The Inferential Model of Concepts Joshua Cowley

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1 The Inferential Model of Concepts Joshua Cowley It is often claimed that concepts are the building blocks of thoughts. If this claim is true, as I think it is, any adequate theory of cognition will require a theory of concepts. Both philosophers and psychologists have attempted to give such a theory, however, as with most of nature s building blocks, developing a theory of concepts has not been an easy task. I want to argue that the difficulty is due, in part, to a fundamental error in how one ought to approach a theory of concepts. Most theories of concepts seek to explain the internal structure or the proper parts of concepts. For example, the classical theory of concepts claims that a concept is a set of necessary and sufficient conditions. On the classical theory, each of these conditions is a part of the concept. As another example, the prototype theory claims a concept is a collection of typical features. Again, each feature is a part of the concept. Borrowing a term from Laurence and Margolis (1999) I will say that a theory which emphasizes the internal structure of concepts fits the containment model of concepts. I think that better theories of concepts can be constructed by placing an emphasis on how concepts relate to one another. In particular one should emphasize the inferential relationships between concepts. Borrowing another term for Laurence and Margolis, I will say that a theory which emphasizes the inferential relationships between concepts fits the inferential model of concepts. That a concept s role in inference should be important part of a theory of concepts is not a novel proposal (e.g., Field, 1977; Block, 1986; Harman, 1987; Pollock, 1989; Peacocke, 1992). However, the proposal has largely been treated as a theory of concepts. My suggestion is that the proposal is not a theory of concepts at all. Rather it is a model in which specific theories of concepts can be constructed. I want to argue that theories constructed under the inferential model are better than theories constructed under the containment model because the inferential model provides tools for constructing a theory of concepts which are not available when constructing theories under the containment model. How a theory makes use of those tools, though, is dependent upon the particular theory of concepts. As a result, my argument will have two parts. First, I will show how to reconstruct two specific theories of concepts, originally constructed under the containment model, to fit the inferential model. This modification results in theories that are both simpler and that have previously unavailable explanatory power. I will then discuss some of the general advantages of adopting the inferential model, using the modified theories as illustrations. The remainder of this paper is comprised of four sections. Section 1 gives a more detailed description of the containment and inferential models. Section 2 discusses the classical theory of concepts. First, I give a detailed account of the classical theory as constructed under the containment model. I then show how to modify the classical theory to fit the inferential model. I conclude the section by discussing the advantages of the modified theory. Section 3 considers the prototype theory of concepts. The structure of Section 3 is otherwise identical to that of Section 2. Finally, Section 4 demonstrates some of the general advantages of the inferential model. 1

2 1 Two models of concepts The impetuous for this paper comes from two paragraphs from Laurence and Margolis (1999) where they point out that there are two general models one might adopt toward conceptual organization. They call these the containment model and the inferential model. The containment model places emphasis on the internal structure of a concept, such as lists of common features or a collection of necessary and sufficient conditions. For theories that fit the containment model, any time a concept is accessed, all of its internal structure is accessed as well. For example, tokening bachelor 1 in the thought This room is full of bachelors, necessarily includes tokening a more basic psychological structure representing male. This aspect of the containment model has important consequences for what the internal structure of a concept can make up the internal structure of a concept. In particular, if a concept is composed of other concepts, then a regress problem develops. Suppose an agent tokens bachelor and therefore tokens another psychological structure representing male. If this structure is the concept male then which ever concepts compose to make male will also be tokened. This would continue until either every concept is tokened, a circle brings the agent back to some previously tokened concepts, or some primitive concepts are tokened. As we will see later, this is one reason for preferring the inferential model over the containment model. The inferential model deemphasizes the internal structure of concepts. Instead, it places the emphasis on how a concept is connected to other concepts. Let s say that whenever two concepts, C 1 and C 2, could be tokened in an inference, there is an inferential connection between C 1 and C 2. What theories cast under the inferential model claim is that a concept, C s, functional role consists of a privileged subset of C s inferential connections to other concepts. Exactly, which inferential connections compose C s functional role depends upon the particular theory of concepts. Unlike the containment model, on the inferential model tokening a concept like bachelor does not necessitate tokening male as well. However, tokening bachelor may generate a disposition to draw an inference that tokens male. Fodor (1998) has gone to great lengths to argue that any inferentialist view of concepts is untenable(see also Fodor, 2004). For the most part I will not be addressing the problems Fodor raises in this paper. This is because, from Fodor s standpoint, the debate between the containment and inferential models is internal to inferentialist views in general. For Fodor, any view which treats concepts as involving relations between any mental representations is an inferentialist view. Both the containment and inferential model meat this criterion. I take, Fodor s characterization of all these views as inferentialist to be a tacit recognition that these theories can be reworked to fit either the containment or inferential model. Where the containment and inferential models pull apart is in the 1. the nature of the representations that concepts are related to and 2. the nature of the relation between the concepts and the representations. The containment model takes concepts to be related to representations which are not themselves concepts (e.g. feature or conditions) and it states 1 It will often be necessary to distinguish between a concept and the English word that represents that concept. Small caps will be used to a specific concept. 2

