Conceptual Combination with Prototype Conceptsosherson/papers/smithOsh1.pdf · 2013-01-22 · COGNITIVE SCIENCE 8, 337-361 (1984) Conceptual Combination with Prototype Concepts

COGNITIVE SCIENCE 8, 337-361 (1984)

Conceptual Combination

with Prototype Concepts*

EDWARDE.SMITH Bolt Beranek and Newman, Inc.

DANIEL N.OSHERSON Massachusetts Institute of Technology

This puper deals with how people combine simple, prototype concepts into

complex ones; e.g., how people combine the prototypes for brown and opple so they con determine the typicality of objects in the conjunction brown apple.

We first consider a proposal from fuzzy-set theory (Zodeh, 1965). namely, that

the typicality of an object in a conjunction is equol to the minimum of that ob-

ject’s typicality in the constituents (e.g., on object’s typicality OS (I brown apple

cannot exceed its typicality as o brown or OS on opple). We evoluoted this

“min rule” agoinst the typicality ratings of naive subjects in two experiments.

For each of numerous pictured objects. one group of subjects rated its typicol-

ity with respect to on adjective concept, a second group rated its typicality vis-

&vis a noun concept, and o third group roted its typicality with respect to the

odiective-noun conjunction. In both studies, most objects were roted OS more

typical of the coniunction than of the noun. These findings violate not only the

min rule but also other simple rules for relating typicality in o conjunction to

typicalities in the constituents. As an alternative to seeking such rules, we

argue for on approach to conceptual combination that starts with the proto-

type representotions themselves. We illustrate one version of this approach in

some detail, ond show how it accounts for the maior findings of the present

experiments.

l We thank Helene Chaikin, Deborah Pease, and Maggie Kean for their invaluable assistance in all phases of the research reported here. We are also greatly indebted to the

following people for their many, many, critical comments, which have helped foster and shape this research: Ken Albert, Susan Carey, Allan Collins, Dedre Gentner, David Israel, Michael

Lipton, Mary Potter, and Lance Rips. The research was supported by U.S. Public Health Ser- vice Grant MH37208 and by the National Institute of Education under Contract NO. US- HEW-C-400-82-0030.

Requests for reprints should be sent to Edward E. Smith, Bolt Beranek and Newman,

Inc., IO Moulton Street. Cambridge, MA 02238.

337

338 SMITH AND OSHERSON

CONCEPTUAL COMBINATION WITH PROTOTYPE CONCEPTS

Recent work on natural concepts like apple, fish, hammer, and shirt has led many researchers to a prototype view of the mental representations of such classes. While proponents of this view have yet to phrase a precise theory, they generally agree upon certain critical issues. Most proponents of the view deny that the mental representation of a natural class specifies neces- sary and sufficient conditions for membership; it is claimed, instead, that the representation specifies properties that are merely characteristic of the class, or perhaps of only some exemplars of the class (see discussion in Smith & Medin, 1981). Prototype theorists usually also assert that entities fall neither sharply in or sharply out of a concept’s extension, the boundary between membership and nonmembership being inherently fuzzy (see discussion in Osherson & Smith, 1981; 1982).’

Until recently, all empirical work on concepts-as-prototypes dealt with “simple” concepts, roughly, concepts denoted by single words such as “fish” and “apple.“’ Issues about the combination of these simple concepts into complex ones, and the use of these composites in categorization and other mental activities, have been largely ignored. As a consequence there are few explicit proposals relevant to the combination problem within prototype theory, and none of these proposals appears to be consistent with strong intuitions about combinatorial phenomena involving adjective-noun conjunctions, disjunctions, logically-empty and logically-universal concepts, and truth conditions for inclusion. The matter is reviewed in Jones (1982), Osherson and Smith (1981; 1982), Smith and Osherson (1982), and Zadeh (1982).

The present paper continues our analysis of prototypes and conceptual combination but differs from our earlier work in three ways. First, in the present paper we deal with a more restricted set of phenomena: we consider only adjective-noun conjunctions, such as pet fish and brown apple, and deal only with their use in categorization. The latter restriction is motivated by the following hypotheses: (a) concepts have a dual structure, with the “identification procedure” being chiefly responsible for categoriz- ing instances of the concept and the “core” of a concept playing a central role in certain reasoning situations; and (b) only the identification procedure conforms to the prototype view (for a defense of these hypotheses, see, e.g., Armstrong, Gleitman, & Gleitman, 1983, and Osherson & Smith,

’ What we are calling the “prototype view corresponds to what Smith and Medin (1981) called the “Probabilistic” and “Exemplar” views.

* We use quotes to indicate words, while reversing italics for the concepts that these

words denote.

CONCEPTUAL COMBINATION 339

1981; 1982). A second difference from our previous work is that rather than rely exclusively on just our own intuitions about a few instance-concept pairs, we consider experimental data derived from numerous instance-concept pairs. The third difference from our previous papers pertains to the kinds of theories we entertain. Though initially we focus once again on fuzzy-set theory, eventually we consider in detail a representational approach to conceptual combination.

In more detail, our exposition is organized as follows. In the next section we briefly summarize the fuzzy-set theory approach to conceptual combination and its attendant problems. Section 3 develops a taxonomy of adjective-noun conjunctions and presents two experimental tests of fuzzy- set theory’s ability to explain certain phenomena associated with various of the taxa. In the final section we consider an alternative to fuzzy-set theory, one that emphasizes representations.

FUZZY-SET THEORY AND CONCEPTUAL COMBINATION

Characteristic Functions

Fuzzy-set theory (e.g., Zadeh, 1965) is of interest to cognitive scientists because it offers a calculus for combining prototype concepts. A key notion in the application of the theory is that of a characteristic function, cA: D- [0, 11. This function maps entities in domain D (the domain of discourse) in- to the real numbers 0 through 1 in a way that indicates the degree to which the entity is a member of concept A. To illustrate, consider the characteristic function, cF, which measures degree of membership in the concept fish (F). When applied to any relevant creature, x, cF (x) yields a number that reflects the degree to which x is member of fish, where the larger cF (x) is the more x belongs to fish. Since our pet guppy is not very typical of fish, it gets a characteristic-function value of .80; our pet dog will get an even lower value, say .05. If we consider the concept pet (I’), and its characteristic function, cp, then our guppy and dog might be assigned the values of .70 and .90.

