Introduction / p. 1
Preliminaries / p. 2
Osherson and Smith – Overview / p. 6
The Conceptual Combination Objection / p. 8
The Truth Conditions Objection / p. 11
Can the Classical Theory account for Prototype Effects? / p. 15

Introduction
In what follows I will mostly discuss the Osherson & Smith paper (henceforth O&S) which is supposed to be an attack on the Prototype Theory of concepts (henceforth PT) on behalf of the Classical Theory of concepts (henceforth CT). O&S argue that the PT fails to account for important phenomena surrounding our conceptual mechanisms. They concentrate on two such phenomena: conceptual combination and truth conditions. An adequate theory of concepts, say O&S, should provide an explanation of the relation between complex concepts and their constituents and of the ways in which we confirm or falsify the truth of our thoughts; the PT is inadequate in both of these accounts and should therefore be rejected, as a theory of concepts, in favor of the CT. In what follows I will argue all of the following (not in this order):

    1. The objection to the PT regarding truth conditions fails.
    2. The objection regarding conceptual combination is successful, but this shortcoming of the PT is not an indication that the CT is preferable, since:
    3. The CT also fails to account for conceptual combination, and this failure is in addition to two other important shortcomings of the CT that, unlike  are not shortcomings of the PT. These two additional problems that the CT faces are:
    4. The objections raised against the CT by Wittgenstein.
    5. Attempts to account for prototype effects within the CT framework fail.

As may already be clear, the following is not so much a defense of the PT as it is a rejection of the implicit assumption made by O&S according to which failures of the PT almost automatically indicate that the CT is preferable. Before addressing the details of the paper, however, I will discuss two preliminary issues: the Wittgensteinian objection to the CT of concepts, and the distinction between Prototype Theory and Prototype Effects.

Preliminaries

Consider … the proceedings that we call “games”. … What is common to them all? – Don’t say: “There must be something common, or they would not be called ‘games’” – but look and see whether there is anything common to all. – For if you look at them you will not see something that is common to all, but similarities, relationships, and a whole series of them at that. To repeat: don’t think, but look!  [Wittgenstein 1953]

    · Reaction time: subjects identify prototypical members of a category as members of the category more quickly than they classify less representative objects.
    · Production of examples: when asked to list or draw examples of category members, subjects are more likely to list or draw the more representative examples.
    · Asymmetry is similarity ratings: Less representative examples are often considered to be more similar to the more representative examples than the converse. Within the category of country, for example, Americans (not very surprisingly) judged Mexico to be more similar to the US than US to Mexico.
    · Asymmetry in generalization: New information about a representative category member is more likely to be generalized to nonrepresentative members than the reverse. For example, it was shown that subjects believed that a disease was more likely to spread from Robins to ducks on an island than from ducks to Robins.

All these, as I said, in themselves, do not attempt to provide any specific alternative theory of concepts. As Rosch put it, PE underdetermine mental representation (and she is very clear on this matter on page 200). PE, however, constrain the possibilities for what concepts might be. They are scientific data, obtained in well replicated experiments, that should be accounted for by our theory of concepts.
    It may now be clear what was the initial intuition regarding the inadequacy of the CT as a theory of concepts when evidence regarding PE started coming in. It seems that for a theory of concepts to adequately account for PE it must posit some additional internal structure of some sort as part of our conceptual structure. It is this structure that, according to critics of the CT, is missing in the CT which makes it an inadequate theory of concepts. PE at least seem to be inconsistent with the CT since if an object falls under a category iff it has some necessary and sufficient conditions than it seems to follow that membership in a category is a binary thing, and it is not clear why there should be PE. (This is of course a very confused way of presenting the objection to the CT based on the PT, but it will do for now, and I also think it’s a fair representation of the kind of reasoning that was applied initially).
    One last thing before we get to O&S: a very general and brief characterization of what are concepts according to the PT, and why the PT seems better suited to account for PE. The following is what I call the minimal PT, and I will fill in the details later on: the PT holds that the concept GAME, for example, is characterized by some set of properties. According to the PT, however, no particular property or subset of properties from this set needs to be necessary or sufficient; rather, an object will be classified as a game if it satisfies a sufficient number of properties from the list (“sufficient” here is not necessarily some particular minimum number of properties; rather, the object must “score” enough “points” as a game in order to be classified as one, and different properties might have different point-value.<3>  So “sufficient” here means sufficient properties for the object to gain the required number of “points” or “weight”). Since different games might qualify as games (accumulate the required number of points) in virtue of satisfying different combinations of properties, there need not be (although there could be) properties that are necessary or sufficient. Furthermore, the sufficient “weight” need not be (although it could be) very precise; thus, the PT allows membership in a group (or classification under a concept) to be graded, or a matter of degree. But as will be clear later, there’s much more to the gradedness issue than meets the eye.

