On the adequacy of prototype theory as a theory of concepts

Cognition, 9 (1981) 35-58 @Elsevier Sequoia S.A., La._sanne- Printed in the Netherlands

On the adequacy of prototype theory as a thmry of concepts* DANIEL N. OSHERSON Massachusetts lnstitu te of Terhnology

EDWARD E. SiUllTH Stanford

University

and Bolt Beranek and

Newman, trlc.

Abstract Prototype theory construes membership in a concept’s extension as graded, detemzined by similarity to the concept’s ‘3e.W exemplar (or by some other measure of central tendency). The present paper is concerned with the compatibi1it.J of this view of concept membership with two criteria of adequacy for theories of concepts. The first criterion Conrorns ;he relationship between complex concepts and their conceptual con;,&:CSrr The second concerns the truth conditions for thoughts correspo,ldi.T? lo simple inclusions.

A novel and ambitious theory has emerged from the last ten ye:rs of psychological research into the concepts that underlie ‘kind” terms like “animal”, “tree”, “tool”, and “clothing”. The distinguishing doctrine of the new theory is that entities fall neither sharply in nor sharply out of a concept’s extension. Rather, an object instances a concept only to the extent that it is similar to the prototype of the concept; the boundary between membership and nonmembership in a concept’s extension is thus fuzzy. We’ll call the new theory (really, class of theories) prototype theory. In this paper we consider two aspects of concepts relevant to choosing between pretcltype theory and its more traditional rivals. One concerns conceptual combination, that is, the process whereby relatively complex concepts are *‘I’%is research was partly supported by U.S. Public Health Service Grant MH-19705. We thank Ned Block, Susan Carey, D&vidIsrael, Louis Narens,Gary Olson, Molly Potter, Lance Rips, WilliamSalter, and Ken Wexler fcr helpful iliscussion of the ideas in rhis paper. Reprint requests should be sent to EdwardE. Smith, Bolt Beranekand Newman, Inc., 50 Moulton Street, Cambridge,MA 02138.

36

D. N. Oshelsonand E. E. Smi2h

forged out of relatively simple ones. The other deals with truth conditions for thoughts, that is, the circumstances under which a thought corresponding to a declarative proposition is true. For both aspects, we argue that the new theory of concepts fares worse than the old. The organization of this paper is as follows. We first present one version of prototype theory. We then show how it might be extended to account for conceptual combination by means of principles derived from ,&zy-ret theory (e.g., Z&h, 1965). This extension is demonstrated to be frautrht with difficulties. We then move on to the issue of truth conditions for thoughts, again using fuzzyset theory as a means of implementing the prototype approach, ar?, again iemonstrating that this implementauon won’t work. In a final seLlion, we establish that our analysis holds for virtually any version of prototype theory, and consider ways of reconciling previous evidence for this theory 1.ith the wisdom of the older kind of theory of concepts. Section 1. Prototype theory

The versron of prototype theory expounded i:i this section appears implicitly and explicitly in several well known papers (e.g., Posner & Keele, 1968; Rosch, Simpson and Miller, 1976), and it is the simplest account we know. However, other work within the prototype theory tradition (e.g., Reed, 1972) directly contradicts it, and there are numerous alternative formulations of the theory that differ from our version in various details. Nevertheless, the version about to be specified is “prototypical” of prototype models in that it captures key ideas of prototype thee-y more successfully than rival versions, and it is only these key ideas that are crucial for our analysis, as we delnonstrate later (see section 4.2). 1. _r I Formal characterization I According to the present version of prototype theory, many concepts can be identified with (mental) representations of quadruples like (*), (*) (A, d, P, c) where A is a set of readily envisionable objects (real or imagined) called a conceptual domain; d is a function from A x A into the positive real numbers, called a distance metric;

On the adequacy of prototype theov

37

p is a member of A, called the concept’s pvofotype; and c is a function frolm A into [0, 11, called the concept’s characteristic function; and such that the folio wing two conditions hold: (1.1)

(A, d> is a metric space, i.e., 0fx-Wfy-U

(la) d(x,y)=Oiffx=y (lb) d(x, y) = ‘;l(y, x) (1.2)

( 1c) d(x, Y) + W, 2) (VxEA)(VyEA) d(x,

2 dk

2)

PI g d(y, PI + C(Y) d c(x).

The second condition requires that the closer an object is to its prototype, the more characteristic it is of the concept. We can illustrate with the concept bird. According to prototype theory it is ic’sntical to a mental representation of the quadruple (B, dbird9 Pbird , Cbird ), where B is the set of all readily envisionable birds (including robins, sparrows, and penguins);’ db+d is a function on pairs of such birds into real numbers (aSSi@iIX smaller numbers to pairs of similar objects, e.g., robin-sparrow, than to pairs of dissimilar ones, e.g., robin-penguin, thereby reflecting relative psychological similarity among elements of B); pbtid is some particular bird in B (usually taken ?o !.e the bird that has the average value on each dimension of the underlying metric space); and ci.,W is a function assigning numbers in [O, 11 to members of B in such a way that closeness to the prototype reflects greater “birdiness.” When construed in this way, the concept bird appears to meet conditions (1.1) and (1.2). And this construal is consistent with a number of empirical findings. In particular, consider the case where subjects give similarity ratings for pairs including either two instances of bird (e.g., “robin-sparrow”) or one instance and the term “bird” itself (e.g., “robin-bird”), and then these &ic space. Assuming that ratings are converted into a multidimensional ‘If you come across an ordinary robin that is new to your experience, is the theory committed to a change in your bird concept, in light of the new element added to B? No, the new robin was already envisionable, so it was in B to begin with. On the other hand, if you happen upon a very queer bird {e.g., one with tusks), not so readily envisioned, your concept may well change by expansion of B.