The question of interest concerns the relation between the characteristic function of a conjunction and those of its constituents. How, for example, do we specify cpaF (the characteristic function of pet fish) in terms of c, and c,? The answer from Zadeh’s 1965 version of fuzzy-set theory is that the conjunction’s characteristic-function value is the minimum of the constituents’ values; i.e., c,,~ (x) is the minimum of cp (x) and cF (x). Applying this min rule to our pet guppie, g, yields,

(1) cpBF (g) = min [c&3), cF (8) 1 = min (.70, .80) = .70.


Assuming that characteristic-function values can be estimated by judgements of typicality, (1) provides a testable claim about the relation between an object’s typicality in a conjunction and its typicalities in the constituents.’

Counterexamples to the Min Rule

The min rule is wrong. For (1) says that our guppy is no more typical of pet fish than it is offish, and as Osherson and Smith (1981) point out, intuition suggests that a guppy will be more typical of the conjunction pet fish than of either constituent, pet orfish. Osherson and Smith further argue that this pet fish example is just one of many counterexamples to the min rule (e.g., a perfectly brown apple seems to be more typical of the conjunction brown apple than of either brown or apple).

However, there are two problems with the foregoing counterexamples. First, they rest only on our own intuitions; such claims need to be tested against typicality ratings of naive subjects. Second, there is no indica- tion of the generality of the failure of fuzzy-set theory; perhaps, the cited counterexamples are of a few types in some underlying taxonomy of conjunctions, where other types might conform to the theory. To deal with these questions, we first present a taxonomy of adjective-noun conjunctions and then describe some relevant experimental work.

TAXONOMY AND EXPERIMENTAL TESTS

A Taxonomy of Adjective-Noun Conjunctions

All counterexamples of the type pet fish and brown apple share the following characteristic: the adjective concept (i.e., the property denoted by the adjective) is negatively diagnostic of the noun; for example, being brown counts against an object being an apple. More precisely, an adjective is “negatively diagnostic” of a noun to the extent that the applicably of the adjective to a given object decreases the probability that the noun is a true description of that object, and the inapplicability of the adjective to the object increases the probability that the noun is true of that object. An adjective is “positively diagnostic” of a noun to the extent that the adjective being

’ Note that there are two interpretations of this min rule: people’s judgements may simply conform to the rule or they mayfoNow it. In the former case, the min rule expresses an abstract constraint between typicalities. In the latter case, the rule offers a processing account of typicality judgements; i.e., people decide on membership in a conjunction by determining membership for each constituent separately. and then combine these two separate determina- tions.


a true (false) description of a given object increases the probability that the noun is a true (false) description of that object. And an adjective is “nondiagnostic” of a noun to the extent that the applicability of the adjective to a given object has no bearing on whether the noun is true or false of that object. As examples: in unsliced apple the adjective is largely nondiagnostic; in red apple the adjective is positively diagnostic; and, in brown apple the adjective is negatively diagnostic. We thus have three distinct cases in our taxonomy.

In addition to the relation between the constituents, we also need to consider the degree to which the conjunction provides a true description of an object that is to be categorized. To keep things simple, we consider only whether the to-be-categorized object manifests the property denoted by the adjective in the conjunction, and we let the object either obviously manifest this property (e.g., an apple that is red coupled with the conjunction red apple), or obviously fail to manifest this property (e.g., an apple that is brown coupled with red apple). We will refer to these two possibilities as a “good match” and a “poor match,” respectively. When these two possibilities are combined with the previously described variations in diagnosticity, we have a total of six cases, and the resulting taxonomy is presented in Table I.

For each case in Table I, consider the relation between typicality judgments for the conjunction and for the constituent concepts. In Case 1, the constituents are relatively independent of one another; accordingly, people may judge separately the degree to which an object is an instance of the adjective and of the noun concepts, and then use some rule to combine the outcomes of these judgments into an overall typicality rating. The min rule (1) is a plausible candidate for this rule since it uniquely meets certain natural conditions on conjunctive operators (see Dubois & Prade, 1980, pp.

TABLE I

Taxonomy of Adjective-Noun Conjunctions

Degr88 fo Whfch object Monlfests Pr0p8rfy

Reloflon of Adiectlve COnC8pt

to Noun Concept

Nondiagnostic

Positively

Diagnostic

N8gOtiV8ly

Good Match

(1) unsliced apple

object is unsliced

(3) red oppfe

object is red

(5)

Poor Match

(2) unslked apple

object is sliced

(4) red Opp18

object is brown

(6) Diagnostic brown apple brown oppfe

object is brown obiect is red


11-12.) This line of reasoning suggests that some variant of fuzzy-set theory may be adequate for Case 1. The same intuitive prediction holds for Case 2. In Case 3, since the adjective and noun concepts are not independent, there is no obvious reason to expect the judgment for the conjunction to be a simple function of the judgments for the constituents. Nor is there reason to expect a violation of fuzzy-set theory. We simply lack firm intuitions for this case (as well as for Case 4). Case 5, where the adjective is negatively diagnostic of the noun, captures the counterexamples used in Osherson and Smith (1981). Here intuition strongly suggests that an object that is a good match (e.g., an apple that is indeed brown) will be rated more typical of the conjunction brown apple than of either constituent, brown or apple. This intuition does not hold when the object is a poor match, as in Case 6 (e.g., an apple that is red makes a better apple than a brown apple).

To summarize our intuitive predictions: For Cases 1 and 2, the only ones where the adjective and noun concepts are largely independent, the min rule of fuzzy-set theory might work, while for Case 5, which includes the Osherson and Smith counterexamples, the min rule should unequivocally fail. The outcomes for the remaining Cases (3,4, and 6) may fall somewhere in between.

Experiment 1

Overview. The purpose of Experiment 1 was to determine the relation between typicality ratings for conjunctions and their constituents for each case in our taxonomy. For each of 48 pictured objects, one group of 20 subjects rated the object’s typicality with respect to an adjective concept, a second group of 20 rated its typicality vis-a-vis a noun concept, and a third group of 20 rated its typicality with respect to the adjective-noun conjunction. The adjective-noun conjunctions were such that all six cases of our taxonomy were tested.