3 that tokening a concept requires tokening the other representations. The inferential model takes concepts to be related to other concepts and it states that tokening a concept only creates a disposition to token other concepts. Before moving on to the classical theory it is worth reiterating a point made already. Neither the containment nor the inferential models are, themselves, theories of concepts. Rather, they are general frameworks for theories of concepts. In what follows I will argue for the inferential model by considering two popular theories of concepts currently cast under the containment model: the classical theory and the prototype theory. I will demonstrate how to modify these theories to fit the inferential model and show that this generates better theories than theories constructed under the containment model. 2 The Classical Theory of Concepts Most philosophers are familiar with the classical theory of concepts. In its most typical formulation the classical theory says that a concept is composed of the set of necessary and sufficient conditions required for the application of the concept. For example, the concept bachelor might be composed of three conditions: adult, unmarried, and male. Other formulations of the theory state that a concept is a mental representation that encodes a definition or that a concept is a list of essential characteristics. In each formulation, conditions, definitions, or characteristics are treated as proper parts of a concept. Thus, it is reasonable to say that these formulations of the classical theory assume the containment model. In the first part of this section I will lay out the classical theory as it is formulated under the containment model and list some of its advantages. In the second part I will show how the classical theory can be modified to fit the inferential model and discuss the advantages of this modification. 2.1 The classical theory under the containment model The classical theory has been the dominant theory, in one guise or another, through much of the history of philosophy. The view still lies hidden in the background of a good deal of contemporary philosophy. For instance, much of the work done on the Gettier problem in epistemology is an attempt to find the fourth condition to add to justified true belief as an analysis of knowledge. In general, any philosopher doing classical conceptual analysis is at least implicitly adopting the classical theory for some concepts. The large role that the classical theory has played in philosophy is enough of a reason to explore the theory. However, my reasons for looking at the classical theory here are 1. it is simple, 2. it is well known, and 3. it is clear that the classical theory is usually cast under the containment model. I do not mean to imply that everyone that has given a theory similar to the classical theory has done so under the containment model. I only mean that the common formulation of the classical theory concepts are sets of necessary and sufficient conditions is a theory which falls under the containment model. For instance, one could read Peacocke (1992) as developing the classical theory under the inferential model. What 3

4 I will now do is go through some of the advantages of the classical theory to point out both why it was such an appealing theory and what more recent theories have to accomplish to compete with the classical theory. I will then turn to showing how the arguments supporting the classical theory are made better when the theory is modified to fit the inferential model. The basic intuition behind the classical view is easy to see. There is something it is for an object to satisfy the concept triangle. 2 If an object is a closed figure with three sides, then it is a triangle. On the other hand, if an object is an open figure or has 4 sides then it is not a triangle. Satisfying the concept triangle just is to satisfy the necessary and sufficient conditions of triangle. The classical theory fits nicely with the means by which we typically communicate concepts to one another. For example, when asked to describe a concept one typically lists some conditions which need to be satisfied for its application. For example, one might describe an airplane as a vehicle with fixed wings and a propulsion system that allows it to move through the air. This looks roughly like the set of necessary and sufficient conditions for the application of airplane. Possibly the most appealing feature of the classical theory is its simplicity. The entire theory can be stated in the single sentence, A concept is composed of the set of necessary and sufficient conditions required for application of the concept. Philosophers have a good grip on what necessary and sufficient conditions are, even if we cannot always find them for the concepts in which we are most interested. Likewise, psychologists have little trouble understanding how a concept could be represented as a definition or list of necessary and sufficient conditions. The classical theory also has a good deal of explanatory power. Philosophers are concerned with issues regarding the reference of concepts and epistemological questions concerning when an agent is justified in applying a concept to an object. According to the classical theory, a concept refers to all the things which satisfy the necessary and sufficient conditions of the concept. An adequate theory of reference will still need to explain what it means for a condition to be satisfied, but this is plausibly an easier problem to solve. By adopting the classical theory we also get a fairly simple analysis of when an agent is justified in her application of a concept. For the classical theory, an agent is justified in believing that C applies to an object o if she is justified in believing that each of the necessary and sufficient conditions which compose C are met by o. Concepts play a role in a number of psychological theories, and a theory of concepts which fits easily into these other psychological theories is exceptionally useful. For instance, one psychological issue intimately tied to concepts is categorization. Categorization is the psychological term used for a person s ability to group objects together. More precisely, it is the psychological process by which an agent applies a concept to an object. According to the classical theory this is accomplished quite easily. To apply a particular concept to an object an agent only needs to judge that each of the concept s necessary and sufficient conditions holds of the object. The classical theory does such a nice job of explaining categorization that it may not look like anything needed explaining in the first place. 2 I use the term object to refer to anything too which concepts might apply. 4