38

D. N. Oshersonand E. E. S with

the point in the space for the term “bird” corresponds to tL2 prototype, the following fmdings emerge: (a) The value of the prototype on each dimension is roughly equal to the average value of all scaled instances on this dimension (Rosch t!kMen&, $975). This provides some justification for identifying the prototype with an average, though it is pot really critical for the present version of prototype theory s (b) In general, the less the metric distance between any bird and the prototype, the more characteristic or prototypical of the concept that bird is judged to be (e.g., Rips, Shoben & Smith, 1973). Since different groups of subjects make the initial pair&e similarity ratings and prototypicality judgments, these results provide good evidence for condition (1 .Z) above, i.e., for the psychological reality of a function mapping distance from a prototype into a characteristic function ‘4at reflects “birdiness”. *z) In general, the more characteristic or prototypical a particular bird: (i) the more efficiently it can be categorized, efficiency being measured by both accuracy and speed (e.g., Rips et al., 1973); (ii) the earlier it will be output when subjects list instances of the concept (e.g., Rosch et al., 1976); and (iii) the earlier a child will learn that it is an instance of the concept (e.g., Rosch, 1973). Thus, the value of the characteristic function for an object can be used to predict aspects of real-time processing and conceptual development. 6.2 Some gap 6s ?he theory While the above fmdine provide some support for one version of prototype theory, there Cirecurrently two substantial gaps in this or any other extant vetion of the theory. First, not all natural concepts succumb to the kind of construal we have given for bird. Prototype theory, LS thus far developed, is bzt suited to “kind” notions (like &g, tree, and animal), to “artifact” notions (like tool and clothing), and to simple descriptive notions like triangular arm red What remains outside the theory’s purview are intentional or otherwise intricate concepts like belieA desire, and justice, as well as the meanings of prepositions, sentence connectives, and a host of other ideas. it is an open question whether or not the theory can be extended to cover these cases, and in this paper we shall say no more about them. The second gap is the inverse of the fast ; namely, many nonnatural concepts do succumb to the kind of construal we habe given for bird. Accord?g to the present version of prototype theory, each concept is idenllcal to some quadruple, with distinct concepts being identical to distinct

On the adequacy of prototype theory

39

quadruples (i.e., quadruples differing in at least one component).2 But many such quadruples will have bizarre domains (and perhaps other defects :,s well). It is possible, for example, to use a single metric space to represerit objects as diverse as dogs, chairs and toothpaste (thereby satisfying conditiot (1 .l), to define a prototype in this space, and to then map distances from this prototype into values of a characteristic function so that small distances go with larger characteristic values (thereby satisfying condition (1.2)). This state of affairs reveals an important incompleteness in the theory. We need to know more about the entities A, d, p, and c than is revealed by conditions (1.1) and (1.2), so that we can limit the theoretically possible concepts to those that are psychologically natural. Again it is an open question whether such an extension of the theory can be made, and again we shall say no more abet? t the issue here. Section 2. Prototype theory and conceptual combination

Despite the gaps ju st mentioned, it is useful to evaluate prototype theory by means of facts about conceptual combination, where the concepts are natural ones and restricted to kind terms. Though there has been some experimental work on this issi;:: (e.g., Hersch & Caramazza, 1976; Oden, 1977), tls present analysis provides a more general outlook. In what follows, we first briefly describe the issue of conceptual combination, &n relate tixns issue 20 legitimate demands tIpon prototype theory, next spell out the principles of fuzzy-set theory oti which prototype theorists rely in order to deal with conceptual combinaticn, and lastly show that this enterprise eventuates in a snarl of contradictions. 2. I The issue One or more concepts combine to form anoth::r whenever the latter has the former as constituents. Grammatical constituency can often sexve as a guide to conceptual constituency. Thus, the words “red” and “table” are constituents of the grammatical structure “red table,” and in parallel fashion the concepts Ired and table are constituents of the conceptual structure red table. The same parAle holds for many other conceptual combinations, e.g.,

‘More precisely, dlffment concepts are to be ideatiied with mentd representations of different differentmental representatonsof the same quadruple to be variants of the same concept). In what follows, we will occasionally suppress thin mental-representationquallller.

quadruples (we% count

40

D. iV. Oshemonnnd 17.E. Smith

square window and tasty onion. 3 There are, of course, cases where the paral-

lelism breaks down. Thus, the concept dark horse (as in political contests) does not have dark and horse as conceptual constituents. Such idioms notwithstanding, it seems safe in what follows to frequently rely on grammatical structure as a guide to conceptual structure. The phenomena surrounding conceptual combi,qation can be used to evaluate prototype theory in the following way. Let’s say that this theory is compatible with conceptual combination if principles can be supplied that correcc;y predict the relation between complex concepts an+: their constituents, when concepts are construed as specified by the theory. Or at least, many such conceptual combinations accord with the theory’s principles and few violate them. 2.2 Criteria of adequacy regarding conceptual combination Suppose a given concept C has concepts C1 and Ca as constituents. For prototype theory to be compatible with this case of conceptual combination, principles must be available to specify the quadruple associated with C on the basis of th,? ,duadruples associated with C1 and C,P Actually, these principles ought to specify the (mental) representation or the former quadruple on the basis of the (mental) representations of the latter quadruples; but in the present em of psychological understandmg, we shall settle for a specification, not of the representation (i.e., not c ‘C itself), but of the associated quadruple (i.e., of what C represents). The desired principles, then, will characterize C’s domain, distance metric, prototype, and characteristic function on the basis of the entities from the quadruples for CI and Cz. However, we shall restrict attention to the problem of specifying C’s characteristic function on the basis of those of C1 and CZ . There are two reasons to so narrow the present inquiry. First. a solurion to the character-i& function problem is necessary for a gene&Asolution to the problem of conceptual combination, and it may be close to sufficient as

?It is unclear whether the parallelholds for grammaticalstructures l&e “very happy”. It may be that very kpp~~ has only one conceptual constituent, namely, Iwrpp~.That is, the word ‘very” may not ¬e a concept in very Iazppy,since it may not correspond to a constituent in the language of thought, but instead be representedsyncategorematic4ly. Simiku remarksapply to UnhuppY.Well extend the ordinary usage of “comtination” to atlow a sin@I collcept to be “combined” into another concept by the use of such devices. 4 Additionally, it must be true that prototype theory successfblly construes C, Cl, and (2% as mental represcntatiok4of quadruplesof form (*) in the firpltplace. Well assume 80 in what follows.