Materials. Table 2 specifies the concepts used. The first column gives the 12 noun concepts (6 natural kinds and 6 artifacts). Column 2-5 give the 48 adjective concepts. Those adjectives in Column 2 are presumed to be positively diagnostic of their corresponding noun concepts, those in Column 3 are presumed to be negatively diagnostic, and those in Columns 4 and 5 are presumed to be largely nondiagnostic. (Later, we will present data that support these presumptions). Each adjective was combined with its corresponding noun to form a conjunctive concept, yielding a total of 48 conjunctions. In addition to designating a conjunction, each intersection of an adjective and noun in a row of Table 2 specifies a pictured object; i.e., 48 pictures were constructed, where one was of a red apple, a second was of a brown apple, a


TABLE II

Materials Used in Experiment 1

Adjective Concepts

Noun Positively

Concepts Diagnostic

apple red

canteloupe round

corn straight

peas green

conory yellow

ostrich feathered

chair symmetric

toble level

car four-wheeled

bicycle metal

jacket sleeved

shoes leather

Negotlvely

Diagnostic

brown

square

bent

yellow

red

unfeathered

asymmetric

tilted

three-wheeled

wooden

sleeveless

wool

Nondiagnostic

sliced

split

husked

shelled

caged

standing

overturned

round

convertible

red

yellow

laced

Nondiagnostic

unsliced

whole

unhusked

unshelled

uncaged

running

upright

square

sedan

brown

red

unlaced

third was of a sliced apple, a fourth was of an unsliced apple, etc. All pictures were hand drawn, photographed, and made into slides.’

Procedure. We will consider the procedure for each of the three rating groups in turn. In the “Noun” group, on each trial the experimenter spoke the name of a noun concept, then 1% s later a pictured object appeared and subjects rated how good an example it was of the concept. Each picture was presented once. Since there were four different pictures for each noun (e.g., four pictures contained apples), each noun occurred four times. This yielded a total of 48 trials. Two quasi-random orders of the trials were created. Each order minimized repetitions of the same concepts on successive trials, and each was used with 10 subjects.

In the “Adjective” group, on every trial the experimenter spoke the name of an adjective concept, then 1 r/z s later a pictured object appeared and subjects rated how good an example the pictured property was of the

’ Inspection of the adjectives in Columns 4 and 5 of Table II suggests that some of them may not be nondiagnostic in the intended sense of independent probabilities. We had intended that the probability of a noun being true of an object be independent of whether a nondiagnostic adjective was true of that object, but such independence seems not to apply when the noun denotes, say, cur and the adjective denotes convertible or sedun. In cases like these, what nondiagnosticity seems to amount to is that the probability of the noun being true is roughly the same regardless of which adjective it is conditionalized on. For other adjectives in Table 11, however, nondiagnosticity in the intended sense does seem to hold; e.g., the probability of an object being a jacket is independent of it being red. Since our experiments failed to reveal any marked difference in results between cases like converfiblecarand redjuckef, we will have little more to say about this problem.


concept (e.g., how good an example a particular red apple was of the concept red). Now each picture was presented twice, once with an adjective denoting a property for which the pictured object was a good match, and once with an adjective denoting a property for which the picture was a poor match. As examples, the picture of the red apple and that of a brown apple were each presented once with “red” and once with “brown.” Each adjective thus occurred twice, for a total of 96 trials. Again, two different quasi- random orders of the trials were used, with 10 subjects per order.

The third group is the “Adj-Noun” group. On each trial the experimenter spoke the name of an adjective-noun conjunction, then 1 l/2 s later a picture was presented and subjects rated how good an example it was of the concept. Each picture was presented twice, once with a conjunction for which it was a good match, and once with a conjunction for which it was a poor match. For example, the picture of a red apple was presented once with “red apple” and once with “brown apple.” Each conjunction occurred twice, with 24 of the conjunctions being positively diagnostic, 24 being negatively diagnostic, and the remaining 48 being nondiagnostic. There was thus a total of 96 trials, and as usual two quasi-random orders were used with ten subjects receiving each order.

All subjects had 10 s to make a judgment, the judgments being made on a 1 l-point scale where “0” indicated the object was not an instance of the concept and “10” indicated it was a perfect example. The subjects, 60 Harvard undergraduates who participated for pay, were given 16 practice trials before rating the test items. The concepts and pictures in the practice trials were different from those employed in the test trials.

Results. The top half of Table III contains the data for the three cases of the taxonomy where the object was a good match, while the bottom half has the results for the three cases where the object was a poor match. For each case, the second, third, and fourth columns give the average typicality ratings for the adjective concept, the noun concept, and the conjunction, respectively. (The huge differences between the entries in the top and bottom halves of the table attest to the effectiveness of our variation in degree-of- match.) The final column in Table III specifies the signed difference between the typicality rating of the conjunction and the minimum value of its constituents, providing a measure of how well the min rule of fuzzy-set theory fits the data. As an aid to understanding we have included, in parentheses, indications of sample pictures and concepts for each case. (The results are averaged over natural-kinds and artifacts since there were no interesting differences between the two types of concepts, nor did this variation in type of concept interact with other factors of interest.)

Let us start with the top half of Table III, with those cases where the pictured objects were good matches. For the nondiagnostic conjunctions,

TABL

E Ill

Mean

Ty

picali

ty Ra

tings

fa

r Ad

jectiv

e Co

ncep

ts,

Noun

Co

ncep

ts an

d Co

njunc

tions

.

Sepo

rotely

fo

r Ea

ch

Case

of

the

Ta

xono

my

(Exp

erim

ent

1).

Case

s (S

ample

Pi

cfure

)

a.

Good

M

atche

s:

Nond

iogno

stic

(unslic

ed

apple

)

Posit

ively

diagn

ostic

(re

d ap

ple)

Nega

tively

dia

gnos

tic

(bro

wn

opple

)

b.

Poor

Matc

hes:

Nond

iogno

stic

(slice

d op

ple)

Posit

ively

Diag

nosti

c (b

rown

op

ple)

Nega

tively

Di

agno

stic

(red

apple

)

Adjec

tive

[Sam

ple

Conc

ept)

8.71

(unslic

ed)

8.50

(red)

6.93

(bro

wn)

0.45

(unslic

ed)

0.02

(red)

0.81

(bro

wn)

Noun

Ad

i-Nou

n Ad

j-Nou

n

(Sam

ple

(Sam

ple

Minu

s

Conc

ept)

Conc

ept)

Mini

mum

7.25

(0pple

J 8.6

5 (un

sliced

ap

ple)

1.40

7.81

(apple

) 8.8

7 (re

d ap

ple)

1.06

3.54

(apple

) 8.5

2 (b

rown

ap

ple)

4.96

7.25O

(ap

ple)

0.52

(unslic

ed

apple

) 0.0

7

3.54’

(apple

) 0.1

0 (re

d op

pleJ

0.08

7.810

(ap

ple)

0.39

(bro

wn

apple

) -0

.42

a Sin

ce

the

vorio

tion

betw

een

good

an

d po

or m

atche

s do

es

not

apply

to

no

un

ratin

gs,

the

mea

ns

here

ore

simply

the

rep

eats

of

the

mea

ns

listed

dir

ectly

ab

ove,

i.e.,

these

thr

ee

meo

ns

ore

not

new

data.

