachelor. In these cases, people do seem able to generate plausible interpretations. It seems that the problem with X that is not an X is not its logical impossibility as a concept, but rather its infelicity in normal communication. We postulate that people combine such concepts by inferring appropriate mediating relations and then substituting atypical features for typical features in the mediating relation (e.g., robot cat is understood by substituting robot for animal, bache- Iorette is understood by substituting f e m a l e for male). The classical view is completely unable to explain people's use or understanding of such concepts.

Finally, it should be mentioned that these supposedly empty concepts also exhibit characteristics of prototype-based concepts: gradations of representativeness and unclear cases. For example, a completely mechanized

42 COHEN AND MURPHY

cat indistinguishable in appearance and behavior from Morris, is typical of robot cat; but the cat with only a mechanical leg is a borderline case. Proto- type theory seems useful here as well.5

3.6. Quantification and Fuzzy Sets

The fuzzy set rule for universal quantification is given by this test for "All A's are B ' s"

(VxeD) [Ax--Bx] iff (VxeD) [TYP.(x)<_ TYPB,(x)]

where D is some domain. This rules stipulates that all A's are B's if and only if every object (in D) has no greater characteristic value in A than in B. By this rule, "All pet fish are fish" is falsified by any object that is more typical of pet fish than offish. As has been observed (e.g., by Osherson & Smith, 1981) this is clearly the wrong truth condition. In fact, we have already noted that the goldfish is a more typical pet fish than a fish.

This objection to the fuzzy truth condition is correct, but in a representational model, these problems need not arise. Both classical and fuzzy set theory require that large or infinite sets of objects be examined in order to evaluate quantificational statements, thus making these theories of ques- tionable psychological interest (see Johnson-Laird, 1980). In the considerable body of psychological research on how people evaluate quantificational statements of the forms "Some A's are B ' s" and "All A's are B ' s , " the basis of subjects' decisions is usually taken to be mentally represented word meanings (which we would extend to include conceptual knowledge), not set extensions (see Smith, 1978, for a review). These theories, which are generally consistent with the knowledge representation model that we describe below, can begin to explain why some true quantified statements are harder to verify than others and why some false statements seem "more false" than others and easier to disconfirm. Classical set theory cannot ex-

'Osherson and Smith (p. 46) claim that prototype theory also fails to correctly explain "logically universal" concepts such as f r u i t that (ei ther) is or is no t an apple. The correct account, they say, is:

(Vx~FRUIT) T Y P ~urr rHAr Is OR IS nor A~, Appt~X)= I.

They base this conclusion on the fact that the class f r u i t that is or is no t an apple "is logically universal (with respect to the domain F) ." But this is simply a mistake. To say that the complex concept f r u i t that is or is no t an apple is logically universal with respect to f r u i t is merely to re- describe the category f r u i t . However, no one believes that all fruit are prototypical (i.e., have a characteristic value of I), which is what the above equation says. Thus , prototype theory 's inability to predict that equation is not an error. The mistake seems to arise from confusing the t ru th value of the universally quantified statement "All fruit either is or is not an apple" with the typical i ty of instances of the disjunctive class f r u i t that e i ther is or is no t an apple.


plain this, since each quantified statement is either true or false with nothing inbetween. More recent versions of fuzzy set theory do seem able to explain such phenomena (Zadeh, 1981). Not surprisingly, these models take a more representational approach than previous models.

3.7. Core and Identification Procedures

Miller and Johnson-Laird (1976, Chapter 4) distinguish between a concept's identification procedure (or key) as separate from the core of a concept. Their idea is that people can learn to identify objects by perceptual tests that are not necessarily central to the concept. Osherson and Smith (1981) suggest that prototype theory may be true of identification procedures in which objects fit perceptual tests to greater or lesser degrees, but false of the conceptual core, where membership must be all-or-none and is presumably represented by necessary and sufficient features. This may be seen as an attempt to reconcile the empirical findings of fuzziness with the elegance and formal adequacy of classical models. The account of concept cores originally proposed by Miller and Johnson-Laird, however, is in clear disagreement with this definitional view of the core: " A conceptual core is an organized representation of general knowledge and beliefs about whatever objects or events the words [in a lexical field] denote- -about what they are and do, what can be done with them, how they are related, what they relate t o " (p. 291). They also say that "a conceptual core is an inchoate theory about something" (p. 291). Similarly, Rumelhart 's (1980) description of schemata stipulates that a schema is akin to a theory of the thing being represented, and that schemata are not definitional.