On the adequacy of protorype theory

41

well? Second, the only explicit account of conceptual combination within the prototype theory tradition (given below) restricts its attention to the characteristic functions of C, C1 , and C1. We may now investigate the compatibility of prototype theory with the facts of conceptual combination, in particular, with the relationship between the characteristic functions of complex concepts and the characteristic functions of their conceptual constituents. Combinatorial principles germane to this problem have been supplied by Zadeh (e.g., 1965) under the name of fuzzy-set theory, and prototype theorists (e.g., Oden, 1977; Rosch and Mervis, 1975) cite fuzzy-set theory in this connection. It is possible that principles other than those provided by Zadeh can better serve prototype theory in accounting for conceptual combination, but no suitable alternative has yet been suggested (so far as we know).6 Since, in addition, fuzzy-set theory is a natural complement to prototype theory, and the former is an appealing theory in its own right, we shall evaluate prototype theory esclusively in the context of fuzzy-set theory. If this ensemble is at variance with the facts of conceptual combination, there is some reason to doubt the compatibility of prototype theory with these facts. 2.3 Fuzzy-set theory The principles of fuzzy set theory are straightforward generalizations of elementary principles of standard set theory.’ We start by enumerating the relevant principles of standard set theory. Let D be the domain of discourse, and A a subset of D. In standard set theory, the characteristic finction for A is defined to be that unique function (2.1) CA:D + (0, I} such that (2.2)

(V x E D)cA(x) = ;iff;;; I

.

s For: (a) C’s domain is likely just the union of those for Cl and C2, (b) C’s distancemetric can be highly constrained by its characteristicfunction if condition (1.2) is strengthened in any of several empirically plausible ways (we omit the details), and (c) c’s prototype can be taken as the “average” memberof C’s domain according to its distance metric. ’ In Sections 3.5 and 4.3 we discussalternaiivesto Zadeh’s(1965) proposals. ‘Zadeh (1%5) develops fuzzy-set theory beyond the limits imposed here, e.g., to deal with the composition of relations. However, we will present enough of the theory to assessits empiricaladequacy.

42

D. N. Osherson and E. E. Smith

Set membership is thus strictly binary; there are no liminal cases of objects falling neither precisely in nor precisely out of a set. The characteristic functions for intersections, unions, and complements of sets can be standardly defined as follows (where A and B are arbitrary subsets of D). (2.3) (2.4) (2.5)

(VXED) (vx~D) (VXED)

c An B(x) = min(c* (x), cg (x)) (Intersection) c A U B(x) = max(cA(xh (union) 63 (x)) (Compiement) Cnd(X)= 1 -CA(X)

The characteristic functions for the empty and universal sets,@ and D, meet conditions (2.6) and (2.7), respectively. (2.6) (2.7)

(V x E D)CS (x) = 0 (Empty set) (V x E D)cn (x) = 1 (Universal set)

Thus, in the domain of animals, suppose that cdW(Rover) = 1 and cfanale (Rover) = 0. Then: cd@f,&(Rover)

= minicd,(Rover), cfati,(Rover)) I*‘” = min(1, 0) = 0; cdaUfenale(Rover) = max(c&,(Rover), cfmale(Rover)) =max(l,O)= 1; = 1 - cd&Rover) = 1 - 1 = 0; c,,tig(Rover) c,demde(Rover) = I - cfmale(Rover) = 1 - 0 = 1. Fuzzy set theory results from expanding the range of characteristic functions from {O, 1) to [O, 1 I , i.e., from the set whose only elements are 0 and 1 to the set of all real numbers between 0 and 1, inclusive. Set rnelmbership thus becomes continuously graded, an element, e, belonging to A to the extent that c,(e) approximates 1. Instead 01 (2.1) and (2.2), we now have (2.8) (2.9)

CA: D+ [0, 1] The larger CA(x), the more x belongs to A; the smaller c,(x), the less x belongs to A; 1 and 0 are limiting cases (for all x e D).

The definitions given by (2.3~(2.7) remain unchanged, but the characteristic functions therein are now taken to be of the fuzzy-set type, (2.8), rather than of the standard type, (2.1). To illustrate, if cdW(Rover) = 0.85 and cfenat(Rover) = 0.10, then:

On theadequacyofprototypetheory

c&g

43

nfemdRoveO= mWdJRover), cfemate(Rover))

CdogUBemale(Rover)

c,,h,(Rover) c,demale(Rover)

= min(0.85,O.lO) = 0.10; = max&,JRover), cfemde(Rover)) = max(0.85,O.lO) = 0.85; = 1 - c&,,(Rover) = 1 - 0.85 = 0.15; = 1 - cfeemale(Rover)= 1 - 0.10 = 0.90.

2.4 Is prototype theory compatible with conceptual combirzation? We are ready to demonstrate that prototype theory in conjunction with fuzzy-set theory contradicts strong intuitions we have about cencepts. Three problem areas are considered: conjunctive concepts, logically empty and logically universal sets, and disjunctive concepts.* 2.4.1 Conjunctive concepts Let the domain of discourse be the set, F, of all fruit, and consider the characteristic functions for the (fuzzy) concepts apple, striped, and striped apple. The concept stripe< apple stands in the same relation to striped and apple as does red house to red and house, square field to square and field, and so on. Striped apple thus qualifies as a conjunctive concept. Within fuzzy-set theory, conjunctive concepts are most naturally represented as fuzzy intersections, as in (2.3). At least, they clearly are not fuzzy unions, nor any yet more complex fuzzy Boolean function of their constituents. So, if there is to be any account of the conceptual combination striped apple within fuzzyset theory, it will likely rest upon (2.10). (2.10) (Vx E F) (c,~ -re(x) = mh-&&&x), c,,l,(x)) Equation (2.10) exhibits striped apple as a familiar kind of combination of the concepts striped and apple. Without (2.10) it is doubtful that fuzzy-set theory can secure the compatibility of prototype theory with this elementary case of conceptual combination. *A comment is in order about the kinds of problems that we will not consider. One kind involves counterexamples to fbaxy intersections that can be generated with concepts like connfe$eit, bogus. tmitutfon, etc. For example, a “good” counterfeit dollar, d, might be a “good” counterfeit, but it will be a “bad”’ dollar; hence, the characteristic function for the relevant fizzy intersection between countetj’kit and dollar will mistakenly declared to be a “bad” counterfeit dollar, since the intersection function yields the minimum value of the constituents. We wig pass over this kind of objection, however, in view of our earlieragreementto treat only simpleconcepts, among which counterfeit and the like cannot be found.