No

te,

howe

ver,

that

the

as

signm

ent

of

meo

ns

to

posit

ively-

on

d ne

gativ

ely-

diagn

ostic

ca

ses

mus

t be

rev

ersed

in

mov

ing

from

the

top

to

the

bo

ttom

of

the

tab

le,

e.g.

, o

pictur

e of

a

red

apple

po

ired

with

ap

ple

corre

-

spon

ds

to

the

posit

ively-

diagn

ostic

ca

se

in the

top

of

the

tab

le bu

t to

the

ne

gativ

ely-d

iagno

stic

case

in

the

botto

m

of

the

table.


we expected the min rule to work because the adjective is largely irrelevant to the noun (though see Footnote 4). The results are otherwise: the conjunction’s typicality clearly exceeded the minimum of its constituents. When we look at the trios of concepts contributing to the averages in this case (where a sample trio consists of unsliced, apple, and unsliced apple), we find that the conjunction’s typicality exceeded the minimum in 18 of 24 trios @c .05 by sign test). There is a comparable deviation from the min rule for the positively-diagnostic conjunctions. For the concept trios contributing to the averages in this case, the min rule was violated 10 of 12 times (p< .05 by sign test). For the negatively-diagnostic conjunctions, where we had expected the largest violations of the min rule, the conjunctions’ typicality exceeded the minimum value of the constituents by virtually half the scale. Inspection of the concept trios contributing to the average indicated the min rule was violated in 11 of 12 trios @< .05 by sign test).

An analysis of variance on the good-match data indicated that the more salient differences in Table III can be taken seriously. Rating groups differed, F(2, 162) = 19.90, p < .Ol, and so did the three cases of our taxonomy, F (2, 162) = 84.12, p< .Ol. More informatively, rating groups and taxonomic cases interacted, F (4, 162) = 20.94, p< .Ol. The bulk of this interaction was due to the noun ratings being much lower in the negatively- diagnostic case, 3.54, than in the other two cases, an average of 7.53. This result attests to the effectiveness of our diagnosticity variation. Adjective ratings were also somewhat lower in the negatively-diagnostic case, 6.93, than in the other two cases, an average of 8.60. This result may reflect a diagnosticity effect-e.g., “it’s less of a brown if it’s an apple”-or an effect of the pictures themselves-e.g., our brown apple may not have been as good a brown as our red apple was a red.

The most important result, then, is that the min rule was violated in all cases where the object was a best match. Nor is there a simple alternative to the min rule within fuzzy-set theory that seems to do a better job. Perhaps the most plausible alternative is that the conjunction’s typicality be the average of its constituents, but this is violated by all three good-match cases (see Table III). The best-fitting post hoc rule is that the conjunction’s typicality is the maximum of its constituents (or equivalently, that the conjunction’s typicality is the same as that of the adjective constituent). The “max” rule works reasonably well for the nondiagnostic and positively-diagnostic cases, but fails for the negatively-diagnostic case (see Table III). And it is not really a serious possibility in fuzzy-set theory, for if conjunctive concepts are represented by a maximum then there is no obvious way to repre- sent disjunctive concepts.

The bottom half of Table III contains the results for cases where the pictured objects were poor matches. For all three cases the min rule works well, but only because subjects in the Adjective and Adj-Noun groups rated


the pictured objects to be unequivocal nonmembers of the relevant concepts. When presented a picture of a brown apple, for example, and asked to judge its typicality for red or red apple, most subjects gave it zero ratings. This finding reflects an uninteresting conformity to fuzzy-set theory; at best it indicates that in tasks such as ours subjects adopt a threshold, below which all objects are rated as nonmembers; at worst the finding suggests that there is something wrong with the prototype view that lies behind fuzzy-set theory.

We suspect that the zero ratings reflect a poor choice on our part of how to experimentally implement our degree-of-match variable, and consequently that the data in the bottom of Table III can not be taken as a sensitive test of the min rule. (For this reason, we will not bother to provide statistical analyses.) Thus, had we used pictures of red apples and reddish-brown apples, it is likely that we would not have obtained so many zero ratings for the concepts red and red apple. This change was made in Experiment 2.

Summary. For the three cases of our taxonomy where the objects are clear- cut members of conjunctive concepts, fuzzy-set theory unequivocally failed to account for the relation between an object’s typicality in the conjunction and its typicalities in the constituents. What remains an open question is whether the theory fares better for items that are poorer matches to concepts.

Experiment 2

Overview. To provide a more sensitive test of how well fuzzy-set theory accounts for conceptual combination when the objects are poor matches, we made two changes from the previous study. First we added pictures that had intermediate values on the properties denoted by the adjectives, e.g., in addition to pictures of a red apple and a brown apple, we included a picture of a reddish-brown apple. We expected that ratings for such intermediate matches would be substantially greater than zero. Second, we had subjects rate the three related pictures-e.g., the red apple, the reddish-brown apple, and the brown apple-on a single trial. This move should also foster non- zero ratings.

Materials. Table IV specifies the concepts used. The first column gives the six noun concepts, all of which are natural kinds. Columns 2-5 give the 24 adjective concepts: those in Column 2 are positively diagnostic, those in Column 3 are negatively diagnostic, and those in Columns 4 and 5 are relatively nondiagnostic of their corresponding noun concepts (though, see Footnote 4). Each adjective was combined with its corresponding noun to


TABLE IV

Materials Used in Experiment 2

Noun Posltlvely

Concepts Diagnostic

apple red

canteloupe round

corn straight

peas green

ostrich feathered canary yellow

Adjective Concepts

Negotlvely

Diognostlc Nondiagnostic

brown peeled

square sliced

bent husked

yellow shelled

unfeathered standing

red caged

Nondiognostlc

unpeeled

unsliced

unhusked

unshelled

walking

uncaged

form a conjunction, yielding a total of 24 conjunctions. Like Experiment 1, each intersection of an adjective and noun in a row of the table also specified a picture. Unlike Experiment 1, the cells in the table do not exhaust the pictures used. Specifically, “between” a pair of pictures that corresponded to related adjectives-e.g., the pictures of red and brown apples-we intro- duced a picture that had an intermediate value on the adjective-e.g., a picture of a reddish-brown apple. There were 12 such intermediate pictures, yielding a total of 36 pictures in all.