5 Another challenge for psychologists is determining the process of concept acquisition. Once again, the classical theory has an easy explanation. A person learns a new concept by having a mental representation which groups together a set of necessary and sufficient conditions. For example, to acquire bachelor an agent need only generate a mental representation composed of the conditions is an adult male and is unmarried. Having shown some of the advantages of the classical theory, I now want to turn to modifying it to fit the inferential model. What I will show is that the modified theory either retains or improves upon the advantages of the classical theory. 2.2 The classic-inferential theory Although the classical theory has traditionally been cast under the containment model, it does not have to be. I now want to turn to the classic-inferential theory, a modification of the classical theory designed to fit the inferential model. Recall that the inferential model states that the functional role of a concept, C, consists of a privileged set of inferential connections between C and other concepts. I now want to suggest that the classical theory is best seen as an attempt to determine which inferential connections are privileged. Under the containment model the classical theory states that a concept is composed of the set of necessary and sufficient conditions that must be satisfied to apply the concept. If the satisfaction of a condition is necessary for the application of a concept, then the application of the concept entails the satisfaction of the condition. Along the same lines, if the satisfaction of a set of conditions is sufficient for the application of a concept, then satisfaction of the set of conditions entails that the concept is applicable. These entailment relations are the key to the classic-inferential theory. The classic-inferential theory proposes that the inferential connections that make up the inferential role of C are entailment relations between C and other concepts. Each condition which composes the necessary and sufficient conditions for C will correspond to some other concept, D, and there will be entailment relations between C and D. For example, x is a bachelor entails x is unmarried and it entails x is male. Likewise, x is unmarried and male entails x is a bachelor. Thus, there are entailment relations from bachelor to male, from bachelor to unmarried, and from the conjunction (male and unmarried) to bachelor. Each of these is an inferential connection or, in other words, a disposition to make the following inferences: 1. From x is a bachelor infer x is unmarried. 2. From x is a bachelor infer x is male. 3. From x is male and x is unmarried infer x is a bachelor. We can now see the major difference between the classical theory as characterized under the containment model and the classic-inferential theory. In the classical theory as characterized under the containment model, a concept literally contains a list of conditions which are mutually necessary and jointly sufficient for the concept s application. The classic-inferential theory does away with conditions completely and simply links concepts via entailment relations. 5

6 Having laid out how one would modify the classical theory to fit the inferential model, I now want turn to the some of the advantages of this modification. The classic-inferential theory has all the advantages of the classical theory as well as a few of its own. Like the classical theory, the classic-inferential theory is very simple and both philosophers and psychologists have an even better grip on entailment than they do on necessary and sufficient conditions. Recall that categorization is the psychologists term for concept application. For the classic-inferential theory, categorization is just an inference. Suppose C is a concept, D 1... D n are concepts representing the necessary and sufficient conditions of C, and o is an object. Since D 1... D n are the necessary and sufficient conditions of C, D 1 x... D n x entails Cx,. Therefore, if an agent believes D 1 o... D n o, then she can infer Co. It is precisely these kinds of inferences that account for categorization. Furthermore, since any complete theory of psychology has to account for inference anyway, the classic-inferential theory actually offers a more parsimonious explanation of categorization than is offered by the classical theory alone. Concept acquisition is also easily explained by classic-inferential theory. To acquire a new concept, C, an agent forms connections between a new mental symbol, ϕ, and other preexisting mental symbols, ρ 1... ρ n, where ρ 1... ρ n correspond to the concepts 1) whose application is entailed by the application of C and 2) which jointly entail C s application. The second clause is important for distinguishing a concept from its subcategories. For example, it is a necessary condition of something s being a square that it have four sides and that it have four right angles. However, these conditions are both necessary and sufficient for a rectangle. So if C plays a role in an inference from x has four sides and four right angles to Cx, then the concept is of a rectangle not a square. The problem of reference for classic-inferential theory is handled essentially as it is for standard classical theory. If D 1 x... D n x entails Cx, then C refers to any x for which a belief in D 1 x... D n x would be true. Likewise, an agent is justified in believing Cx just in case she is justified in believing D 1 x... D n x. The classic-inferential theory s approach to justification is, in fact, more elegant than the classical theories. Epistemology is typically concerned with the justification of beliefs, not the justification of concept application. Under the classic-inferential theory the structure of a concept just is a set of reason schemas for moving from one belief to another. No additional structure is needed to understand how a belief in the application of a concept is justified. The classical theory is a well known and relatively simple theory of concepts. These qualities make it a useful theory to demonstrate how we might modify a theory to fit the inferential model. However, because the classical theory is simple the advantages of the classic-inferential theory over the classical theory are not very dramatic. In the next section I will consider a more complicated theory of concepts, prototype theory. Modifying prototype theory to fit the inferential model is much more difficult than modifying the classical theory, but the payoffs are higher. 6