D. M. Oshemn and E. E. Smith

44

(b)

(cl

But (2.10) cannot be maintained. Let (a) in Figure 1 be a particular apple in F. There can be no doubt that it is psychologically less prototypical of an apple (whose prototype looks more like (b)) than of an apple-with-stripes; SO, :.2-11)

&biped

appie 00 > Gppk (a).

heqrsa?ity (2.11) asserts, simply, that (a) is a better illustration of the concept strii~ed-apple than it is of the concept apple (just as the reverse is true of (b)). 33ut(23 1) contradicts (2 .lO) since the latter implies

Tk

ht?Wd

a&W? (a) = mh

which in turn implies

h&&ed

(ah %Wle (a))

On the adequacyof prototypetheory

(2.12)

cstrlped apple

45

(a) Q capple (a).

And (2.11) and (2.12) are inconsistent. Given the above, it becomes clear that prototype theory conjoined to fuzzy-set theory will lead to a contradiction whenever an object is more prototypical of a conjunction than of its constituents. When phrased this way, numerous familiar conjunctions seem to provide counterexamples, e.g., a guppie is more prototypical of the conjunctive concept pet fish than it is of either pet or &kg We conclude that prototype theory cum f&y-set theory is not compatible with strong intuitions about conjunctive concepts. 2.4.2 Logically empty and logically ur~ivcrsad ccncep is The concept apple that is not an apple is logically empty since it can apply to nothing.‘* Its characteristic function within fuzzy-set theory should reflect this peculiar property; that is, its function should conform to conditiol? (2.6), which in this case can be written as: (2.13)

(Vx E FSc apple that id not an

apple(X)

= 0.

However, within fuzzy-set theory, the concept apple that is not an apple would be repre:;en:ed 1s the (fuzzy) intersection of apple and nonappZe yielding (by (2.3) and (2.5)): (2.14)

(V XE F’)Capple that is not an apple(X) = mh(Cappk (X), 1 - CappIe(XI)* Consider again apple (a) of Figure 1. Since (a) is a “better” apple than (c), but a “worse” one than (b), we have, by condition (1.2):

(2-l 5) cappie(c) < cappIe(a) < CappIe(b). (2.15) implies (2.16) (in the presence of (2.8). (2.16)

O< cappP,(a)< 1

gIn derivingour counterexample, we chose to work with an unfamiliarconjunction (stripedapple) rathtr than with a familiarone (e.g., per fi&) so as to make it less likely that the readerwould have representedthe conjunction as a single conceptual constituent. The latter could be the case for many familiarconjunctions (Bolinger, 1975; Potter & Faulconer, 1979), and for such cases representationIn terms of intersection is misleading. “Some might object on the grounds that people frequently use locutions of the form “-and not a -” to describec me object. For example, one might describetomatoes by saying: (i) They are both fruit and not fruit. But (i) seems to be a case in which grammaticalstructure is a misleadingguide to conceptual structure; (i) is probably idiomatic for asserting that tomatoes have some properties of fruit but lack others. isa_-and not a ---*’ that we can think of More generally, all locutions df the form “seem to be idiomatic.

46

D, N. OshersonandE. E. Smith

But if the appleness value of (a) is between 0 and 1, then so is one minus that value. Therefore both cagple
thereby contradicting (2.13).11 In the above, the contradiction rested critically on the fuzzy-set theory assu.mption that the value of a conjunctive combination cannot be less than that of rts minimum-valued constituent. We now derive a similar contradiction for logica~ulyuniversal concepts, this time using the fuzzy-set theory assumption that the value of a disjunctive combination cannot exceed that of its maximumualued constituent. The concept fruit that either is 0r is not an apple is logically universal (with respect to the domain F), and therefore its characteristic function should conform to (2.73, which in this case can be written as: (2.17) (VXE F)c fruitthat either is or is not an apple (x) = 1. Within fuzzy-set theory &heconcept fruit that either is or is not an apple is represented as the (fuzzy) union of apple and nonapple, yielding (by (2.4) and (2.5)): (2.18)

(VX E F)c druitthat

either is or if4not an apple (XI = max

(Capple (XI,

1 -

cappie(x)).

Again, consider apple (a). Since its characteristic function value for -apple, an one minus that value, both li,: between 0 and 1 (as shown by (2.1 S)), the maximum of these two values must be less than 1. Hence (2.18) must be less than 1, which contradicts the condition stated in (2.17). Similar counterexamples are easily generated. We conclude that fuzzy-set theory does not render prototype th.:ory compatible with strong intuitions pertaining to logically empty and logically universal concepts. 2.4.3 Disjunctive concepts As an example of a disjunctive concept, consider (financial) wealth. It seems clear that wealth is conceptually connected to liquidity and investment in I1Indeed, since in the case depicted by Figure 1, apple a fails to neither Capple that ia not an apple(a) amr~~ably

to meeting condition (2.13).

extremeof appleness.

exceeds 0. SO Capple ttmt ia not an apple d*s

not come ~10%

On theadequacyof protovpe theory

Table 1.

47

Wealthof threepersons

A. B. C.

Liquidity

Investment

$105,000 $100,000 $5,000

$5,000 $100,000 $105,000

this way: the degree to which one enjoys either or both of the latter determines the degree to which one is wealthy. In particular, it is clear that of the three persons whose assets are given in Table 1, person (A) enjoys the greatest liquidity, person (C) the greatest investment, and person (B) the greatest wealth. Given the appropriate domain, D, and subsets for liquidztv, investment, and wealth, we thus have: (2.19)

cliquidity

(2.20) (2.2

1)

(A)

>

Cliwaitity

Gweatment(C)

>

Gnve*tment(~)

(i) ~wealth(B)

>

Cwealth (A)

(6)

>

Cwealth (C?.