Procedure. Separate groups of subjects (16 Harvard undergraduates per group) gave typicality ratings in the Noun, Adjective, and Adj-Noun groups. The procedure was the same as in Experiment 1 save the following excep- tions: (a) On each trial, 1 l/1 s after the experimenter spoke the name of a concept, all three related pictures were presented. Subjects had 20 s to rate all the pictures for typicality. (b) Prior to each trial, subjects were presented for 2 s each the three pictures they would have to rate.

Results. The average typicality ratings are in Table V. The top third of the table contains the data for good matches, the middle third for intermediate matches, and the bottom third for poor matches. The substantial differences between the entries in the top, middle, and bottom of the table attest to the effectiveness of our variation in degree-of-match. As an aid to understanding, we have included in parentheses sample pictures and concepts for each case.

Let us start with the good-match cases. These data replicate the major findings in Experiment 1, and if anything provide an even stronger refuta- tion of fuzzy-set theory. For all three cases, the typicality ratings for the conjunction substantially exceeded the minimum and maximum of its constituents. The difference between the conjunction’s typicality and the minimum typicality of its constituents is significant at the .05 level for all three cases by sign tests. In addition, an analysis of variance yielded results similar to those obtained in Experiment 1. There were significant effects due

TABL

E V

Mean

Ty

picali

ty Ra

tings

fo

r Ad

jectiv

e Co

ncep

ts,

Noun

Co

ncep

ts an

d Co

njunc

tions

,

Sepa

rately

fo

r Ea

ch

Case

of

the

Ta

xono

my

(Exp

erim

ent

2)

Case

s (S

ample

Pi

cture

s)

Adjec

tive

Noun

(Sam

ple

(Som

ple

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) 4.5

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to taxonomic cases, F (2, 135) = 58.73, p< .Ol and to rating groups, F (2, 135) = 87.24, p-z .Ol. (This time rating for conjunctions significantly exceeded those for both nouns and adjectives; for the planned comparison between conjunctions and nouns, F (1, 141) =66.27, p< .Ol; for the comparison between conjunctions and adjectives, F (1, 141) = 12.64, p< .Ol). And again, rating groups interacted with taxonomic cases, F (4, 135)= 30.57, p c .Ol., reflecting the fact that both adjective and noun ratings were lower in the negatively-diagnostic case than in the other two cases.

Let us move on to the middle of Table V, and the data for intermediate matches. Our use of intermediate pictures eliminated the all-or-none responding of Experiment 1, as all ratings in the middle of Table V are substantially greater than zero. A glance at the last column indicates that, for all three cases, the conjunction’s typicality exceeded the minimum of its constituents. These deviations from the min rule, however, are less pro- nounced than those observed with good matches. An analysis of variance yielded a main effect of rating groups, F (2,135) = 29.42, p < .Ol , and planned comparisons indicated that ratings for the conjunction exceeded those for the adjective, F (2, 135) = 5.64, p-c .Ol, which is sufficient to reject the min rule. There was also an effect of taxonomic cases, F(2, 135) = 12.75, p< .Ol, and an interaction between taxonomic case and rating group, F(4, 135) = 12.85, p< .Ol. Both of these effects are explicable in terms of our diagnosticity variation.

The bottom of Table V contains the data for the poor matches. Our variations in materials and procedure seemed to have induced the subjects to treat our variation in degree-of-match as a continuum, i.e., they no longer responded to the poor matches as out-and-out nonmembers. Still, a couple of the numbers at the bottom of Table V are perilously close to zero. With this caution in mind, inspection of the last column of these data indicates that the min rule of fuzzy-set theory works rather well here. An analysis of variance supports this conclusion. While there was a main effect of rating groups, F (2, 135) = 3 13.7 1, p< .Ol, it reflected only higher ratings for nouns, as ratings for conjunctions were clearly close to those for adjectives (the minimum constituent). There was also an effect of taxonomic cases, F (2, 135) = 37.78, p< .Ol, and an interaction between case and rating group, F (4, 135) = 51.95, p< .Ol. As usual these two effects are explicable in terms of our diagnosticity variation.

Summary. We have replicated the good-match results of Experiment 1: when the objects were clear-cut members of conjunctions, the min rule of fuzzy-set theory unequivocally failed. Fuzzy-set theory also failed when the objects were intermediate matches to the relevant concepts. Indeed, the only cases where the min rule correctly described the relation between typicality in a conjunction and typicality in its constituents was when the objects were


judged to be very poor members of the relevant concepts. This seems, at best, a minor victory. A theory that can predict membership ratings only for objects that are close to nonmembers is clearly missing much of what is go- ing on. One should be able to construct theories that capture more of the data, and in what follows we attempt to do SO.~

A REPRESENTATIONAL APPROACH TO CONCEPTUAL COMBINATION

Rationale for a Representational Approach

The min rule does not account for typicality judgments in conjunctive concepts. This is the burden of the two experiments just reported. We now wish to argue that the difficulty for fuzzy-set theory lies not in the min function itself, but in the use of characteristic functions to the exclusion of the mental representations that they summarize.

Let us briefly sketch the case against such exclusive dependence on characteristic functions. Our experimental data are inconsistent with the use of min, average, and maximum rules to explain judgments of typicality in conjunctive concepts on the basis of judgments of typicality in their constituents. What about other simple rules or functions? In a previous paper (Osherson & Smith, 1982), we presented a formal argument against there being any simple function that relates the typicality properties of conjunctive concepts to those of its constituents. (See Tversky and Kahneman, 1983, for a comparable argument regarding the similarity properties of conjunctive concepts.) If our argument is correct, proponents of fuzzy-set approaches will have to revert to “complex” functions (that is, functions that consider more than just the typicality of constituent concepts).

Zadeh (1982) and Jones (1982) have done precisely this, although the uses made of characteristic functions by these two researchers are very different. Unfortunately, both approaches appear to be fraught with problems

’ One might, however, question the definitiveness of our data on the following grounds.