7 3 Prototype Theory In the 1970 s psychologists took note of a growing body of evidence pointing towards a statistical structure to concepts. This body evidence, called typicality effects, is focused on the finding that subjects will regard some features or instances of a concept as more typical than others (Rosch, 1973b; Rosch & Mervis, 1975; Mervis, Catlin, & Rosch, 1976). A number of effects, such as the speed and accuracy of categorization, vary with the how typical a subject views a feature or instance of a concept (Smith, Shoben, & Rips, 1974). Rosch (1973a) and later Rosch and Mervis (1975) were the first to develop a statistical theory of concepts as an explanation for typicality effects. Rosch s work led to two theories of concepts called prototype theory and exemplar theory. According to prototype theory, a concept is a list of typical features; while according to exemplar theory a concept is composed of representations of its most typical instances (Medin & Schaffer, 1978; Nosofsky, 1986). In this section, I will discuss prototype theory and how to augment it to fit inferential model. 3.1 Prototype theory under the containment model As the name suggests, prototype theory claims that a concept is a prototype. Exactly what constitutes a prototype depends upon the particular theory, but in its weakest form, a prototype is simply a collection of features (or properties) that are typical of instances of the concept. Thus, the prototype for bird might include: {feathers, wings, beak, sings}. In some ways, this is similar to the classical theory. In both theories, a concept is a list of features/conditions to be satisfied by instances of the concept. The difference is in how these features are selected. According to prototype theory, an agent will categorize an object under bird precisely to the degree that the object is similar to (has the same features as) the prototype. An object that has feathers, wings, a beak, and sings is almost certainly a bird, while an object that only has wings and sings is only probably a bird. In other words, none of the features that compose the prototype are necessary to apply the concept. Most versions of prototype theory also include some kind of weight that is associated with each feature (Smith, Osherson, Rips, & Keane, 1988; Hampton, 1995). These weights are intended to reflect exactly how typical a feature is of a given concept. Bird, for instance, may have a greater weight associated with feathered than with sings. The primary motivation for prototype theory has been its ability to explain typicality effects. To find typicality effects, subjects are first asked to rank how typical a feature or instance is of a given concept. For instance, a subject may be asked to rank, on a 1 to 10 scale, how typical the feature green is of the concept apple. Presumably, green will be ranked above yellow but below red. Once the rankings have been made, a separate task is used to look for an effect which corresponds to the rankings. Typicality ratings are quite similar across subjects (though see (Barsalou, 1987), for some evidence to the contrary) and can be demonstrated even for concepts for which the necessary and sufficient conditions can be made explicit. As an illustration, a whole number not divisible by 2 constitutes the necessary and sufficient conditions for odd number. Despite there being no ambiguity in whether something fits these conditions, subjects happily 7