We agree with Miller and Johnson-Laird that the conceputal core is tantamount to a theory about some real-world domain, and that like most theories (and unlike standard set theory), it will be somewhat fuzzy. Some objects will fit into it easily, and others may require special explanations. We think it will be very difficult to capture all of the information in the conceptual core using a simple definitional format. Below we will propose a richer knowledge representation structure that is consistent with both conceptual cores and perceptual identification procedures. Identification procedures are probably not independent of the core of a concept, but are related to the perceptually salient features of the core (Miller & Johnson- Laird, 1976).

To make the concept core conform to classical set theory and to make identification procedures conform to prototype theory introduces a radical dichotomy into conceptual structure. Osherson and Smith (1981)justify this dichotomy, in part, by citing the numerous objections to the extensional fuzzy model. But we shall see that these objections may be avoided by a more representational approach.

44 COHEN AND MURPHY

3.8. Set Theory Reconsidered

After reading our discussion of fuzzy set theory, one may question whether we have been attacking a straw man: Does anyone really believe that concepts can be described by sets, fuzzy or otheiwise? As we have already seen, logicians and semanticists seem to believe that words can be represented as set extensions, and these lexical items are closely related to concepts. Studies in semantic memory and categorization have often cited Zadeh's model as being consistent with their ~'esults (e.g., Rosch & Mervis, 1975; Smith, et al., 1974). Some investigators have explicitly considered fuzzy set models as descriptions of psychological representations (Hersch & Caramazza, 1976; Oden, 1977a), or fuzzy logic as a description of reasoning with concepts or evaluating propositions (Brownell & Caramazza, 1978; Goguen, 1981; Oden, 1977b; Smith et al., 1974). (Interestingly, Brownell & Caramazza, 1978, and Oden, 1977b, found that people do not seem to use the fuzzy set rules of combination.)

None of these investigators is very clear about the connection between conceptual representations and the fuzzy set model, but they all believe that the model has direct implications for the use of concepts. Just as the older tradition of concept formation arising from Bruner et al. (1956) depended on traditional logic as a model of logical competence and complex concepts, work deriving from prototype theory has implicitly depended on fuzzy set theory. Osherson and Smith (1981) claim that other than fuzzy rules, no rules for conceptual combination have been suggested by prototype theorists. We agree and conclude that the fuzzy set model is not a straw man. The next section describes our alternative to it.

4. A KNOWLEDGE REPRESENTATION MODEL

Our representational model of prototype theory is non-extensional and knowledge-dependent, that is, it explicitly attempts to incorporate as much domain knowledge as possible. Our goal in this section is to show how AI knowledge representation techniques and psychological schema theory can provide an account of people's comprehension of complex concepts as well as an account of experimental findings. The discussions presented so far lead us to construct a model that differs in several ways from current knowledge representation schemes (e.g., Brachman, 1979). However, it remains compatible with the work in psychological schema theory.