In both our experiments type of rating was varied between subjects, which leaves open the possibility that different groups used the rating scale differentially. Subjects in the Adj-Noun

group, for example, might have considered different domains of objects when making their ratings than did subjects in the Noun group, and this might have affected how the subjects divided up the rating scale (L. Rips, personal communication, June, 1983). This possibility suggests the need for a study similar to Experiments I and 2 but with rating-type as a within-

subject variable. We performed a pilot study along these lines and found that the min rule still failed to account for the typicality ratings for best matches, which argues against the above-

mentioned scaling problem. Our pilot results also suggested that the within-subject variation in rating-type led to the use of task specific strategies (e.g., “always order the three ratings in the

same way”), which suggests that our original between-subjects design is preferable.


as severe as those besetting the original fuzzy-set theory. The matter is dis- cussed in Osherson & Smith (1982). These negative findings suggest that it is time to explore an approach to conceptual combination that starts with the prototype representations themselves. Such a shift of emphasis has previously been argued for by Cohen and Murphy (1984). Hampton (1973), and Thagard (1983).

The following representation-based analyses are primarily suggestive (a more formal account is presented in Smith, Osherson, Rips, Albert, & Keane, forthcoming). In developing these analyses, we have focused on two critical findings from the foregoing experiments (re-presented in Table VI): (a) when an object matched the adjective in a conjunctive concept, the object was judged more typical of that conjunctive concept than of the noun constituent; (b) in contrast, if an object did not match the adjective in a conjunctive concept it was judged less typical of that conjunctive concept than of the noun. Thus, red apples were more typical of red apple than of apple, while brown apples were less typical of red apple than of apple-and similarly for the ‘concepts brown apple and peeled apple (see Table VI).

TABLE VI

Mean Typicality Ratings for Noun Concepts ond Conjunctions,

Separotely for Objects that Match or Mismatched the Adjective in the Conjunction

Experiment I Experlmenf 2

Noun Adi-Noun NOWI Adi-Noun

o. Obiect Matched Adjective:

Nondiagnostic

Positively Diagnostic

Negatively Diagnostic

Average

b. Object Mismatched Adjectiv

Nondiagnostic

Positively Diagnostic

Negatively Diagnostic

Averaae

fe:

7.25 8.65 6.50 7.75

7.81 0.07 7.00 a.75

3.54 8.52 2.17 8.42

6.20 8.60 5.28 8.31

7.25 0.52 7.25 1.25

3.54 0.10 2.17 1.17

7.81 0.39 7.00 0.50

6.20 0.34 5.28 0.97

Prototype Representations of Noun Concepts

In constructing a representational account of typicality judgements for conjunctive concepts, a useful starting point is to consider the comparable account for simple noun concepts. According to recent work: (a) both a simple concept and an object can be represented by a set of attributes and their associated values or “features,” (e.g., Smith & Medin, 1981); and (b) the typicality of an object in a concept is a direct function of the object’s similarity to the concept, where similarity is assessed in terms of a contrast between common and distinctive features (Tversky, 1977). Figure 1 illustrates the key assumptions.


Apple (A)

I red*

3 color white

I brown

- -

I round* 1 shape

I

square - -

I smooth* 2 texture rough

1 - -

- -

a red apple (0,)

I red*

color white

t brown \ - -

I round* shape

I

square - -

I smooth* texture

1

rough - -

- -

Sim. (0,) A) = 6 -0 -0 =6

a brown apple (OJ

I red

color ) white

t brown* t - -

shape I round*

I

square - -

texture

- -

I smooth’

I

rough - -

Sim (02, A) = 3 -3 -3 =- 3

Figure 1. Illustration of attribute-value representations of noun concept (apple) and rele-

vant objects (o red apple and a brown apple): beneath each object representotion is the

similarity between the object and the concept.

The left-most panel of the figure contains a sample representation for the concept apple. The representation specifies: (a) a set of relevant attributes (color, shape, texture, etc.), and (b) for each attribute, a set of possible values, or features, that instances of the concept can assume on that attribute (e.g., for color, the features include red, white, and brown). In addition, the representation also specifies (c) the most likely feature ‘for each attribute, as indicated by the position of an asterisk, as well as (d) the diagnosticity of each attribute for the concept, as indicated by the number to the left of each attribute. This kind of representation is essentially a frame (e.g., Minsky, 1975; Winston & Horn, 1981), with attribues being slot-names, features being values, and most-likely features being default values.

In the remaining panels of Figure 1, we have given sample representations for two specific objects, one a typical red apple and the other a brown apple. Note that an object representation is simpler than a concept representation, as the former specifies only a set of relevant attributes, and the actual value of each attribute for that object. Beneath each object representation we have calculated its similarity to the concept apple using Tversky’s (1977) “contrast” model. In terms of this model, the similarity between the most- likely features of apple (labelled “A”) and the actual features of the object (labelled “0”) is given by:


(2) Sim (A, O)=af (AnO)-bf (A-0)-cf (O-A).

Here, An0 designates the set of features common to the concept and object, A - 0 designates the set of features distinct to the concept, and 0 -A designates the set of features distinct to the object. In addition, f is a function that measures the salience or prominence of each feature, and a, b and c are parameters that determine the relative contribution of the three feature sets. The basic idea is that similarity is an increasing function of the features common to the concept and object, and a decreasing function of the features distinct to the concept and of those distinct to the object. In making the computations in Figure 1 (as well as in subsequent figures), we have assumed that a, b, and c are equal to one. We have further assumed that f is the weighted sum of the features in that set. That is, to determine f (AnO), we simply add the weights of all features that are common to the object and concept, and similarly for F (A - 0) and F (0 - A) (where the weight of a feature is given by the weight of the corresponding attribute in the concept.) For the examples provided in Figure 1, application of the contrast model correctly predicts that the red apple should be judged to be more typical of apple than is the brown apple.6

Prototype Representations for Conjunctions

We now want to extend this kind of account to adjective-noun conjunctions. The first order of business is to specify how the adjective will interact with the noun concept. There are two obvious possibilities. One is that the adjective is represented in a similar way to the noun and that the two representations are intersected. Alternatively, the adjective may play a different role than that of the noun, in that it is used to modify the noun. There are a number of reasons for favoring the modification approach over that of intersection (see Cohen & Murphy, 1984, for a discussion of some of these issues), and modification is the track that we will take.’