8 rate 7 as a better example of an odd number than 91(Armstrong, Gleitman, & Gleitman, 1983). One might be tempted to think that a subject s ability to rank instances of a concept as more or less typical might have little to do with the actual structure of a concept. But the influence of typicality can be seen in more than just a subject s ratings. If a subject is asked to judge whether a concept, C, applies to some object o, both the response time and the chance of error are correlated with how typical the subject would rate o as an instance of C (Rosch, 1973b; Smith et al., 1974). That is to say, a subject is likely to judge o as C both faster and with a reduced chance of error the more typical o is of C. Typicality ratings can also be correlated with biases in probabilistic reasoning (Sloman, 1993). Our use of natural language can also be influenced by typicality. If one is willing to say o is mostly C then the typicality rating of o as a C is lower than something for which the subject would not use mostly. Rosch gives the example, It is correct to say that a penguin is technically a bird but not that a robin is technically a bird, because a robin is more than just technically a bird; it is a real bird, a bird par excellence (Rosch, 1978, p. 39). In order to reconstruct prototype theory under the inferential model, it will be worthwhile to consider a specific example of prototype theory. Smith et al. (1988) present one of the most complete versions of the theory. They say, In our view, a prototype is a prestored representation of the usual [features] associated with the concept s instances (Smith et al., 1988, p. 487). On this view, a feature has four parts: an attribute, a value, and two numbers. An attribute names the class to which the feature belongs. Examples of attributes might be color, shape, and taste. Paired with each attribute is a value. A value is a specific instance of a given attribute. Red, round, and sweet would be possible values corresponding to the attributes color, shape, and weight. A feature then is an ordered pair consisting of an attribute and a value. An example may help to clarify the view. A prototype for the concept apple would be a list of (attribute, value) pairs such as: (color, red) (color, green) (shape, round) (taste, sweet) For each attribute and value, there is also a number that roughly represents how important the attribute or value is to the concept. For attributes, this number represents the diagnosticity of the attribute. Diagnosticity reflects how useful the attribute is in judging whether the concept applies to an object. For the concept apple, taste and shape might have high diagnosticity, while smell might not. A value, on the other hand, has a degree of salience. Smith et al. do not explicitly state what they take the salience of a value to be. As examples, they suggest that the red in apple is more salient than the round in apple and it is also more salient than the red in brick. My interpretation of salience is that it represents how typical a feature is of the objects that satisfy the concept. This interpretation makes it clear 8

9 why salience is different from diagnosticity. Knowing that an object is red is not very helpful in determining that the object is an apple. However, knowing that an object is an apple strongly suggests that the object is red. It is clear from their description that Smith et al. are committed to the containment model. For them, a concept is a complex structure consisting of a number of proper parts, features, which are, themselves, composed of more proper parts, attributes and values. This places a heavy emphasis on the structure of concepts as opposed to the functional role of concepts. Furthermore, there is no mention of how concepts are related to one another. Having laid out the prototype theory, I now want to show how it can be modified to fit the inferential model. Among other things, it will turn out that the modified theory is significantly simpler than the theory proposed by Smith et al. 3.2 Prototype-inferential theory As a first approximation, we can augment prototype theory to fit the inferential model simply by changing each feature to a concept. Let s call the resulting theory the prototype-inferential theory. According to the prototype-inferential theory a concept, C, has inferential connections to the concepts representing C s typical features. This would suggest the following definition of the structure of a concept: Definition 3.1 The conceptual structure of a concept, C, with typical features, F 1... F n, is a set of inferential connections that link C and one or more F i. The problem with Definition 3.1 is that there are an infinite number of inferences which will token C and one or more of C s typical features. Let o be an object and suppose an agent forms the belief Co on grounds that are independent from F 1... F n. For each F i, she can infer any of the following F i o, F i o F i+1 o, or F i o Gx. The first two inferences seem fine. The third, however, has a free variable that could be substituted with anything. Although it is an inference which tokens both C and some of C s typical features, it is not an inference prototype theory would want to include in the inferential role of C. The obvious solution to this problem is to limit the inferences in the inferential role to those that token only C and one or more of the F i. Modifying Definition 3.1 to reflect this suggestion results in the follow improved definition: Definition 3.2 The conceptual structure of a concept, C, with typical features F 1... F n, is a set of inferential connections that link C and only one or more F i. One striking difference between modifying prototype theory to fit the inferential model and doing the same for the classical theory is that Co does not necessarily provide a conclusive reason for any F i o. Because features are merely typical of a concept, it is possible to defeat an inference from an instance of the concept to an instance of its features. The second and more difficult step in changing prototype theory to the inferential model is dealing with the weights that are associated with the features. It seems clear that one of the weights associated with F i in prototype theory ought to be related to the strength 9