4.1. The Basic Model

We draw on four notions from knowledge representation systems. One is a generalization of the familiar lexical notion of cases on verbs to include


roles or slots on nouns (see Charniak, 1981). For example, the concept piano may include the roles SIZE, COLOR, FUNCTION, SOUND, and LOCATION, which are filled by feature lists, called "values ." These roles are the representational counterpart of the notion of mediating relation in- troducted earlier. (We attempt to distinguish concepts from roles and their values by italicizing the former and capitalizing the latter. However, it should be clear that these values are usually concepts themselves.) The second notion is the organization of concepts into a lattice or taxonomy supporting inheritance of roles (Bobrow & Winograd, 1977; Brachman, 1979; Collins & Quillia n, 1969; Fahlman, 1979). So, the piano concept may have inherited the SOUND role from its superordinate, musical instrument. Both of those concepts may have inherited roles from artifact or physical object, such as COLOR, FUNCTION, and SIZE. The third notion is that roles do not accept arbitrary values--rather they are restricted to particular values. Thus, the values that fill the LOCATION of a piano would not fill its FUNC- TION or COLOR. The fourth notion is that concepts have individual instances or tokens corresponding to actual members of the concept. For example, the concept ship has descriptions of individual ships as its instances. We will expand on these four notions below.

First of all, we should point out the differences between the representation that we are suggesting and the more traditional network models of knowledge representation. Like many schemes since Collins and Quillian (1969), we depend on inheritance of roles and role-values. This explains how someone can learn a great deal about a concept just by learning what superordinate concept it falls under. Knowing that a lark is a bird leads one to suppose that larks have wings, can fly, migrate in winter, and so on. However, unlike some past models, this inheritance is only a default mecha- n i s m - i t operates freely in most situations, but it can be overridden by con- travening information. Thus, one might, at one point, think that penguins can fly, by virtue of their being birds, yet later override that inference when contradictory information is received. Thus, unlike the position taken by Brachman (1979; and Brachman & Levesque, 1982), the taxonomy is not to be considered as embodying "analytic t ru ths ," definitions, or meaning postulates, or as supporting logical entailments. Rather, the structure represents people's domain knowledge--a flexible web of belief that can be revised in the face of new evidence and can adapt to exceptions and atypical cases (see also Fahlman, 1979).

Also, unlike early feature-based and network models of concepts, this version specifies the relations between different role-value sets and between role-values for a given role. For example, the values listed for a particular role are known to be mutually exclusive alternatives, since an object generally cannot have two values filling the same role. Values for different roles may be statistically correlated, or there may be causal, numerical, functional, or logical dependencies between values. What is most important

46 COHEN AND MURPHY

for present concerns is that values for a given role may be ordered by typicality.

Figure 1 shows a portion of the representation of the concept bird and its subordinates. The subordinates are ordered by typicality. Following Rumelhart and Ortony (1977; and Rumelhart , 1980), we have filled in "defaul t values" for some of the roles, but we shall also assume that each role has not just the default value shown, but access to a data structure con- taining values that are weighted by typicality.

How are the typicality orderings of subordinates of a given concept derived? Most generally, they could be derived from family resemblance, as specified by Rosch and Mervis (1975). Degree of family resemblance is computed by counting the number of features an instance has in common with other instances and subtracting the number it shares with non-instances of the concept. Thus, subordinates that have the most common features and are the least similar to non-instances will be more central to the concept. "Fea tu res" in the semantic memory and categorization literature correspond to our roles and role-values, and typicality h la Rosch and Mervis may be computed by a matching procedure that operates on the roles.

Prototype Summary

(ISA: Animal ..) (SPECIES: Robin Chicken ..) (EXTERIOR: Wings Beak i")sm

D 4~

Animal

{3 Robin

{3 Wings

{3 Files

El Animal

r' l Penguin

El Wings

El Walks

Figure 1. A schematic depiction of the concept bird and some of its subordinates, ordered by typicality.


Another account of typicality (for some categories, at least) is that it relates to the ability of an object to fulfill the function typically associated with the category (see Barsalou, 1981). In some cases, this will be directly represented in FUNCTION or USED-FOR roles. In other cases, one would have to search the other roles for values that indicate the fulfillment of some function. For example, for the concept of desserts, one might look for certain values in the TASTE role to calculate the typicality of an object. The point is that the concept must represent all this information in order to be accessible to these processes--a simple ranking of typicality will not be sufficient. Further evidence for this position was given by Barsalou (1981), who showed that people agreed on typicality rankings of objects in novel categories. This demonstrates that people use their knowledge of categories and objects in order to judge typicality, rather than depending on prestored characteristic values.

