City Research Online Conceptual Combination and Negation 1 Conceptual Combination: Conjunction and Negation of Natural Concepts Conceptual Combination and Negation 2

The operation of negation on combinations of natural categories was examined in two experiments. In the first, category membership ratings of lists of items were obtained for pairs of concepts considered individually, and in two logical combinations: conjunctions (for example, Tools which are also Weapons) and "negated conjunctions" -forms of those conjunctions in which the modifier noun category was negated (Tools which are not Weapons). For conjunctions, results supported earlier findings of overextension, and the geometric averaging of constituent membership values (Hampton, 1988b). Previous findings of concept dominance and non-commutativity within conjunctions were also replicated, both for typicality ratings and for probability of class membership. For negated conjunctions, the pattern of dominance was similar, but interacted with order within the conjunction. Negated conjunctions were also overextended. The second experiment explored how the attributes of negated conjunctions are derived from those of the two component concepts. Frequency of generation of attributes expressed positively (has wheels) or negatively (has no wheels) followed rated frequency in the negated category. The distinctiveness of an attribute to distinguish the complement from the head noun class was associated with the generation of attributes, particularly when there was relatively high overlap between the two categories. Conceptual combination and negation 3 Conceptual Combination: Conjunction and Negation of Natural Concepts The study of conceptual combination has recently come to assume considerable theoretical importance for psychological theories of concepts (Hampton, 1996a; Rips, 1995). The prototype theory of concepts (Rosch, 1978; Rosch & Mervis, 1975) postulated that the classification of objects in categories such as Furniture or Sports is based on the overall similarity structure of the items composing the category. The theory proposed that the category is represented by a prototype, which is an idealized representation of the set of attributes positively associated with category membership. Items are judged to belong in the category if they are sufficiently similar to the prototype (and dissimilar from the prototypes of contrasting categories). Against the prototype theory, it has been argued (Fodor, 1994; Osherson & Smith, 1981, 1982) that the lack of any clear set of rules for combining prototype concepts in logical combinations casts grave doubt on the general value of prototype theory as a theory of human concepts. Hence research on how logical functions operate on prototype concept categories is of considerable theoretical interest. The arguments surrounding this issue have been widely aired (Cohen & Murphy, 1984; Hampton, 1987, 1988b, 1991, 1996a, 1996b; Jones, 1982; Murphy, 1988; Murphy & Spalding, 1995; Osherson & Smith, 1981, 1982; Rips, 1995; Smith & Osherson, 1984; Thagard, 1983; Zadeh, 1982). 1 The problem addressed in this article is how similarity-based prototype concepts enter into logically constructed complex concepts. In particular the focus here is on the logical operations of conjunction and negation. In order to study these operations, the studies to be reported used a head noun plus relative clause construction as a means of expressing conjunction ("Tools which are weapons"), and a similar construction with a negated modifier noun ("Tools which are not weapons") as a means of studying the effect of negation. The questions raised were first, to what extent is category membership in such classes predictable from category membership in each of the constituent classes, and second, to what extent are the attributes which are considered descriptive of the negated conjunctions also true of the constituent classes. Conceptual combination and negation 4 Much current theorizing on the formation of conjunctive concepts has arrived at the view that the most fruitful theoretical approach is an intensional one (Cohen and Murphy, 1984; Hampton, 1987; Smith & Osherson, 1984; Smith, Osherson, Rips, & Keane, 1988). This is to say that rather than modelling membership in the conjunction extensionally in terms of some function of degree of membership in the constituent classes (see for example the fuzzy logic approach, Zadeh, 1965, 1982, or the statistical approach proposed by Huttenlocher & Hedges, 1994), models of conceptual conjunction should aim to define how the two prototypes (or schemas) representing the two concepts become combined into a modified or composite representation of the conjunctive class. (Dissenting accounts are offered by Chater, Lyon & Myers, 1990, and Huttenlocher & Hedges, 1994.) Two intensional models have been developed for concept conjunctions with some degree of detail. Smith et al. (1988) proposed a selective modification model for adjectivenoun combinations such as "Red Apple". In their model a head noun such as "Apple" is represented by a frame (Minsky, 1975) composed of attributes such as COLOR, SIZE or TASTE, each of which can take values, such as red, large, or sweet respectively. Specifically, the representation of the head noun "Apple" would possess an attribute for COLOR which would normally take a range of values -red, green, yellow et cetera -each with an associated number of votes, reflecting its frequency of occurrence as the color of an apple. According to the model, this head noun frame becomes selectively modified in the combination "Red Apple", by switching all the votes for COLOR to the value red, while at the same time increasing the overall weight of color in the determination of similarity to the concept schema. The second intensional model was proposed by Hampton (1987, 1988b) in order to account for the way people understand the conjunction of two noun concepts in phrases such as "Sports that are also games", or "Tools that are also weapons". Hampton's composite prototype model for conjunctions proposed that each noun concept is represented by a prototype, consisting of a list of attributes or properties. When the concepts are conjoined, Conceptual combination and negation 5 then a new composite prototype is constructed by merging together the two sets of attributes defining the two constituent noun concept prototypes. This composite list of attributes is then subject to further modification in order to satisfy various constraints, such as Necessity (an attribute that is considered necessary for a constituent is also considered necessary for the conjunction), Impossibility (an attribute that is considered impossible for a constituent is also considered impossible for the conjunction), and Coherence (the composite prototype may not contain two incompatible attributes). A similar formal approach has been suggested by Thagard (1983, 1995). Apart from being directed at different forms of conjunction, an important difference between the Smith et al. and Hampton models lies in their assumptions concerning the determination of set membership in the conjunctive class. Whereas Hampton (1987, 1988b) explicitly proposed that membership in the conjunction is determined by similarity of instances to the composite prototype, Smith et al. (1988) chose to limit their model to the determination of typicality or representativeness of instances in the conjunction. They recognised that, as constituted, their model failed to pick out a conjunctive concept category which would actually be the logical