10 of Cx as a reason for F i x in the prototype-inferential theory. What is less clear is how to move from a belief in F i x to Cx. To determine this we need to look more closely at how weights are meant to be used in prototype theory. Smith et al. provide two numbers here which we might want to associate with reason strength. Recall that diagnosticity reflects how useful an attribute is in judging whether a concept applies to an object, while salience represents how typical a feature is of the objects that satisfy the concept. Of these two numbers, the diagnosticity seems to more closely capture the strength of a reason from F i x to Cx. If an attribute has a high diagnosticity then, presumably, the presence of that attribute in o would give an agent a better reason for believing Co than the presence of an attribute with a low diagnosticity. The strength of the reason from F i x to Cx, cannot be related only to diagnosticity. Consider the concept apple again. Color may have a high diagnosticity for apple. However, believing o is red (along with some other beliefs) gives one a stronger reason for thinking o is an apple than believing o is green. If o is red and o is green give reasons of different strength for o is an apple then the salience must also be related to the reason strength from F i x to Cx. At this point it might be useful to consider why Smith et al. give properties both an attribute and a value. Their explanation is that the attribute is needed for an agent to know how to compare two values. For example, an agent needs a way of knowing that green means not blue. Each attribute can only be filled by one value at a time. Therefore, if (color, green) is a feature of an object, then the agent knows that (color, red) is not. All of this seems overly complicated. More importantly, though, it is inaccurate. Consider the color of an apple. That an object is red, round, and smooth is a reason to think that it is an apple. That an object is green, round, and smooth, is a weaker reason for thinking that it is an apple because most apples are red and because lots of other fruits are green. For the concept apple, color has a lower diagnosticity when the color is green and a higher diagnosticity when the color is red. It seems then, that diagnosticity is not really a function of the attribute, but a function of the value. The attributes in Smith et al. s theory are playing two roles. The first is to keep track of how useful the property is in determining whether the concept can be applied to an object, (i.e., the feature s diagnosticity). The second role is to classify two features (e.g., red and green) as mutually incompatible. We have seen that diagnosticity is really a function of values, not attributes. Now let s tackle the second role an attribute is to play. Suppose an agent knows that o is smooth, round and purple. Being smooth and round is a reason for thinking that o is an apple. However, being purple should be a reason for thinking that o is not an apple. However, to get to this conclusion an agent needs a way of moving from o s being purple to its not being red. Smith et al. handle this by having the feature contain the information that red is a color. Since, purple is a different color and different values of the same attribute are always mutually incompatible, o cannot be red if it is purple. On the inferential model, this is completely unnecessary. An obvious aspect of the concept red is that it is a color. And an obvious aspect of the concept color is that if something is one color, then it is not another color. All that is really needed is an inferential connection from apple to red. The inferential connections between purple, red and color already build 10

11 in that if something is purple, it is not red. By modifying prototype theory to the prototypeinferential theory we see that attributes are not needed to play either of the roles for which Smith et al. proposed them. Furthermore, features no longer have internal structure as they did in prototype theory. This lack of internal structure makes it more obvious that the features of one concept are just other concepts. Smith et al. do have one important insight. There does seem to be a difference between the diagnosticity of a feature and its salience. There are cases where a feature can be very salient for a concept, while not necessarily being particularly useful in applying the concept to an object. For example, one of the salient properties of maple leaves is that they are green. However, most leaves and plants are green, so green may have very little diagnostic value. Smith et al. say the following about salience: When asked to verify that a property is true of a particular concept, people respond faster to properties that have previously been rated as more related or associated to the concept than to those rated less related (e.g., Glass & Holyoak, 1975). Thus, people are faster at deciding that apples are red than that apples are round, suggesting that red is more salient than round in the prototype for apple. (Smith et al., 1988, p. 487) In this passage they are suggesting that two things determine the salience of a feature: the subjective frequency with which the value occurs in instances of the concept and the perceptibility of the value. If there is a difference between salience and diagnosticity, then how would that be represented under the inferential model? In prototype theory, diagnosticity determines how one moves from features to application of the concept. In the prototype-inferential theory this will be reflected in the reason strengths for inferences from (sets of) features to concepts. But one can also make inferences from a concept to a feature, and this is where salience enters the picture. Salient features are just those which can be inferred easily from the application of the concept. In many cases features which have a high salience also have a high diagnosticity. But in some cases, (e.g., the greenness of maple leaves) the diagnosticity is much lower than the salience. To say that the salience and diagnosticity of prototype theory are related to reason strength in the prototype-inferential theory is a little unclear. I do not mean to suggest that there is a function which will convert salience and diagnosticity into an appropriate representation of reason strengths. Smith et al. have psychological evidence for two values. However, when designing their experiments they had prototype theory in mind, not the prototype-inferential theory. The values they get for salience and diagnosticity are influenced by the reason strengths, but they may not be the only things influencing those values. I now want to consider the advantages of the prototype-inferential theory. As with the classical theory, prototype theory s advantages come in two forms: psychological advantages and philosophical advantages. The first thing to recall is that prototype theory was intended as a response to classical theory. Classical theory got into trouble because there just do not seem to be necessary and sufficient conditions for most concepts. Prototype theory does 11