Osherson and Smith – Overview and Preliminaries
O&S method of arguing against the PT is as follows: any adequate theory of concepts must be at least compatible with important phenomena surrounding our conceptual mechanisms. O&S concentrate on two such phenomena: conceptual combination and truth conditions. An adequate theory of concepts should provide an explanation of the relation between complex concepts and their constituents and of the ways in which we confirm or falsify the truth of our thought. The PT is inadequate in both of these accounts and should therefore be rejected, as a theory of concepts, in favor of the CT. O&S recognize that PE are a constraint on any theory of concepts and attempt to reconcile them with the CT by distinguishing between a concept’s core and its identification procedure.
    O&S’s argument can be divided into three parts, and so will be the discussion that follows, and my responses:

    1. The PT fails to account for conceptual combination. Response: the PT is guilty as charged, but so is the CT.

    2. The PT fails to account for truth-conditions. Response: not true. O&S’s result is a consequence of applying a model that is inconsistent with the PT.

    3. The CT can account for PE by distinguishing between a concept’s core and it’s identification procedures. Response: not true. First, Wittgenstein has argued persuasively that many concepts do not have cores; second, PE cannot be classified as merely a matter of rapid identification procedures.

    Before discussing the first objection (conceptual combination) another preliminary issue: standard and fuzzy set theories. O&S seem to think it’s useful to present their objections by modeling the relevant theories of concepts on set theories (I myself think it’s a very unfortunate idea as a general method in philosophy, and in this context as well, for reasons that will become clear, but anyway…). Here are the relevant essentials of Standard Set Theory (henceforth SST) and Fuzzy Set Theory (henceforth FZT<4> ): According to SST, membership in a set is strictly a binary matter. For every object x and every set A, either x e A or x e/ A is true ["e" here is supposed to be epsilon, "e/" negated epsilon]; there are no borderline cases. To facilitate discussion of the standard set-theoretical operations of intersection, union, and complement, it is helpful to define a function, F, that assigns a value, C, to an object according to its status as a member (or not a member) of a given set. Membership value of 1 is assigned to an object x relative to a set A iff x e A, and membership value 0 is assigned to an object relative to a set A iff x e/ A.
    Notation: C(x)A=1 when x e A; C(x)A=0 when x e/ A.
    Now membership in complex sets can be intuitively characterized and formally defined using SST operations of intersection and union as follows:
    Intersection: for any two sets A and B, an object x will be a member of the set AÇB ["Ç" is here used for intersection] iff x is a member of A and x is a member of B. Using the binary function F we can formalize it as follows: "VxC_{(x) AÇB}=min (C_(x)A, C_(x)B). Thus, if x is not a member of either A or B, the value of C_(x)AÇB will be 0 -- that is, x is not a member of the intersection set AÇB. And this is the desired outcome.
    Union: for any two sets A and B, an object x will be a member of the set AÈB ["È" is here used for union] iff x is a member of A or x is a member of B. Formally: "VxC_{(x) AÈB}=max (C_(x)A, C_(x)B).<5>
    FZT differs from SST in that the values that C_A(x) can take are not the binary 0/1 only, but all the real numbers between 0 and 1, inclusive; and the more CA(x) is closer to 1, the more x belongs to A.<6>