One might raise the same issues for the list of values of each role: they are probably also ordered by typicality (e.g., the SIZE of birds is typically fairly small, although some birds are medium-sized or large), but how is this ordering derived? There has been little empirical work on this question, but a number of possibilities seem likely: most frequently encountered values could be most typical (see Rosch & Mervis, 1975); values that "f i t in" with the values of other roles could be more typical (see Malt & Smith, in press); or values that are consistent with the values of the superordinate categories could be more typical. It would be premature to argue for any of these, but the knowledge representation model appears to be the appropriate arena in which to raise these possibilities.

The model outlined can capture the major insights of prototype theory. Since the role-values of a concept are not taken to be definitional, the concept does not specify necessary and sufficient criteria for membership. Objects that have the most typical role-values will be considered " g o o d " members; objects that have atypical or unacceptable values will be considered "border l ine" or " p o o r " members. Category nonmembers will also vary in the extent to which they fit the category representation (Rosch & Mervis, 1975). To borrow an example from Ortony (1979), a blood vessel will fit the concept aqueduct (as in the metaphor "Blood vessels are aqueducts") better than a rock will, because of shared roles (e.g., blood vessels and aqueducts both have the FUNCTION of guiding the FLOW of a LI- QUID; whereas a rock has few, if any, roles in common with an aqueduct). When the processes involved in object recognition and categorization are better understood (see Murphy & Smith, 1982, and Smith, 1978, for discussions), this model should provide appropriate representations for a theory of categorization. We now show how this basic model may be extended to handle disjunctive concepts, quantification, and complex concepts in general.

48 COHEN AND MURPHY

4.2. Disjunctive Concepts

We distinguish two sorts of disjunction. The first is the logical disjunction of arbitrary concepts, such as horse-or-chair and P-or-Q. These disjunc- tions seem very different from more natural concepts that have been argued to be disjunctive (by Miller & Johnson-Laird, 1976; Newport & Bellugi, 1978), but whose constituents bear a family resemblance to each other. For example, the different members of furniture may be represented by a disjunction of the individual concepts chair, table, dresser, stereo, etc. How- ever, the individual concepts are not arbitrarily strung together; they are related by overlapping attributes (Rosch & Mervis, 1975; Wittgenstein, 1953). The classical approach does not distinguish arbitrary logical disjunction from disjunctive categories that are based on overlapping attributes, and is therefore unable to explain disjunctive concepts. Thus it is difficult for the classical view to explain why chair, dresser, table, and stereo, is not an unrelated list of categories, since as sets they are nonoverlapping.

How does the knowledge representation account deal with these two sorts of disjunction? For the first, a logical disjunction of arbitrary concepts such as horse-or-chair could be created by linking the component concepts to a dummy disjunctive concept by IS-A links. In this case, the concept would not have roles of its own, and the typicality of an instance would be determined completely by how typical it is of either of the disjuncts. As long as the combination is purely logical, the max rule may be appropriate: an object is a good example of horse-or-chair to the extent that it is a good example of horse or of chair.

For the second sort, exemplified by natural language superordinates, we must assume that people know something about the superordinate in general-- the information is not completely included in the disjuncts. For example, while subordinate concepts of furniture are connected to furniture by IS-A links, we assume that people also know many things about furniture in general: its typical uses, location, and construction. To decide how typical some object is of furniture, it may be examined to see how well it matches the roles of the furniture prototype, or it may be compared to subordinate categories, as in the case of logical disjunction, or both. Of course, which method people actually use is an empirical quest ion--perhaps both rules are tried simultaneously. This appeal to knowledge of superordinates avoids the problems with the max rule. It may be that psychologists have given less attention to the information stored in disjunctive concepts because they assumed that such information must be criterial or definitional. But, in keeping with our prior assumptions, we claim that people might store the role-values USED-TO-SIT-ON or USED-TO-COVER-FLOOR with their concept of furniture, even though such features are certainly not necessary for membership.