12 not require necessary and sufficient conditions for a concept, and neither does the prototypeinferential theory. The prototype-inferential theory says that the functional role of a concept is the set of inferential connections which satisfy Rule 3.2. Neither individual features nor sets of them need to be necessary and/or sufficient to satisfy this constraint. By building concepts out of inferential connections, the prototype-inferential theory also has an easy time explaining much of the psychological evidence that motivated prototype theory in the first place. First of all, prototype theory has to posit some kind of psychological process whereby a concept s features can be extracted and manipulated from the concept. In the prototype-inferential theory this is already handled by adopting a theory of reasoning. A theory of reasoning provides some very powerful explanatory tools for dealing with typicality effects. For example, the reason a subject is more likely to guess that an apple is red than to guess that it is green is that x is an apple is a stronger reason for x is red than it is for x is green. Each of these inferences counts as a defeater for the other, but since the first inference is stronger, it ultimately defeats the second. The prototype-inferential theory is also simpler than prototype theory. According to prototype theory, a concept is a list of typical features. Each feature is an (attribute, value) pair and associated with each member of the pair is a diagnosticity or salience. According to the prototype-inferential theory, however, there are concepts, inferential connections between the concepts, and reason strengths. These three items not only capture all of the important elements of prototype theory, they also capture facts about concepts that are not handled by prototype theory. For example, the concept red presumably bears some relationship to the concept apple. Prototype theory, however, only discusses a relationship between the feature red and the concept apple. Exactly how features are connected to their corresponding concepts is still in need of explaining. Although Smith et al. present one of the most complete versions of prototype theory, it is not the only version on the market. I think it is clear that their version benefits greatly when reconstructed under the inferential model. The basic techniques I have used to modify Smith et al. s version of prototype theory to the inferential model could be applied to other versions as well and doing so would produce theories that have many of the same advantages. However, these advantages are specific to modifying prototype theory. The same benefits may not be available after reconstructing other theories under the inferential model. In the next section I will consider some of the general advantages of the inferential model. 4 Conclusion The classical theory and prototype theory are representative examples of theories of concepts cast under the containment model. I have argued that these two theories can be modified to fit the inferential model and that doing so makes them better theories. However, there are many other theories of concepts, and I have not given a general argument that all of them would benefit from the same modification. The inferential model does not solve many problems that are common to all theories of concepts. Rather, it provides tools for solving the specific problems confronting a given theory. In particular, the inferential model 12

13 enables us to utilize our general theory of reasoning in developing a theory of concepts. In converting prototype theory to prototype-inferential theory, for instance, we saw that a theory of reasoning can be used to explain why a subject is more likely to guess that an apple is red than to guess it is green. However, this specific problem does not come up for the classical theory so it cannot be considered a general advantage of the inferential model. For the most part, how a theory of reasoning will be useful depends on the theory of concepts being developed. However, this is not always the case and I will conclude this paper by considering several general advantages of adopting the inferential model. The first advantage concerns the application of concepts to objects while the second concerns issues of parsimony in a general theory of mind. A common explanatory problem for theories of concept application are cases when an agent has two concepts which seem to apply to an object, but those concepts are inconsistent with one another. For instance, seen from a distance a horse standing on a hill at night may look as much like a cow as a horse. However, an agent in this situation, can t apply both horse and cow to the object. On the inferential model, this situation results from an agent having reasons for believing Ho and reasons for believing Co. Since horse and cow are incompatible, Ho is a reason for Co and Co is a reason for Ho. These are cases of rebutting defeaters. Any theory of reasoning will have a solution for this kind of situation. For example, on Pollock (1995) view, if the agent has a stronger reason for believing Ho than for believing Co, then Ho defeats Co. If, on the other hand, the reason strengths for Ho and Co are of equal strength, then Ho and Co suffer from collective defeat and neither should be believed. I don t mean to profess that most theories of concepts can t deal with the application of conflicting concepts. It is such a basic issue that any reasonable theory of concepts will have a solution. However, any theory cast under the inferential model gets the solution for free. I take this to be an advantage of constructing theories of concepts under the inferential model. Another issue, frequently overlooked in developing a theory of concepts, is retracting the application of a concept. We are often in the position of having applied a concept to an object and then after further reasoning or acquiring new information, we find that we must unapply the concept. This situation can crop up for several reasons. Suppose an agent applies C to o. One reason to unapply C is if, at a later time, the agent applies a new concept, H, to o and H is incompatible with C. Notice though, that applying H to o isn t enough to retract the application of C to o. The agent must also recognize that H is incompatible with C and go back to unapply C to o. Another case where an agent may need to unapply a concept is when she has retracted something that led her to apply the concept in the first place. To illustrate, suppose that an agent applies apple to o partially on the basis of o s being red. If she later learns that o was illuminated by red light, she no longer has a reason to think that o is red. If she does not have reason to believe that o is red, then should also unapply apple. This example, is importantly different from the previous. In the previous case that o is H is a reason to think that o is not C. In the latter case, there is not reason to think that o is not an apple. Rather, the reason for thinking it is an apple 13