The Conceptual Combination Objection
Now to the objections. First, the PT, modeled on FZT, fails to account for some important features of conceptual combination. Consider the complex category STRIPED APPLE and a normal looking apple with stripes on it. This apple will be a very representative member of the category STRIPED APPLE, and its membership value in this category, C, will be high. The very same object, however, will score poorly as a member of the category APPLE (since apples are rarely striped) and it will also score poorly as a member of the category STRIPED (since striped objects are rarely apples or apple-shaped). But now here is the problem: according to the operation of intersection defined in FZT, there is no way for an object that is a “bad” member of a set A and a “bad” member of a set B to become, all of a sudden, a very good member of the set AÇB (remember that membership in intersection of sets is defined as "VxC_(x)AÇB=min (C_(x)A, C_(x)B), and the minimum of two low values is a low value). Thus, the PT makes an incorrect prediction: it predicts that an excellent example of a striped apple will have a low value in that category.<7>
    O&S charge, then, that the PT cannot handle at least some cases of conceptual combination, and they’re right. There is a major problem for the PT to account for conceptual combination since in many cases objects that are good examples of some complex category are bad examples  of the categories that the complex category is composed of. It is obvious that in many case of complex concepts our theory of concepts must allow at least gestalt effects and contextual information, since both seems to play major role in every-day semantics. There is no clear way in which these components can be accommodated within PT. However, O&S also imply that the CT is better off in that regard, and it is this implicit assumption that I consider to be most questionable.
    Notice that this objection implicitly assumes that most complex concepts are like STRIPED APPLE, RED HOUSE, and SQUARE FIELD, in that they are conjunctive concepts (a complex concept GF is conjunctive iff an object is GF iff the object is G and the object is F), best represented as intersections in the set-theoretical meaning of the term.<8>  It is very easy to show, however, that this view is highly optimistic. For look what happens to the CT when non-conjunctive concepts are considered, of which there are very many.<9>  The intersection model doesn't work well for CT either for concepts such as LUNCH BOX (which doesn’t include the set of objects that are both lunches and boxes), DESK CHAIR, KITCHEN CABINET, BUS STATION, COFFEE TABLE, LOGIC BOOK, PHILOSOPHY PROFESSOR, ELECTRICAL ENGINEER, FAKE FUR etc. What these examples have in common is that they are all  non-conjunctive, and therefore “non-intersective”, in the sense of not being accounted for by the intersection model of conceptual combination. There are many different types of non-conjunctive concept.  One type is generated by nullifying modifiers such as "counterfeit," "fake," et al.  Another type involves so-called "attributive adjectives" such as "small."  A small G is not something that is small and is a G, but rather something that is a G and is small relative to G’s. – hence SMALL GALAXY, SMALL ELEPHANT etc. A third type involves functional modification:  A lunch box is a box used for [carrying] lunch, not something that is both a lunch and a box; and there are many more types here. Notice also that just as conjunctive concepts such as RED SQUARE can be systematically multiplied (BLUE SQUARE, RED DESK, and so on), so can non-conjunctive concepts such as LOGIC BOOK (BIOLOGY BOOK, MATH BOOK, LOGIC CLASS, and so on). The CT, modeled on SST, offers no adequate account of all these many different types of non-conjunctive complex concepts, and just as O&S comment on FZT, there is no reason to think that they could be ultimately accounted for by traditional set theory, or any simple extension of it that will remain psychologically plausible.
    The upshot is that while O&S are right in pointing out that for conjunctive concepts the CT does better than the PT, we are still very far from concluding that the CT is an adequate theory of concepts as far as conceptual combination is considered. First, there could be types of complex concepts for which the PT may offer a better account. But second, and more importantly, once we see how many conceptual combinations are non-conjunctive, the CT’s victory over the PT is small to the point of insignificance.