4.3. Simple Quantifieational Propositions

The model we have sketched has three mechanisms to explain how people evaluate simple quantificational propositions. First, some prototypes may be connected by IS-A links, as in theories of semantic memory since Collins & Quillian (1969). The standard method to verify the proposition that "All elephants are mammals" is to see if the elephant prototype is IS-A-linked to mammal. However, we should re-emphasize that these links are default l inks--they do not establish "analytically t rue" concept relations, as is done, for example, in Brachman's KL-ONE (1979). (See Hampton, 1982, for empirical evidence regarding such default links, and Woods, 1975, for a classic discussion of the difficulties of arriving at a consistent interpretation of IS-A links.)

Second, role-values and value restrictions on prototypes can be used by verification procedures. For example, if the LOCOMOTION slot on fish includes only the value SWIMS, then "All fish swim" would be verified. In other cases, a role might be restricted to a certain set of values. For example, the possible values for the HABITAT slot on fish may be restricted to BODIES OF WATER. Here, although there are a number of alternative role-values, such as OCEAN, LAKE, or RIVER, the value restriction justi- fies such propositions as "All fish live in water ."

Third, people may confirm or disconfirm a qualified proposition by accessing the representations of individual instances of the quantified concepts. For example, "All pets are fur ry" could be disconfirmed by finding an instance of pet that did not inherit the FURRY role-value, e.g., an instance of pet fish. Since this method uses instances rather than prototypes, it has more in common with formal semantic accounts of quantification than the first two methods (see e.g., Dowty et al., 1981). Unlike formal semantics, however, we do not assume that procedures to confirm universal quantifications somehow have access to every instance in the (possibly infinite) extension of a concept. Instead access is restricted to a small representative sample of remembered instances.

These three proposals make verification of category membership seem like a very simple process, but as researchers in both AI and semantic memory well know, it is not. We believe, however, that the concept representations suggested can account for many of the empirical findings. For example, the subordinates to a category are stipulated to be ordered by typicality, which could be partly realized by varying the strength of the IS-A link con- necting the categories (Collins & Loftus, 1975). When the IS-A link is weak, people may have to resort to comparing the concept representations directly, as suggested by Smith et al. (1974). Furthermore, people can make decisions about membership of new concepts, which would require processes other than looking for ISoA links. Presumably, this would work by search-

50 COHEN AND MURPHY

ing for similar roles and role-values of the two concepts and then matching to see how well they compare. Once again, the empirical issues are not wholly settled here.

The main point is that it is entirely possible to have an account of simple quantificational propositions that is compatible with prototype theory and does not have fatal problems (of the sort mentioned by Osherson and Smith). For example, the existence of objects that are more typical of pet fish than they are offish is irrelevant to the verification of "All pet fish are f ish" in this system, although such examples disconfirm the fuzzy set rule.

The reader should note that our account only deals with simple quantificational propositions involving natural concepts, and it is not meant to handle complex sentences with embedded quantifiers, scoping ambiguities, and the like. Nor will it explain syllogistic reasoning or other issues of logical performance in formal contexts. Finally, it is not intended to account for quantified expressions in discourse. Such expressions are often not literal universal predications, but are used to emphasize a point in the discourse. "Everybody knows tha t ! " is more a statement about the obviousness of some fact rather than a literal predication. "All professors are absent- minded" may be a statement about prototypical professors rather than about each and every one of them. Cohen (1982) discusses representations needed for the literal interpretation of quantified NPs such as "all beavers," as well as related generic plural statements such as "Beavers build dams," which apparently carry the prototypical interpretation. How people decide when quantified statements are to be taken literally is a question for the theory of discourse processes, not for the theory of concepts. But both interpretations can be dealt with in the framework we have described.