14 has been removed. In other words, she ought to withhold judgment on the issue. Unapplying concepts is not often dealt with in theories of concepts. However, it is clear that it ought to be a fundamental part of any complete theory of concepts. One advantage of the inferential model is that the process by which a concept is unapplied comes free with a theory of defeasible reasoning. Any adequate theory of defeasible reasoning will include methods for retracting beliefs either because of stronger arguments for the negation of the belief or because the original argument for the belief has been undercut. On the inferential model, the application of C to o just is the formation of an argument for believing Co. Likewise, unapplying C to o just is defeating the argument for Co. This ability to utilize a general theory of reasoning to solve specific problems for one s theory of concepts also creates a more parsimonious theory of the mind as a whole. A complete theory of the mind requires both a theory of concepts and a general theory of reasoning. If one of these theories can be used to explain all or part of the other, then the overall theory of mind will be simpler. The inferential model accomplishes this by enabling a theory of reasoning to be used in constructing a theory of concepts. As I mentioned before, the benefits of constructing a theory of concepts under the inferential model are largely specific to the theory. In this section, I have shown two of the general advantages of constructing a theory under the inferential model. Primarily, these advantages are created by the ability to use one s theory of reasoning while building the theory of concepts. The classical theory and prototype theory, discussed in the previous two sections, are examples of how a theory can be improved when modified to fit the inferential model. I take this to be a reason to think that the correct theory of concepts will fit the inferential model as well. 14

15 References Armstrong, S., Gleitman, L., & Gleitman, H. (1983). What some concepts might not be. Cognition, Barsalou, L. (1987). The instability of graded structure: Implications for the nature of concepts. In U. Neisser (Ed.), Concepts and conceptual development: Ecological and intellectual factors in categorization (pp ). NY: Cambridge University Press. Block, N. (1986). Advertisement for a semantics of psychology. Midwest Studies in Philosophy, X, Field, H. (1977). Logic, meaning, and conceptual role. Journal of Philosophy, 74, Fodor, J. (1998). Concepts: Where cognitive science went wrong. NY: Oxford. Fodor, J. (2004). Having concepts: A brief refutation of the 20th century. (forthcoming in Mind and Language) Glass, A., & Holyoak, K. J. (1975). Alternative conceptions of semantic memory. Cognition, 3, Hampton, J. (1995). Testing the prototype theory of concepts. Journal of Memory & Language, 34, Harman, G. (1987). (nonsolipsistic) conceptual role semantics. In E. LePore (Ed.), New directions in semantics (pp ). London:: Academic Press. Laurence, S., & Margolis, E. (1999). Concepts and cognitive science. In E. Margolis & S. Laurence (Eds.), Concepts: Core readings (pp. 3 83). Cambridge, Massachusetts: MIT Press. Medin, D. L., & Schaffer, M. M. (1978). Context theory of classification learning. Psychological Review, 85, Mervis, C., Catlin, J., & Rosch, E. (1976). Relationships among goodness-of-example, category norms, and word frequency. In Bulletin of the psychonomic society (Vol. 7, pp ). Nosofsky, R. (1986). Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 115, Peacocke, C. (1992). A study of concepts. Cambridge, MA: MIT Press. Pollock, J. (1989). How to build a person: A prolegomenon. Cambridge, MA: MIT Press. Pollock, J. (1995). Cognitive carpentry. Cambridge, MA: MIT Press. Rosch, E. (1973a). Natural categories. Cognitive Psychology, 4, Rosch, E. (1973b). On the internal structure of perceptual and semantic categories. In T. E. Moore (Ed.), Cognitive development and acquisition of language (pp ). New York: Academic. Rosch, E. (1978). Principles of categorization. In E. Rosch & B. Lloyd (Eds.), Cognition and categorization (pp ). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Publishers. Rosch, E., & Mervis, E. (1975). Family resemblances: Studies in the internal structure of categories. Cognitive Psychology, Sloman, S. (1993). Feature-based induction. Cognitive Psychology, 25,

16 Smith, E., Osherson, E., Rips, L., & Keane, M. (1988). Combining prototypes: A selective modification model. Cognitive Science(12). Smith, E., Shoben, E., & Rips, L. (1974). Structure and process in semantic memory: A featural model for semantic decisions. Psycholgical Review,

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