The Truth Conditions Objection
I will now turn to discuss the grizzly bears objection, otherwise knows as the objection regarding truth conditions. O&S argue that the PT, modeled on FZT, runs into trouble also in accounting for the truth conditions of statements of the form All A’s are B’s, Some A’s are B’s and so forth. Here’s why. Using SST, the truth conditions of the statement All the members of set A are members of set B can be formally written as the following inclusion: "Vx(C_(x)A < C_(x)B). This will ensure that every member of set A (that is objects with C_(x)A=1) will also be a member of B (that is its C_(x)B will also be 1 since its  C_(x)A  must be smaller or equal to C_(x)B). All this is straightforwardly generalizable to FZT, the only difference again being the broadening of the possible values of C. But consider now the following inclusion: All grizzly bears are inhabitants of North America. It’s truth condition is supposedly captured by: "Vx(C_{(x)GRIZZLY BEARS} < C_{(x)LIVES IN NORTH AMERICA}). But when we try to use the PT, this formulation fails to account for our intuitions about what conditions ensure the falsehood of the inclusion. Consider a squirrel found on Mars, and let accept, for the sake of the argument, that a squirrel is a better member of the set of grizzly bears than living on Mars is a member of the set of objects that exist in North America. From that it now follows that we have found an object for which "Vx(C_{(x)GRIZZLY BEARS} > C_{(x)LIVES IN NORTH AMERICA}). This should serve, therefore, as disconfirmation of the claim that all grizzly bears are inhabitants of North America. But this is absurd – the existence of squirrels on Mars should have nothing to do with the truth or falsity of that statement.
    The immediate intuitive response from prototype theorists will of course be that squirrels are not grizzly bears to any extent at all, so the existence of squirrels on Mars indeed has nothing to do with the truth of the statement regarding the natural habitat of grizzly bears. So what went wrong with O&S’s analysis? It is certainly time to note that throughout their discussion O&S commit the PT to two assumptions with regard to graded membership in a category, both of which are logically independent from the PT, and the second assumption is also utterly implausible.<10>  The first assumption is that according to the PT membership in a category is graded. The second assumption is that membership in a category is continuously graded, in the sense that I will explain shortly. This assumption is the problematic one, but let me start with the first assumption first.
    As I said earlier the PT holds that some concepts are characterized by sets of properties from which no particular property or subset of properties needs to be necessary or sufficient; rather, an object will be classified as falling under the category if it satisfies a sufficient number of properties from the list, “sufficient” being enough weight. The object must “score” enough “points” as a chair, or a game, or whatever, in order to be classified as one. This is enough to respond both to the Wittgensteinian objection and to the objection based on Prototype Effects. Since different objects might qualify as chairs, or games (that is, accumulate the required number of points) in virtue of satisfying different combinations of properties, there need not be (although there could be) properties that are necessary or sufficient (the Wittgensteinian objection). And since some objects will score higher than others in virtue of satisfying more or different properties, we can expect Prototype Effects.
    It is clear that nothing of what has been said thus far implies that membership in a category is graded. Everything is perfectly compatible with binary interpretation of membership, as long as the required “weight” is a precise fixed number that marks a clear distinction between objects that fall under the category, and objects that do not. If, for some concepts, an object needs to score, say, at least 10 out of 20 possible points, then objects that score 10 or more will fall under the category, and objects that do not, will not. Objects that score 15 points will be better examples of the category than objects that score 11 points,<11>  but we are nowhere obliged to say that objects that score 15 are more members of the category than objects that score 11. I conclude then that nothing in the details of the PT logically necessitates graded membership; the PT need not be committed to this assumption, and PE might arise just as well.
    This, however, is merely a logical point. While the PT can use very precise “sufficient weights”, I see no reason to deny the intuition that in many cases membership in a category is a matter of degree. Still, the grizzly bears objection doesn’t go through since O&S need the PT to be committed not only to graded membership but to continuously graded membership and this further assumption is utterly implausible. O&S’s understanding of gradedness seems to be something of the following sort: if, for instance, there are 40 properties that are associated with some concept, and each property weighs, say, 1 point, then an object that has one of these properties is a member of the category to the extent of 0.025, and an object that has 10 properties is 0.25 member, and so on. This is the kind of structure of gradedness that is required before we can accept the dubious claim that squirrels are, to some extent, grizzly bears and that Mars is, to some extent, in North America. But no prototype theorist needs to commit herself to this kind of gradedness. It commits us to the absurd view that squirrels are to some extent grizzly bears, or that grizzly bears with no hair on their left ear are less grizzly bears than bears with two hairy ears. The much more plausible structure of gradedness must be something along the following lines: objects that have 10 or less properties of the relevant 40 are absolutely non-members (that is, they all share the same membership value = 0), objects that have 30 or more properties are absolutely members (they all share the same membership value = 1), and membership may be graded for objects that have, say, between 20 to 29 of the relevant properties. Objects that have 20 will be 0.1 members, and so on.
    It is easy to show now why the grizzly bears objection does not, and cannot, go through. The specific objection that O&S mention fails simply because even though squirrels share some properties with grizzly bears they cannot be categorized as grizzly bears to no degree whatsoever, and even though Mars is closer to North America than the sun is, they are both in North America to the same degree, that is not at all. More importantly, however, the form of the argument fails. While it is still technically possible, within the gradedness framework just described, to find instances such that C_{(x)LIVES IN NORTH AMERICA} < C_{(x)GRIZZLY BEAR},<12>  the objection loses its force since if we find animals that are close enough to being grizzly bears, even if their degree of membership is 0.1, and we find these animals in places that are arguably in North America, even if only to the degree of 0.05, then the claim that findings of that sort disconfirm, at least to some degree, the statement that all grizzly bears live in North America doesn’t seem ridiculous any more; actually, it sounds pretty plausible (especially so if confirmation and disconfirmation themselves are taken to be graded, as they should be<13> ). I conclude, then, that the truth conditions objection to the PT fails.