Figure 2 illustrates our assumptions about adjective modification (to keep things simple, we consider only adjectives that presumably contain a single feature, e.g., red and brown). In essence, the adjective does three things: (1) it picks out the relevant attribute in the noun (e.g., color); (2) it dictates where the asterisk for the relevant attribute should be positioned (e.g., red); and (3) it increases the diagnosticity weight associated with the relevant attribute. In the example at the top of Figure 2, the critical asterisk

b Technically. we are using a generalized version of the contrast model where the weight assigned to a feature depends on the concept as well as the feature. For example, the weight for

red may differ for apple and brick. Tversky’s formalism, given in [2], is a stronger theory in that it assigns the same weight to a feature regardless of what concept that feature is part of.

’ We are indebted to Amos Tversky for highlighting the intersection-modification dis- tinction to us.


- - - -

Brown Apple Brown Apple

bZiLji?i+ie brown

+ 6 color { Zite brown*

- - - -

Figure 2. Illustration of three aspects of adjective modification.

was already on the feature specified by the adjective, red, and hence no change in the asterisk’s position is needed. This situation is the hallmark of positively-diagnostic conjunctions like red apple. There is, however, a change in the weights, as the diagnosticity of the attribute color has doubled. In the example at the bottom of Figure 2, the critical asterisk has to be moved from the default feature, red, to an atypical value, brown. This situation is the hallmark of negatively-diagnostic conjunctions like brown apple. In addition, there is again a doubling of the diagnosticity of color.

Figure 3 illustrates the implications of the above changes for typicality ratings with positively-diagnostic conjunctions. The left-most panel of the figure contains the representation for the conjunction red apple. In keeping with our assumptions about modification, the only difference between this representation and that of apple is in the diagnostic importance of being red. The effects of this difference on typicality are illustrated in the remaining panels of Figure 3. There we have repeated the object representations for our red and brown apples, and computed the similarity for each of these objects in the conjunction. When these similarity scores are compared to those in the Figure 1, the result is that the red apple is more similar to red apple than it is to apple, while the brown apple is less similar to red apple than it is to apple. We have therefore reconstructed our major findings for the positively-diagnostic case, that an object that matches the adjective in a conjunctive concept is more typical of the conjunction than of its noun constituent, while an object that does not match the adjective in a conjunction is more typical of the noun constituent.

Figure 4 illustrates a comparable analysis for negatively-diagnostic conjunctions. The left-most panel of the figure contains the representation for brown apple. In keeping with our modification proposal, this represen-

The Positively - Diagnostic Case

Red Apple (RA)

6 color

1 shape

2 texture

- -

red* white brown

- -

round* square

- -

smooth l

rough - -

a red apple (0 1

1 red*

color ) white ) brown I - -

I round+ *ape square

- -

I smooth l

texture rough

I - -

Sim. (0, , RA) = 9-O-O Sim. (02, RA)=3-6-6 = 9 E-g

a brown apple (02)

i red

color white brown*

I round* shape

I

square - -

smooth* texture rough

- -

Figure 3. Illustration of attribute-value representations of red apple and relevant objects;

beneath each object representation is the similarity between the object and the conjunction.

The Negatively - Diagnostic Case

Brown Apple (BA)

1 red

6 color white brown’

I round* 1 shape square

- -

- -

a red apple (0,)

red*

color white brown

- -

I round+ shape square

- -


- -

Sim. (0, , BA) = 3-6-1 E-g

a brown apple (Og)

I red

color white brown*

-

round* shape square

- -


- -

- -

Sim. (Og , BA) = 9 -0 -0 =9

Figure 4. Illustration of attribute-value representations of brown apple and relevant ab-

jects; beneath each object representation is the similarity between the abject and the con-

junction.

356


tation differs from that of apple in two ways: (1) color is more diagnostic, and (2) the value of color is now brown rather than red. The effects of these changes on typicality are shown in the remaining panels of Figure 4. Again, we have reproduced our object representations for the red and brown apples, and computed the similarity for each object in the conjunction. Now, in comparison to our original computations in Figure 1, we find that the brown apple is more similar to brown apple than it is to apple, while the red apple is less similar to brown apple than it is to upple.Again, we have reconstructed our major findings. Note, however, that in this case the results reflect changes in both (a) diagnosticity weights and (b) the number of common and distinctive features (e.g., the brown apple shares more features with brown apple that with apple).

The last case to be considered is the nondiagnostic one, and it is illustrated in Figure 5. The left-most panel of the figure contains the representation for peeled apple. This representation differs from that of apple in three ways: (1) peeledness, and attribute that presumably was only implicit

Peeled Apple (PA)

3 color

1 shape

2 texture

3 peeledness

- -

red* white brown

- -

round* square

- -

smooth* rough

- -

peeled* unpeeled

The Nondiagnostic Case

a peeled apple (03 1

I red*

color white

I brown

- -

I round l

shape

I

square - -

I smooth l

texture rough

I - -

I peeled l

leeledness unpeeled

1 - -

- -

Sim. (OS, PA) = 9-O-O Sim. (0, , PA) = 6 -3-3

=9 = 0

Figure 5. Illustration of attribute-value representations of peeled apple ond relevant ob-

jects; beneath each object representation is the similarity between the object and the con-

junction

an unpeeled apple (04 1

I red*

color white

I brown

- -

I round+ shape

1

square - -

I smooth* texture rou&

I =

I Peel& mledness unpeeled*

1 - -

- -


in apple, is now represented explicitly;8 (2) peeledness has become quite diagnostic; and (3) the value of peeledness is now peeled. (Presumably, there was no asterisk in the relevant attribute of the noun before modification: this situation is the hallmark of nondiagnostic conjunctions.) The consequences of these changes are displayed in the remaining panels of Figure 5. There, we have given object representations for our peeled and unpeeled apples and have computed the similarity for each object in the conjunction. To see how well these results fit with our major findings, we need to know the similarity of peeled and unpeeled apple to the concept of apple. These similarities turn out to be identical to those of a red apple, namely 6, (see Figure l).’ Thus, the peeled apple is found to be more similar to peeled apple than to apple, while the unpeeled apple is found to be less similar to peeled apple than to apple. Once more we have captured our major findings.

The results of these illustrative analyses are summarized in Table VII. These results make it clear that when we combine proposals about attribute- value representations for nouns and adjectives with our specific proposals about modification, the resulting package captures the major qualitative findings of our experiments.