4.4. A Representational Model of Complex Concepts

The question of complex concepts in this model is twofold: (a) given a compound noun phrase, how can the complex concept be constructed? and (b) how is the typicality of the complex concept determined? (The second is the compositionality criterion.) The basic idea in the knowledge representation account of constructing complex concepts is that sub-concepts may special- ize a super-concept by role-modification--specifying a particular role-value for an inherited role (e.g., Brachman, 1979). The complex concept engine repair can be formed as a specialization of repair by modifying an appropriate role that takes ENGINE as its value. If the concept repair has OBJECT as its role, and the value ENGINE (actually another concept) satisfies the value-restriction on the OBJECT role, then the complex concept engine repair can be formed as a specialization of repair by modifying its OBJECT role.


Unlike the min rule, this method is asymmetric: the concept of boat house is not the same as that of house boat in this framework. A boat house is a house that CONTAINS or STORES boats. A house boat is a boat that has the FUNCTION of a house. Thus, different concepts and concept roles are specialized in the two cases. Since intersection is commutative, extensional models based on set intersection must claim that each case of non- commutativity is idiomatic or a special case. Yet, this seems to be the rule rather than the exception.

Linguistic analyses of compound NPs seem much more consistent with the role-modification scheme than with the intersective analyses. Kay and Zimmer (1976) discuss the ambiguity of phrases such as "finger cup ," arguing that they have an indefinite number of possible interpretations, up to the number of relations that one can imagine between the words in the compound. For example, "f inger cup , " might mean a cup held between the tips of the fingers, a cup that holds a finger of whiskey, a cup shaped like a finger, a cup used for washing fingers, and so on. Any one of these or other interpretations might be chosen in context. This indicates that the nominal compound cannot be assigned a single reading based on one rule, as the set intersection analysis assumes. Rather, the combination of the two concepts must be knowledge-dependent; where "knowledge" is expanded to include not only the information associated with each concept but also the information contained in the discourse. If one is talking about washing, then the "cup used for washing fingers" interpretation will be chosen; if one is talking about collecting unusual porcelains, then "cup shaped like a finger" might be activated. In each case, the context may activate one role of the head noun more than others, and the matching process tries to find a way to modify that role with the other concepts in the NP. When different roles are activated, different complex concepts are constructed.

What little experimental data there are seem to support this explanation. Gleitman and Gleitman (1970) studied people's interpretations of compound noun phrases. They noted that there are intonational and syntactic rules (i.e., word order) that should control the interpretation of compounds. But when they tested people's use of those rules, they found large numbers of deviations for some cases. A major cause of those errors was found to be semantic implausibility and, to a lesser degree, prior familiarity of the compound. This occurred even though subjects knew fully that some of the compounds presented would not make perfect sense. This result suggests the use of knowledge-dependent interpretation strategies in understanding such complex concepts.

On our account, concept formation is a process of combining representations according to certain generative rules within a domain of knowledge. We abbreviate the result of role modification by inserting a '1' between the head concept and the modifier as in repairlENGINE. Formally we can define '1' as a concept forming operator:

52 COHEN AND MURPHY

If X and Y are concepts, and if R is a role on X that Y fits, then XJY is a complex concept specializing X with modified value Y for role R.

Algorithms for constructing a complex concept XIY vary in complexity depending on the complexity of the notion of " f i t t ing" a role. All such algorithms involve matching roles and role-values (see Cohen, 1982; Finin, 1980). For example, the role matching process must know enough to decide for expert repair that E X P E R T is not the OBJECT, LOCATION, or FUNC- T I O N or repair, but the AGENT. ha some cases, the complex concept XIY may be constructed by checking role-value sets on X to see if Y is already included there. For example, one might find E X P E R T listed as a possible A G E N T of repair. In a more complex case, the notion of fitting may involve comparing role-values for the "best f i t" based on the ordering of role-values. For example, if M E C H A N I C and P L U M B E R are typical AGENTs of repair, and if E X P E R T is seen as similar to them, then it will be chosen as a specialization of the A G E N T role.