Can the Classical Theory account for Prototype Effects?
Having concluded that the PT is an inadequate theory of concepts, O&S still need to make sense of PE. As I said earlier, PE place constraints on any theory of concepts. So O&S suggest to do that by distinguishing between a concept’s core and its identification procedure. The core, correctly characterized by the CT, is supposed to account for the true internal structure of our concepts, including their relation to other concepts and the way we reason with them. PE, and then a suitable modified PT, only deal with the kind of things that happen when we have to make rapid decisions about membership. They illustrate their proposed distinction with the concept WOMAN. Its core might contain information about the presence of a reproductive system, while its identification procedure might contain information about body shape, hair length, and voice pitch.
    Now this is a common maneuver used by CT, and I will mention three reasons why this move fails. The first one is a kind of ad-hominem against O&S which deals (for the last time!) with FZT. The other two are more substantial objections: I will argue that both components of the O&S proposal are flawed. First, we seem to be unable to come up with “cores” for most every-day concepts, regardless of how much time we are spending on it. Second, PE cannot be dismissed as identification procedures because they play important part in our reasoning.
    Notice, first, the possible incompatibility of O&S’s own proposal with the idea of modeling the PT on FZT. If the following objection works, the modeling of the PT on FZT will turn out to be not only misconceived but also self-defeating in a sense. Suppose, then, that the PT only describes identification procedures. How does the PT, as describing identification procedure, deal with identification procedures for complex concepts? It seems clear that the very same objections that were raised by O&S against the adequacy of the PT, as a theory of concepts, to account for conceptual combination can be raised against the adequacy of the PT, as a theory of identification procedure, to account for identification of complex concepts. Thus, the PT, as a theory of identification procedures, modeled on FZT, will predict slow identification of a normally shaped striped apple as a STRIPED APPLE for the very same considerations that were offered by O&S in the discussion above. And this prediction is probably false since people can easily identify a normally shaped striped apple as an example of STRIPED APPLE. If this is true, then O&S face a dilemma: they can admit that the PT is not even an adequate account of identification procedures, but then they will have no account for PE that they themselves do not wish to deny or ignore; or they can say that the PT, as a theory of identification procedures, should not be modeled on FZT. But then, why not adopt the same strategy with regard to the PT as a theory of concepts?
    The second objection has to do with the Wittgensteinian insight. Notice that it has nothing to do with rapid identification procedures; rather, after slow, careful, and prolonged examination of the concept GAME we find out that it has no core; so even if O&S are right about WOMAN, we are very likely to fail in finding cores for many other every-day concepts, for the considerations mentioned earlier. It is not surprising, but still quite annoying, that all the classical theorists constantly use their very unrepresentative pet-concepts of WOMAN, BACHELOR, etc.