TABLE VII

Resulting Similarity Scores in Illustrative Analysis

Concepts red o&e

Objects

brown omle

Apple

Red Apple

Apple

Brown Apple

Apple

Peeled Apple

6 -3

9 -9

6 -3 -9 9

peeled apple unpeeled apple

6 6 9 0

Some Extensions

The preceding analyses considered only good and poor matches. We can extend these analyses to intermediate matches by enriching our representation of default features. The asterisk approach we have been employing assumes

* To assume that peeledness was implicit in apple is to assume that though possibly relevant to being an apple, the attribute had no default feature nor weight, and consequently

played no role in judgments of typicality for upple. These assumptions go beyond the specifics of our modification proposal. An alternative assumption is the peeledness is added to the representation of upple during modification.

’ The only representational difference between, say, our peeled apple and our red apple is that the former is implicitly marked for peeledness, but since this attribute presumably has no weight in upple it has no effect on the similarity computations.


that each attribute has a single default, and that the default for any one attribute is as important as that for any other attribute. Suppose, instead, that we assume the representation is like that in Figure 6. The first two panels contain representations for the concepts apple and red apple. For every attribute, instead of an asterisk we have a distribution of numbers, or weights, over features, where these weights indicate the “default mixture” of the relevant features. For example, with regard to color, the prototypical apple is mainly red but has some white and brown in it, while the prototypical red apple is weighted only for red. In the last two panels of the figure we have object representations for our red apple and our reddish-brown apple (the latter being an intermediate match). We have assumed that the red apple is weighted heavily for red while the reddish-brown apple is half red and half brown. Taking these feature weights into account, it is apparent that regardless of the specifics of the similarity computations, the reddish-brown apple is less similar to either apple or red apple than is the red apple; this accounts for why, in comparison to good matches, intermediates were judged less typical of both apple and red apple. In essence, intermediate matches do not match closely either of the above concepts because each concept requires a different weighting of colors than the object in fact manifests. In Smith, Osherson, Rips, Albert, & Kean, 1984, we provide a thorough treatment of such feature weights and their consequences for similarity.

Another restriction of the present paper is that it dealt only with adjective-noun conjunctions. Our representational approach, however, can readily be extended to other kinds of combinations. One relatively straight- forward extension involves adverbs like very and slightly. Such adverbs can be represented as “scalars” (e.g., Cliff, 1959), which multiply feature weights. Thus, in very red apple, essentially very multiplies the weight of red in red apple. In Smith et al. (forthcoming), we provide an extensive treatment of such adverbs.

Other possible extensions could be mentioned, but it seems best to close on a cautionary note. We doubt that any approach based only on prototype representations can provide a complete account of conceptual combination. In particular, it seems unlikely (at least to us) that the approach illustrated here can illuminate the nature of logically-empty concepts (e.g., apple and non apple) and logically-universal concepts (e.g., apple or nof an apple), or inclusion relations between concepts, (e.g., ail coconuts arefruits). There is more to a concept than its prototype. As we mentioned at the outset, a concept may contain an identification procedure, which has a prototype structure, as well as a core, which does not, and it may be the core that is critical in representing logically-empty and logically-universal concepts and in accounting for inclusion relations. We expressed these same reservations when we examined fuzzy-set theory as a calculus for prototype representations (Osherson & Smith, 1981; 1982), and we do not think that the calculus we have proposed here obviates these fundamental problems.

App

le

I red

8

3 co

lor

whi

te

1

1 br

own

1 \

- -

I roun

d 9

1 sh

ape

I

squa

re

1 - -

2 te

xtur

e

- -

smoo

th

9 ro

ugh

1 - -

Red

A

pple

6 co

lor

1 sh

ape

red

10

whi

te

brow

n - -

roun

d 9

squa

re

1 - -

I smoo

th

9 2

text

ure

roug

h 1

I - -

- -

a re

d ap

ple

(0,)

I red

9

colo

r w

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1

I br

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I roun

d 9

shap

e sq

uare

1

I

text

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- -

smoo

th

9 ro

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1

a re

d -

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n ap

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(OS)

red

5

colo

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1 br

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5

roun

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shap

e I

squa

re

1

text

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- -

smoo

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9

I

roug

h 1

Figur

e 6.

Illu

strati

on

of

attrib

ute-va

lue

repres

entat

ions

of

apple

, red

ap

ple,

and

relev

ant

objec

ts,

with

fea

ture

weigh

ts ra

ther

than

aster

isks.


REFERENCES

Armstrong, S. L., Gleitman. L. R., & Gleitman. H. (1983). What some concepts might not be.

Cognition. 13, 263-308.

Cliff, N. (1959). Adverbs as multipliers. fsyc/ro/ogicu/ Review, 66, 27-44.

Cohen, B., & Murphy, G. L. (1984). Models of concepts. Cognirive Science, 8, 27-60.

Dubois, D., & Padre, H. (1980). Fuzzy sets andsysfems: Theory and applications. New York:

Academic Press.

Hampton, J. A. (1983). A composite prototype model of conceptual conjunction. Unpub-

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Jones, G. V. (1982). Stacks not fuzzy sets: An ordinal basis for prototype theory of concepts.

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Minsky, M. A. (1975). A framework for representing knowledge. In P. H. Winston (Ed.), The

Psychology of Cornpurer Vision. New York: McGraw-Hill.

Osherson, D. N., & Smith, E. E. (1981). On the adequacy of prototype theory as a theory of

concepts. Cognition, 9, 35-58.

Osherson, D. N.. &Smith, E. E. (1982). Gradedness and conceptual combination. Cognirion,

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Smith, E. E., & Medin. D. L. (1981). Curegoriesondconceprs. Cambridge, MA: Harvard Uni-

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Smith, E. E., & Osherson, D. N. (1982). Conceptual combination and fuzzy set theory. Pro-

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Smith, E. E., Osherson, D. N., Rips, L. J.. Albert, K., and Kean, M. (1984). A theory of con-

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Thagard. P. (1983). Conceptual combination: A frame-based theory. Paper presented at the

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Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction

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Winston, P. & Horn, B. (1981). Lisp. Reading, MA: Addison-Wesley.

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Zedeh. L. A. (1982). A note on prototype theory and fuzzy sets. Cognirion. 12, 291-297.

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Conceptual Combination with Prototype Concepts*osherson/papers/smithOsh1.pdf · 2013-01-22 · COGNITIVE SCIENCE 8, 337-361 (1984) Conceptual Combination with Prototype Concepts*

Conceptual Combination with Prototype Conceptsosherson/papers/smithOsh1.pdf · 2013-01-22 · COGNITIVE SCIENCE 8, 337-361 (1984) Conceptual Combination with Prototype Concepts