Cwea~th (B)

(B)

Now if fuzzy-set theory is to represent the conceptual connections among liquidity, investment, and wl,alth, it would seem that its only option is to employ fuzzy union, as in (2.22). (2*22)

(Vx E D)cW~alth(X) = Cliquidity

U hvestm-nt

(Xl-

In light of (2.4), defining union, (2.23)

(V x E D)c,,titb 00 = max(CUquidty(xl,

Ci*ve&nent(X))-

But (2.19~(2.23) are inconsistent. To see this, focus on Cam*. According to (2.23), cwedti (B) is the larger of cliqUiddt,,(B) and chvestment(B). Suppose that cUqUidity (B) is the larger, then: (2.24) Now

than

cliquiaity

(W

= cwealth

(B).

focus on cwsolth(A). Since, by (2.23) again, cwealth(A) cannot be smaller cuquimy

(2.25)

(A),

c,mm(A)

2

Cu,uiaity(A)-

Combining this information with (2.19) yields: (2.26) And

cwarlth (2.26)

(A)

implies

> cUq&ity

(A)

> ctiquiciity (B)

= cweelth

(B).

48

D. N. Oshenonand E. E. Smith

Cweatth

(A) >

Cwealth

@It

co;,tradicting (2.2 1) (i). The only alternative is that Cbvesment(B)is equal to or larger than Cgguid&y (B), making the former the value of C,ede(B). But then, by b parallel argument to that just given, we have: which contradicts (2.21) (ii). Hence fuzzy-set theory does not properly represent the relation betv een wealth on the one hand, and liquidity and inves#me~t orI the other. We conclude that fuzzy-set theory renders prototype theory in::ompatible with strong intuitions about disjunctive conceptual combination. Section 3. Prototype theory and the truth conditions of thoughts

3. I The issue We take concepts to be the immediate constituents of thoughts. Given

this, it seems reasonable to ask a prototype theory of concepts for principles that informatively characterize the circumstances under which thoughts aE true, (when the thoughts correspond to declardtive propositions). In particular we are concerned wrth whether prototype theory, coupled with fuzzyset theory, can offer a cxnxt accourlt of simple quantificational thoughts like All A ‘s are B’s, Somt A ‘s are B’s, No A ‘s are B’s, and so forth. We can restrict attention to All A ‘s are W’s since our remarks will apply mutatis mutandz3 to other simple qsantificational thoughts. 3.2 Criteria of adequacy regarding t&h conditions

Let A and B be concepts compatible with the claims of prototype theory, and let T be the thought expressed by (3.1). (3.1)

All A’s are B’s_

On prototype theory, T consists, in part, of (mental) representations of the quadruples for A and B. A fully adequate version of the theory would supply principles that specify the truth conditions of T on the basis of these latter representations. Since so little is known about (such mental) representations, we shall require only that prototype theory specify the truth conditions of T on the basis of the quadruples themselves, i.e., on the basis of A and B’s domains, distance metrics, prototypes, and characteristic functions. And for reasons similar to those advanced in Section 2,2, we shall focus on the

On the adequacy of prototype theory

49

availability of principles that relate A and B’s characteristic functions to T’s truth conditions. In summary, then, we require of prototype theory a specification of what makes (3.1) true on the basis of the characteristic functions for A and B. Once again it is fuzzy-set theory that saves prototype theory from inexplicitness. Zadeh (1965) offers an explicit principle relating inclusions like (3.1) to the characteristic functions for A and B. Since Zadeh’s principle naturally extends the principles reviewed earlier, since his proposal has interest in its o In right, and since no alternative principle has been advanced to fill the prest& theoretical need (so far as we know), we feel justified in pinning the 1 lpes o? prototype theory to the fuzzy inclusion principle. If this ensemble is at variance with strong intuitions pertaining to the truth conditions of thoughts, there is reason to doubt the compatibility of prototype theory with those intuitions. 3.3 Fuzzy-set theory again Fuzzy inclusion is a straightforward generalization of standard inclusion. Standardly, the inclusion (3.1) is assigned the truth condition (3.2). (3.2)

(vx E D) (c*(x) Q cg(x))

To illustrate in the domain of animals, (3.3)

All females are dogs

is true just in case (3.4) is true. (3.4)

(Vx E animals) (cism&x) Q ci&)).

\~~‘,co;~T

,

)ou\false in light of a female nondog, a, for whom cfemale(a) =

The fuzz”v=mclusion principle results as before from generalizing the notion of a’characteristic function from’(2.1) and (2.2) to (2.8) and (2.,9) The fuzzy truth condition on (3.1) may still be stated as (3.2). Th,.:n, (3.3) remains false in fuzzy-set theory, in the light of a female rabbit, r, for whom chrnd,,(r) = 0.80 and cd&) = 0.20, contradicting (3.4). .3.4 Is prototype theory compatible with the truth conditions 2finclusion ? We now present a counterintuitive result to which prototype theory, m the context of the fuzzy inclusion principles, is committed.12 In this example, we use the domain, A, of animals, and the subsets associated with grizzly bear and inhabitant of North America. Consider the following inclusion. l2For footnote please see overleaf.

50

D. ik! Oshersonand E. 6 Smith

(3.5)

All gti&

!..JZJBan inhabitants of North America.

Within fuzzy-set theory, the truth condition of (3.5) is

(3.6)

(V XE A) (cglri~zly bear (x) d &habitant of North Amer~a 00). One thing we want from (3.6) is that it capture our iM.d:ions about what conditions insure 1:hefalsehood of the inclusion (3.5). P rt (3.6) fails to do this. For: according to (3.6), the existence of a squ’;rel on Mars is sufficient to falsify the proposition that all grizzly bears live in North America. The argument unfolds as follows. The characteristic function for grizzly bear is supposeo to represent the degree to which animals count as grizzly bears. Thus, polar bears will be assigned a numbqer close to 1, frogs ‘1number closer to 0, and earthworms a number yet closer to 0. Let Sam be a squirrel, and suppose that (3.7)

c wziY b&Sam) = p (where p > 0).