    But interestingly, the CT is wrong even about concepts such as WOMAN and BACHELOR. As far as the concept WOMAN is considered, notice first that there’s much more, explicitly and implicitly, to the concept WOMAN than the presence of a reproductive system. For example, at least some aspects of body shape are certainly part of the core – a reproductive system inside a box is not a woman; and we will have problems categorizing heavily deformed human bodies as well, as a matter of conceptual determination, not as a matter of identification. And finally, but just as obviously, even reproductive systems come in degrees. It seems that we cannot find “core” neither for the real, socially situated, concept of WOMAN, nor for the technically narrow biological sense.<14>  And as far as the concept BACHELOR  is considered, Lakoff [1988] (following Fillmore 1982) offers the examples of Moslems that are allowed four wives but only have three bachelors, priests, and openly gay people in long term relationships as borderlines cases of the concept. And again, it seems that these cases have nothing to do with rapid identification procedures (remember, however, that nothing much hinges on these considerations since the technical, biological, sense of WOMAN and BACHELOR are far from being good representatives of most of our every-day concepts).
    Finally, the O&S proposal for accounting for PE by distinguishing between core and identification procedures seems problematic even in a world without Moslems, priests, and gays. The distinction between “core” and “identification procedures” might very well be a useful distinction to make; but surely the classification of some cognitive phenomena as those that belong to the “core”, and therefore qualify as the proper subject matter of a theory of concepts, and those cognitive phenomena that belong to the less dignified “identification procedures” cannot be arbitrary, and it cannot be invoked in an ad-hoc way to save some theory of concepts. But this is the impression one might get from O&S’s use of the “core” “identification procedures” distinction. Some clearer criteria as to what should be considered as the proper subject matter of a theory of concepts is needed, and O&S offer no such general theory or criteria, nor any convincing analysis of this matter. Furthermore, although I will not argue in length for this point here, I believe that according to some plausible such criteria, including some criteria that O&S themselves are offering, it will be difficult to exclude many of the Prototype Effects from our theories of concept.
    O&S themselves offer something that can be interpreted as criteria for distinguishing issues that belong to theories of concepts and issues that don’t: “the core is concerned with those aspects of a concept that explicate its relation to other concepts, and to thoughts” [277]. In other places they seem to suggest that the core will account for inferences and “the way we reason with” concepts. Rey says similar things (although his distinction is not between core and rapid identification procedures but rather between core and stereotypical reasoning): “If it is not the conditions either for conceptual identity or conceptual competence that [the PT] can be taken to be addressing, what is its import? Perhaps it is merely this: People rely, more heavily than we might have originally supposed, on stereotypical information on making category judgments. The methods, that is, that people are often criticized for employing in making judgments of e.g., racial or sexual category seem to be instances of a cognitive strategy people employ in making category judgments generally. … However, as most of them would readily admit, they are hardly committed to those properties or exemplars being defining of these concepts” [297].
    But if “playing part in reasoning” is what qualifies some phenomena to be considered a suitable subject matter for a theory of concepts, then as was demonstrated on pages 4-5 above, and as Rey himself admits, prototypes play a very important part in reasoning. Only an extremely narrow conception of reasoning as logical inference to which people stay committed even after reflection will maybe exclude PE from the domain of reasoning.<15>  But then our theory of concepts will leave much that is important out; after all, stereotypical reasoning is a kind of reasoning (there are analytic truths after all!), and we use it very often (most of the time stereotypical reasoning is very important and not at all morally objectionable; in this sense, Rey’s examples are very unfair to stereotypical thinking). And insofar as stereotypical thinking is based on prototypes, exemplars, etc., PE will have to be dealt with within our theory of concepts.
 