Given appropriate matching mechanisms, the rule of role modification can be used to generate interpretations of a wide class of complex NPs" in red apple, RED modifies the C O L O R of apple; in large mosquito, L A R G E modifies the SIZE of mosquito within the value restriction on the SIZE role, which accounts for the difference between large mosquitos and large houses; in ocean drive, OCEAN modifies the LOCATION of drive. Notice also that this rule is general enough to allow for the construction of highly atypical compounds such as virgin birth. The atypicality of such constructs is explained by the fact that the role that the value "best f i ts" is usually filled by other more typical values; that is, the fit in this case is a relatively bad fit.

Does this model o f complex concepts satisfy compositionali ty? Are there simple rules analogous to the min rule for determining the typicality of complex concepts CIX, I.. IX,? This question has two parts. First, is the typicality of a complex concept cIs , I.. Is~ relative to its superordinate C a simple function of its typicality on the components C, X, .... X~? That is, are there simple rules to tell us that the concept red apple is a more typical apple than green apple, or that conservative banker is more typical than radical banker? Second, is the typicality of an individual object x as an instance of a complex concept a simple function of its typicality as an instance of the components? Although these two questions appear to be different, in our model the representation of an instance x is very similar to the representation of the subordinate concept that describes x. Hence, the typicality of an individual red apple x as an instance of the concept apple is determined by virtually the same mechanisms and knowledge structures as the typicality of a subordinate concept appleJRED relative to its superordinate apple.

Is the typicality of an instance or subordinate x of a complex concept a simple function of the component concepts? In general, the answer is no. The typicality of an object or subordinate x as an instance of a complex con-


cept C[X, [.. JAn is determined by matching the description of x to the roles and role-values X, of the complex concept. Complex concepts, instances, subordinates, and components are all represented in the same formalism. However, the actual roles and role-values of a complex concept need not be the same as those of its component concepts. Therefore, an object or subordinate may be more (less) typical of the complex concept than of any of the components. For example, pets are usually furry and cuddly, but pet fish are not. When constructing the complex concept pettish, we claim that people use their domain knowledge to modify some of the role-values inherited from the component concept pet. Thus, in the pet fish prototype, the role-values SCALES and SLIMY will have greater weight than the role- values FURRY and CUDDLY inherited from pet. This explains why a guppy is a better pettish than a pet: it more closely matches the role-values of the complex concept. 6

If the foregoing is correct, there is no simple rule that properly deter- mines the typicality of instances (or subordinates) of complex concepts purely by reference to component concepts. But, suppose that a mechanism is available that takes domain knowledge into account in constructing complex concepts such as pet fish. Assuming the complex concept C has been constructed by this mechanism, are there simple rules for computing the correct typicality of an instance (or subordinate) of C? (Note that such a rule would not take the component concepts into account, except indirectly.) A simple rule is to take the average typicality of the modified role-values R,(x) for each modified role R,. on C as the typicality measure TYPe(x):

TYPe(x) =

n ~(R,(x))

1=1

where n is the number o f the roles R, (i _< n) on the concept C, and w(R,(x) ) is the weight of the modified role-value R~(x) for role R, on x. Thus, if x is a shiny red apple, x is a typical apple because SHINY and RED are typical values of their roles on apple. And if x is a guppy, x is a more typical pettish than a pet because, e.g., SCALES is a more typical value than FURRY for pet fish.

This rule works for simple cases in which role-value sets are independent, for example when there is only one role being modified. But it is inadequate to handle concepts when role-value sets are nonindependent. For example, a bald younger man is a less typical man than a bald older m a n - - even though older men are no more typical than younger men. Assuming that BALD has the same weight in both cases, the rule above incorrectly

6We owe this example to James Hampt on who has a somewhat different explanation o f it.