The characteristic function for inhubitant ofNorth America, if it is to accord with our intuition. must assign prcgressivc-ly smaller numbers to locations that are progressively further from the North American continent. Thus, it assigns Cuba a higher number than Scotland, which in turn is assigned a higher number than is Egypt. Now, is there a place that Sam could live so that cinhabitant would assign Sam a number smaller than p? If of North America so, we will contradict (3.6) and hence falsify (3.5). Surely, if Sam lived far enough from North America, the value of cinhabitant of North America (Sam) would be forced below p. l3 Perhaps a home in Egypt would suffice; if not, we can imagine he lives in Pakistan or Indonesia, or, if necessary, on Mars or “Another comment about kinds of problemsnot to be considered: It is possible to produce cmntelc examples to fuzzy inclusions by suitable choice of the domain of discourse. In a domain including both galaxiesand radishes,for example, the false inclusion All smallgalaxiesare small things

.

would be declaredtrue by fuzzy set theory because it is a theorem of the theory that for all subsets ,7B B,ofD,andallxED, We choose r,ot to rely on th& class of counterexample since the problem of relative ad(jectlveslike s??uJZZ is too severe a burden to impose on a theory with so few competitors. Consequently, in the examples to follow, we employ natural(in particular,intuitively homogeneous) domains. ‘junless cinaobitant of Noxth Amer k a asymptotes above p. But this is an arbitraryand counterintuitive restriction on the behavior of ciuhpbit& of North Am&.a; it would moreover, complicate the repzsentation of subjects* relative similarity judgments, since the psychological distance between objects far away from North America does not seem suMciently foreshortened compared to those nearby for such an elevated asymptote.

On the adequacy ofpmtotype theory

51

Pluto, in the vicinity of a nearby star, or within some distant galaxy. Let’s suppose that Mars is far enough. Then we have: (3.8)

%habiwt

Of

North America(Sam)

< Pv

Putting (3.7) and (3.8) together, we have: (3mg) c&zzly bear (Sam) > Cfnhabitant of North Amerlea(Sam)9 which contradicts (3.6). Since it is easy to generate an indefinite number of examples like this, it seems that fuzzy-set theory does not render prototype theory compatible with the truth conditions of inclusions.14 3.5 Partial falsification to the rescue ? One response to the difficulty just exhibited for fuzzy inclusion is to embrace fuzzy truth, that is, an (uncountable) infinity of truth-values between 0 and 1. On this view, the inclusion (3.5) is not outrightly falsified by Martian squirrels but only slightly weakened, some scheme being envisioned for computing the falsifying effects of counterexamples to truth conditions of form (3.2). To support this proposal it is pointed out that the discovery of Martian squirrels would indeed lower our confidence that all grizzly bears inhabit North America (if Martian squirrels, why not other Martian mammals?). So, psychologically, at least, (3.5) is partly falsified by (3.9), as the new approach predicts. Nene of this is well taken. We give thEe reasons for our distrust of a fuzzy-truth remedy to the problems associated with fuzzy inclusion. Note first that the new pr*-rposalis no mere adjustment of truth condition (3.2), found inadequate i,! the last subsection. To replace (3.2) with a “partial falsification” scheme so as to allow for truth values between truth and fatsity requires nontrivial trleory construction of a kind apparently not yet undertaken. Such a theory will need to include principles governing the degrees of falsification associated with (partial) counterexamples to inclusions. Reverting to our example, the theory must decide (a) how much addi-

t4An inleresting class of examples results from considering inclusions, A 5 B, such that A is a very atypical kind of B, e.g., All penguins are birds. All tomatoes are fruit. All bats are mammals etc. For, there may well be some quintessential penguin that is a better examplarof penguin than of bird, thereby falsifying the generalisation that all penguins are bir 3 (and similarly for the other castes).

52

D, IV. Oshersonand E. E. Smifh

tional falsification is engendered by the existence of n + 1 Martian squirrels instead of n (for all n), (b) whether some number of Martian squirrels falsify (3.5) as much as one European grizzly bear, (c) whether, if giraffes count as grizzly bears no more nor less than do squirrels, then a Martian squirrel and a Martian giraffe falsify (3.5) to the same extent as two Martian squirrels or two Martian giraffes, and so forth for many obvious questions. Until such a calculus cif partial falsification is offered that provides principled answers to questions like these, no explicit alternative to truth condition (3.2) is available to colrjoin to prototype theory. Qcond, and in the same vein, allowing propositions to admit of an inftity of possible truth values raises problems for the interpretation of sentential connectives and quantifiers. This issue is analogous to that for conceptual combinatiion, only now it is the combination of simple propositions into complex propositions that concerns us, instead of the combination of simple concepts into complex concepts. In particular, the familiar semantical apparatus of classical logic needs to be replaced since standard semantics is appropriate for interpreting its connectives and quantifiers only in the context of binarily valued propositions. To interpret, e.g., the conjunction of infmitarily-valued propositions, an appropriate infinite-valued Logic needs to be invoked. Many such logics have been devised; see Rescher (1969, pp. 3645) for a survey, as well as Zadeh’s (1975) own preliminary paper. The problem is that infinite valued logics generally violate strong intuitions about truth, validity, and consistency.15 At the least, their psychological suitability is not obvious. The advocate of fuzzy-truth, then, incurs the responsibility of selecting and defending an appropriate infinite-valued logic, and this promises to be a nontrivial undertaking. But until such a logic is presented, a serious alternative to fuzzy-set theory is not available to conjoin to prototype theory. Finally, we suspect that the partial falsification proposal results from mistaking degrees of belief for degrees of truth. We share the intuition that “To illustrate with an influential system, consider Eukasiewicz’sinfinite valued logic C-aleph. The intuitively valid sentence If John is happy, and if John is happy only if business is good, then business is good. if NM nontautologou s in .JUeph. (A tautology within R-aleph is any formula that assumesmwimal truth value under all truth assignmeutsto atomic subformulas.)Also, explicit contradictions (of the form (p and -p)) may differ in truth-value.Many other counterintuitlve results issue fromX-aleph The rivalinfinite valued lo&s in the literatureappearto yield aimibuparadoxes. None of these paradoxical results are features of classical (two valued) logic, although it is well known that classicallogic yields some counterintuitive results of its own (for discussion, see Kreuger and Osherson,forthcoming).