Footnotes
1  That a-prioricity is a matter of degree I learned, of course, from Bill Lycan.

2  I am aware of the fact that strictly speaking the entire paragraph is about words and meanings, not about concepts. The “translation” will have to say something of the sort: there is no set of necessary and sufficient properties that an object must have in order to be perceived, or categorized, as a game, or a chair, etc. But this is surely not enough and at some point Dorit’s doubts regarding the interchangeability of meanings and concepts will have to be considered. I, however, will not discuss this issue in this paper.

3  And satisfaction of some properties might have different point-value depending on the satisfaction of some other particular properties, etc. The structure of the point-assigning can be very complex.

4  I know it technically should be FST, but FZT seems more suggestive and user-friendly.

5  The complement operation can also be defined using the same function and notation, but we are not going to use it.

6  It might be clear now (maybe…) what is the temptation of modeling the CT on SST and the PT on FZT by translating “membership” into “falling under a category”: it seems that according to the CT objects fall under categories in a binary manner while according to the PT categorization is graded.

7  O&S also offer a very similar objection against the PT with regard to logically empty and logically universal concepts (268-269). The counter-arguments that I discuss in the text apply to these objections as well, but also, specific to this context, one might reasonably question the intuition that O&S seem to take for granted, according to which "VxC_{(x)APPLE THAT IS NOT AN APPLE} must be zero and that "VxC_{(x)FRUIT THAT EITHER IS OR IS NOT AN APPLE} must be 1.

8  In various footnotes O&S admit that there are conceptual combinations that do not admit to such analysis, but they consider such cases to be either idiomatic and relatively unimportant (such as DARK HORSE), or just as further cases that could embarrass the PT (such as GOOD COUNTERFEIT DOLLAR and SMALL GALAXY). As I argue in the text, these cases are numerous, as they embarrass the CT theory of concepts just as badly as they embarrass the PT.

9  I thank Bill Lycan for very helpful comments on this issue.

10  My suspicion is that O&S were led to both of these assumptions at least partly by sloppy application of FZT, and this is yet another unfortunate consequence of modeling the PT on FZT. As Bill Lycan noted, however, these assumptions are not only logically independent from the essentials of the PT, they are also logically independent from the essentials of FZT. Bill Lycan also showed me that there’s more to the gradedness issue than I discuss in the text; but these further aspects of gradedness are not discussed in the text…

11  The better examples will be mentioned more frequently as examples of the category, will be identified more quickly, will serve as better examples for teaching the category, etc. See Section 2 above.

12  This will happen when we have an animal (or whatever) that is close enough to be considered a grizzly bear so that it enters the area of gradedness for the concept GRIZZLY BEAR, and we have a location that is close enough (not necessarily literally speaking!) to be considered to be in North America so that it enters the area of gradedness for the relevant concept, and the location is deeper in its respective gradedness area (in the direction of full membership) than the animal is in its gradedness area.

13  This comment should not be understood as adopting a completely graded notion of truth, an option which O&S discuss and dismiss. All that is needed is a somewhat graded notion of confirmation and disconfirmation, parallel to and to the extent in which I propose adopting a somewhat graded notion of membership. (Incidentally, Bill Lycan noted that O&S’s dismissal of the graded notion of truth is puzzling given the fact that graded truth was devised to accompany FZT, O&S’s favorite model).

14  This is not a conclusive argument against the possibility of coming up with a definition for WOMAN;  even if we cannot succeed using O&S idea of a reproductive system, other attempts might be more successful (Bill Lycan suggested a definition of the biological-technical notion of a woman using the genetic make-up of women. I doubt that there is a set of necessary and sufficient genes that will make up a woman, but I will not pursue this issue any further here. As I said, even if this attempt is successful, it lends very little hope for optimism regarding many other every-day concepts).

15  “Maybe” because , as I was arguing all along, it seems that there is no way of getting rid of PE even within the “core” of concepts.
 

References
Osherson D., and Smith E., “On the Adequacy of Prototype Theory as a Theory of Concepts” [1981]
Lakoff G., Women, Fire, and Dangerous Things [1986]
Rey G., “Concepts and Stereotypes” [1983]
Rosch E., “Principles of Categorization” [1978]