54 COHEN AND MURPHY

predicts that the two complex concepts man I OLD[BALD and man lYOUNG [BALD will be equivalent in typicality. Here it seems that people know that certain role-values are correlated: baldness is more frequent among older men than it is among younger men. These facts are part of domain knowledge and cannot be derived by combining the representations of the concepts bald, young, old, and man. Instead, one must have access to knowledge of the relations between the objects represented by these concepts. This example further explains why simple compositional rules are inadequate to generate the correct complex concepts. Also, it illustrates a general phenomenon of how domain knowledge enters into what seems to be a closed semantic domain subject to simple algebraic rules of composition. The trou- ble with all such rules is that they depend on the fiction that typicality can be determined independently of domain knowledge. Yet people's estimates of typicality are known to be sensitive to causal and correlational knowledge, to how common objects are, and to how well they serve certain goals (Ash- craft, 1978; Barsalou, 1981; Medin, Altom, Edelson, & Freko, 1982).

Given access to a large database of domain knowledge, can current inference and representation techniques be extended to construct the appropriate complex concepts? This is an open problem that we are optimistic can be solved by developing a theory of concept acquisition in the framework that we have outlined. ' However it is solved, one fact seems indisputable: compositionality is too strong a constraint to put on an empirically adequate theory of concepts. 8

5. CONCLUSION

At the outset of this article we suggested that a theory of concepts bears on a variety of enterprises. The classical set theory adopted by Osherson and Smith and other psychologists (Bourne, 1966, 1982; Bruner et al., 1956) finds considerable support in logic and formal semantics (Dowty et al., 1981). Yet, if we want a theory of concepts to have implications for psy-cho-

' In current work on implementing the system we have outlined, one of us (Cohen) has developed a learning algorithm that uses a family resemblance analysis to construct prototype representations from an instance space. By construction of appropriate subspaces o f the instance space, the algorithm is able to handle the complex concepts needed for nonindependent role-value sets, as in the "bald m a n " case.

'Finally, we note that some complex NPs are idiomatic and not at all compos i t iona l - - even knowledge-dependent rules of combinat ion cannot create the correct concept from the individual words. Linguists including Nunberg (1981) have argued that phrases such as country music refer to unitary concepts, though they may have once been composit ional. Country music is not rural music played by people from the country (witness the subgenre o f truck- drivin' songs). Yet, similar phrases like New York loft jazz do seem (somewhat) composit ional, though clearly not intersective. Differentiating these cases is a serious problem for language- understanding mechanisms whatever one 's theory of concepts. These unitary concepts add to the evidence that the compositionality criterion is too strong.


logical models of understanding and for linguistic accounts of semantic structures, then the representations it proposes must also be consistent with models in those areas. Recent work in cognitive science seems much more consistent with the schema-based approaches to these problems than all-or- none extensional approaches (see Bobrow & Winograd, 1977; Collins, Brown, & Larkin, 1980; Norman, 1981; Rumelhart, 1980; Schank & Abel- son, 1977; among many others).

We reiterate that logic and set theory are important both as part of the background of a theory of concepts and as tools of analysis in many areas of cognitive science. To be fair to the proponents of the extensional model, we do not doubt that many of them also believe that a schema-based knowledge representation system is a useful way to model people's concepts. However, they may argue, set theory is also necessary to explain semantic and logical competence, especially since most psychological representations have not been developed to deal with them.

Certainly, psychologists have neglected the issues of quantification and complex concept formation in their studies of concepts. But this need not lead us to revert to classical set theory to answer these questions, as some have suggested (Osherson & Smith, 1981). Our main goal has been to show how a fuzzy representational approach is capable of addressing these issues without falling into the traps that purely extensional fuzzy models do. Assuming we are correct, is there any reason to retain the set model as part of the theory of concepts, as opposed to the metatheory? Using set models encourages the use of logical rules and semantic theories that appear to have little psychological validity (see Clark, 1983; Johnson-Laird, 1975, 1980; Tversky & Kahneman, 1982). By Occam's Razor, we should try to eliminate the extensional component of our theory, unless it generates more realistic and testable predictions about representations than it has thus far.

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