On the adequacy of prototype theory

53

the discovery of Martian squirrels should lower confidence in the truth of (3.5), but it is the confidence that is partially weakened, not the truth. Analogously, evidence migh? be adduced that alters our confidence in some as yet undecided mathem-+: ,-_,cal assertion; but the assertion, irrespective of our doubts and conjectunus, is either flatly true or flatly false. In other ‘words, the effect on our beliefs of the discovery of Martian squirrels is a problem relevant to inductive logic, not to the analysis of the truth conditions of inclusions. The “supporting intuition” of the fuzzy-truth advocate thus explained away, it is helpful to appeal to another intuition, namely, that it is logically possible both for all grizzly bdars, without exception, to dwell in North America ant’ for Mars to be teeming with squirrels; the irrelevance of Martian squirrels to the inclusion (3.5) - pace fuzzy-set theory and partial falsification schemes - is thereby revealed. We conclude that programmatic suggestions about partial falsification do not rescue prototype theory from its dependence on fuzzy-set theory; in particular, truth condition (3.2) is still the most natural and explicit interpretation of the inclusion relation within the prototype theory tradition; and (3.2), we saw, has unwelcome consequences. Section 4. Implication!f In this final section we consider three issues: (1) the “immunity” of standard set theory from the contradictions we derived, (2) the susceptibility of all versions of prototype theory to the derived contradictions, and (3) what can be salvaged from prototype theory in the light of these difficulties. 4. I Standard set theory and traditional theories of concep ts Standard set theory, with its binarily valued characteristic functions, is the natural complement of traditional theories of concepts -- at least insofar as those theories enforce a sharp boundary between membership and nonmembership in a concept’s extension. We now wish to point out that this amalgamation does not lead to the contradictions we derived from the amalgamation of fuzzy+et theory and our version of prototype theory. Hence, the former amalgamation is superior to the latter when it comes to accounting for conce#ual combination and the truth conditions of thoughts. The point is trivial to demonstrate. For, every contradiction we derived rests partly on the assumption that constituents of complex concepts can be assigned characteristic function values that fall between 0 and 1, i.e., this assumption is necessary (though not sufficient) for all our derivations. TO

54

D. N. OshersonandE. E. Smith

take the most obvious example, consider the concept we used in discussing logically empty sets, namely, apple that is not un apple. Recall that in fuzzyset theory the characteristic function of this concept meets (2.14): (2.14)

(Vx E F)

Capple that, in not an apple (XI

= mh

@apple (XL

1 -

cappie(x))

Recall further that the contradiction arose because there exists an apple whose appleness value is between 0 and 1, thereby insuring that one minus this value is also between 0 and 1: which in turn insures that the minimum is greater than 0. In standard set theory the characteristic function of apple that is not an apple must also meet (2.14), but since the appleness value of any apple must be either 8 or 1, for any x, one of CappIe( 1 - CappIe equaIs 0, thereby insuring that the minimum will be 0. A similar analysis in terms of standard set theory will remove the contradiction in the other cases we considered. 4.2 Exrension of the present results to other versions of prototype theory It was noted at the outset that there are versions of prototype theory alternative to the one we presented; indeed, some of these alternative versions enjcy greater empirical confirmatio:. than that we presented. There is, however, good reason to believe that the contradictions we derived apply to other versions of prototype theory as well. In the version we chose, concepts were identified with mental representations of quadruples like (*), (*) (A, Id,P, c), where A is a conceptual domain of envisionable objects, d is a metric function, p is a particular object in A, and c is a characteristic function. Most other versions of prototype theory differ from the above in their choices for the second and third coordinates of (*). Instead of a function that measures the psychological similarity between objects in terms of distance, many prototype theorists favor a function that measures this similarity in terms of number of common features (e.g., Rosch & Mervis, 1975; Smith, Shoben, and Rips, 1974), or in terms of a weighted contrast between common and distinctive features (Tversky, 1977). (The latter function has the virtue of not requiring that the similarity btitween two objects be symmetric.) And instead of the prototype being a particular envisionable object in the domain, some have assumed it is an abstraction from objects in the domain, e.g., a collection of features that occur most frequently in the domain (Smith et al., 1974). But such variations would not influence the contradict;?ps we derived. For as long as a version of (*) postulates that (a) there is some proximity

On the adequacy of pm tovpe theory

55

function that computes similarity between an object and a protot;:pical description, and (b) this computation results in graded similarity values that are mapped into a characteristic function such that high similarity goes with high characteristic values, then (c) the characteristic functions associated with the constituents of complex concepts can assume values between 0 and 1. And point (c), in conjunction with the formalism of fuzzy-set theory, is what produced all our contradictions. This can again be seen most easily by considenlag the logically empty concept: apple that is not an apple. Even if the prototypical apple is an abstraction, and even if similarity to a particular apple is computed by summing common features, it will still be the case that a particular apple can have an intermediate similarity value that results in a characteristic function value betwen 0 and 1. And this is all it takes to insure that the mirimum constituent of apple that is not an apple is greater than 0, which in turn insures that this logically empty concept is not associated with a chancteristic function that is identically zero. Similar remarks apply to other cases of conceptual combination we considerea, and to inclusion. As best we can tell, then, no current version of prototype theory is immune from the problems we have raised in this paper. 4.3 General implications.for pro totype theory One thing is clear. Amalgamation of any of a number of current versions of

prototype theory with Zadeh’s (1965) rendition of fuzzy-set theory will not handle strong intuitions about the way concepts combine to form complex concepts and propositions. This is an important failing because the ability to construct thoughts and complex concepts out of some basic stock of concepts seems to lie near the heart of human mentation. Where does this leave prototype theory? Three answers are worth considering. The first is that the theory is a sound rendition of concepts, but that it should beware of associating with Zadeh’s (1965) specific vers,ion of fuzzy-set theory since the fatter brings with it the kinds of problems we have discussed. This possibility receives some support iq Oden’s (1977) work. He noted that within the general class of fuzzyset tr:?ories (i.e., a set theory that conforms to (2.8) and (2.9)) there are alternatives to Zadeh’s characterization of fuzzy intersection and union. Thus, instead of the minim,Jm rule for fuzzy intersection, (2.3), we can use a multiplicative expression (due to Gougen, P969): ( v x E D) (CAnB (x) = CA (Xh (X)h And applying DeMorgan’s law to the above yields an alternative to the minimum rule for union, (2.4), namely,

D. N. Usherson and E. E. Smith

56

D) (cAu B(x) = CA(x) + cg (x) - cA (x)‘ca (x)). Oden (1977) showed that these alternative formulations provide a better account of subjects judgments of truthfulness of complex statements than do Zadeh Tformulations. But the above alternatives still yield almost all the contradictions we derived; e.g., as long as the two constituents of a logically empty set have characteristic function values between 0 and 1, their product must be greater than 0. Similar remarks apply to all the contradictions we derived, with the sole exception of that concerning the disjunctive concept of wealth (Section 2.4.3). (Referring back tc Table 1, given a reasonable choice of characteristic function values corresponc!ing to the various liquidity and investment amounts, the above expression for union correctly selects B as the wealthiest individual.) In short, we know of no alternative set theory that can be joined with prototype theory to account for all the evidence about conceptual combinations and truth conditions. Nor does it seem promising to fuzz the notion of truth to allow for partial falsification of thoughts, as was seen in Section 3.5. Perhaps a novel set theory and logic can be developed whose association with prototype theory will be free of the difficulties raised in this paper; such a development would be of considerable theoretical interest, in our opinion. Until a viable alternative to fuzzy-set theory materializes, however, prototype theory cariI;ot be suitably extended; and the possibility that no such alternative can be developed ought not to be minimized. A second possibility is to forget about prototype theory entirely. We count this as too extreme. For one thing, the empirical research that has resulted from prototype theorizing points to the use of similarity in establishing membership for many simple concepts. We mentioned some of the relevant findings in Section 1.l, and there are many other interlocking results of this kind: (see, e.g., Rosch, 1978). So prototype theory seems to capture something about the natural use of simple concepts. Furthermore, the leading intuition behind prototype theory - that concepts are often vague and apply to different objects to different extents - is quite cornpelling, and central to theories of reasoning. It is well known, for example, that the principle of mathematical induction cannot be reliably applied to vague predicates. To see this, define F to be the numerical predicate ( ‘tr

X E

grains of sand brought together do not constitute a heap. Note that F(O), and if F(k) then F(k + 1) (0 grains of sand brought together do not constitute a heap, and if k grains won’t do the job neither will k + 1). So mathematical induction leads to the false conclusion that no matter how large k gets, k grains of sand brought together do not constitute a h2ap. An

On the

adequacyof prototypetheor)-

57

adequate theory of concepts should provide an illuminating characterization of the class of vague predicates, and this is the goal of prototype theory. The third possibility is thrt prototype theory is by its nature incomplete because it is about only a lin.,;ed aspect of concepts. To make this clearer, we can distinguish between a concept’s core and its identffication prmedure; the core is concerned with those aspects of a concept that explicate its relation to other concepts, dnd to thoughts, while the identification procedunz specifies the kind of inform&ion used to make rapid decisions about membership. (This distinction is similar to one proposed by Milrer and Johnson-Laird, 1976).We can illustrate with the concept woman. Its core mi&t contain information about thz presence of a reproductive s/stem, while its identification procedures might contain information about body shape, hair length, and voice pitch. Given this distinction it is possible that some traditional theory of concepts correctly characterizes the core, whereas prototype theory characterizes an important identification procedure. This would explain why prototype theory does well in explicating the real-time process of determining category membership (a job for identifjcation procedures), but fares badly in explicating conceptual combination and the truth conditions of thoughts (a job for concept cores). This seems to us to offer the most satisfactory reconciliation of the present failures of prototype theory with its previous successes. References Bolinger, D. L. (1975) Agwcfs of Lmguuge, (2nd Edttion). New York, Harcourt Brace Jovanovieh. Cougen, J. A. l1%9) The logic of inexact concepts, Synthese, 19, 325-373. Hersch, H. M. and Caramazza,A., (1976) A !kzy set approach to modifiers ard vaguenessin natural language. J. #per. Rvchd.: Gen., IOS, 254-276. MUler. C. A. and Johnson-l&d, P. N. (1976) knguage and Perception. Cambridge,Hward Univcrsity Prew. Qdeu, G. C. (1977) Integratiai of firzzy logical information. J. exptx Pgvchol.: Hum. percep. pefwm,3, 565-575. PsBner,M. 1. and Keele, S. W. (1968) Qn the genesis of abstractideas. J. exper. Psych& 77,353-363. Potter, M. and Paulconer, B. (1979) Understandingnoun phrasca,J. verb. I&e@%vub. Behav., I& (5). m9=522, Reed, S. K. (1972) Pattern recognltlon and categorization, Cw P@‘chsl., .%382407. Rmb, N. (1%9)Many=wiuedlo~cs, McGrawHiU. Rips, L, J., Shoben, E. ,I. and Smith, Z;.E. (1973) Semantic distance and the verification of semantic relations. J. verb Lcucm.verb. khav,, lc?, l-20. Rosch, E. (1973) On the inter&4 structure of perceptual andeemanticcategories. In T. E. Moore (Ed.) Co@Mve development and the acquisitim of lhguag~. New York, Academic Press Rosch, E. (1978) Principles of categorization. In E. Roscl: and B. B. Lloyd (Eds.) Cognition und cutegotiutlon. Potomac, MD, Erlbaum.

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Rosch, E. and Mervis,C. B. (1975) Family resemblance: Studies in the internal stntctum of categories. Cog. Pwchol., 7,573-605. Rosch, E. H., Simon, C. and Miller, R. S. (1976) Structural bases of ty&ality effects. J. expefi Pqy= chd.: Hum. Percep. Perform., 2,491-502. Smith, E. E., Shohen, E. J. and Rips, L. J. (2974) Structure and process in semantic memory: pi featural I,todel for semantic decisions. Psychd. Rev., 81,214-241. Tvenky, A., (1977) Features of similarity, Psychol. Rev., 84,327-352. Zadeh, L. (1965) Fuzzy sets.hform. confrcd, 8, 33S-353. Zadeh, L. A. (1975) Calculus of fuzzy restrictions, in L. A. Zadeh, K. Fu, JC.Tanada and M. Shimura (Eds.), I;trzzy se& and their applications to cognitive and decision processes, Academic Press.

La thCorie du prototype considbre qu’il existe des de& d’appattenance i l’extension d’un concept dktermink par ?a similitude avec le “meilleur” exemplaire de ce concept IOUpar quelqu’autre mesure de tendance centrale). Cet article envisage la compatibilitd de cette proposition avec deux critkes d’adiquation concernant * s thiories des concepts. Le premier critke conceme la relation entm les concepts complexes et leers contribuants conceptuels. La seconde a trait aux conditions devirit6 pour les propositions portant sur les inclusions simples.