intersection of the two constituent sets (the same is true of Hampton's model). Quite simply, a logical intersection requires that membership in the conjunction should depend on the level of similarity to each constituent independently. Degree of Redness and degree of Appleness for example should form independent criteria, both of which need to be achieved for something to count as a Red Apple. However if membership of the conjunction Red Apple is based on overall similarity to the conjunctively modified schema, then this independence of criteria will not be possible (see Ashby & Gott, 1988, for discussion of this issue.). For Hampton (1988b), this failure of intensional models to generate intersective conjunctions was taken as a virtue, in as much as his data apparently showed that people's classification of instances in conjunctions was not in fact purely intersective, but showed the kind of interdependence predicted by the models. Smith et al. (1988) in contrast, argued that Conceptual combination and negation 6 typicality and membership depend on two different types of semantic information. They suggested that concepts may have a core meaning some central definitional component of the attribute structure associated with the concept which is used in making class membership judgments, (see also Miller & Johnson-Laird, 1976). If this core is defined as a necessary and sufficient set of common elements, then Boolean set logic can be used to determine how concepts combine. Effectively an object is actually classified as a red apple, only if it has the "core" features of both redness and appleness (no pun intended). Typicality judgments however would be based on similarity using the full range of prototype attributes, and so would not follow logical intersection. Their model for concept conjunction was therefore explicitly restricted in its scope to intuitions of typicality. The primary aim of the present research is to replicate and to extend the range of data considered by such models by exploring the use of negation in conceptual combinations. There has been very little research on how people interpret negated concepts. One obvious reason is that single negated terms have little meaning. People cannot sensibly rate items for their typicality as "not sports". The category is infinite and indefinitely heterogeneous. Within a conjunctive phrase however the task is quite meaningful. Thus "Games which are not sports" is a concept for which participants can sensibly judge the membership and typicality of items. Two experiments are presented which explore both the extensional (category membership) and intensional (attribute listing) aspects of negated constituents in a conjunction. The aims of the research are primarily exploratory, as no previous work has looked at these issues within the framework of the current research tradition. Hypotheses can however be derived on the basis of earlier work within this paradigm. Hampton (1988b, Experiments 2, 3 and 4) asked people to make three categorization decisions about lists of items. In stage 1 of the experiment participants in the experiment made decisions first about whether a list of activities were (for example) "sports", and then whether they were "games". One week later the participants rated the same items again, this time for whether they were "Sports which are also games" (or "Games which are also sports") an apparently explicit Conceptual combination and negation 7 conjunction of the two sets. The main finding was that membership in the conjunction was not determined by Boolean set intersection. Rather, membership in the conjunction was generally a geometric average of the two constituent membership values. In addition the criterion for membership in the conjunction was set quite low, with the result that the conjunctions were often overextended. Chess, for example, was judged by many participants to be in the category "Game which is a sport", even though they had said before that it was not a Sport. The overextension effect occurred across a range of different conjunctions. These inconsistent class membership decisions were interpreted by Hampton (1987, 1988b) as evidence for a similarity based categorization process in which similarity to the Composite Prototype was the basis for categorization in the conjunction. Hampton (1988b) used a combined typicality and membership rating scale, in which participants first decided whether an item was a member of the category or not. If the answer was "yes" they then judged the item's typicality on a three point scale. If the answer was "no" they then judged its relatedness to the category on a three point scale. The two judgments were then combined into a seven point scale from +3 (highly typical) to -3 (unrelated). Hampton (1996b) extended these results to visually presented category materials such as cartoon faces or ambiguous colored letter shapes. Hampton (1988b) also found an asymmetry, or more properly a non-commutativity, in the conjunctions. Converse pairs of conjunctions like "Sports which are also games", and "Games which are also sports", had different graded structures in that the regression weight for a constituent concept predicting membership in the conjunction was higher when the concept was in the relative clause qualifier position than when it was the head noun. Over and above this positional effect, in several cases there was also an imbalance between the two constituent concepts. One concept (the dominant concept) tended to have a higher regression weight than the other, regardless of the order of the terms. This dominance effect has since been replicated in studies by Storms, De Boeck, van Mechelen, and Geeraerts (1993), (see also Storms, De Boeck, van Mechelen & Ruts, 1996). Conceptual combination and negation 8 Consider then what may be expected when the second constituent is negated (as in the class "sports which are not also games"). On the basis of the previous results (Hampton, 1988b) similar patterns may be expected to occur. The regression weight of the second constituent should of course have a negative sign, since the better an activity is as an example of Games, the worse it should be as an example of "not Games". Otherwise if the previous results are generalizable then membership in the conjunctive category should not be predictable simply on the basis of set intersection or set complementation, there should be a greater regression weight for a category when it is in the relative clause qualifier position than in the head noun position, and there should be dominance effects between the two constituent categories as before. Experiment 1 therefore had the aim of testing the degree to which the pattern of results obtained with conjunctions can be generalised to negated conjunctions. Experiment 1 had the secondary aim of testing two predictions of other models concerning conjunctive categorization through introducing two methodological changes. First, a between subjects design was used so that the probability of categorization could be used to provide a more direct test of overextension of conjunctions. If categorization probability for an item is measured independently for two constituent categories and for their conjunction, then if the conjunction is treated as an intersection of the two constituents, as predicted by classical models of concepts (Armstrong et al., 1983; Osherson & Smith, 1981), the probability with which an item is judged to belong in the conjunction should lie within certain limits. Let the measured proportion of the population sampled who believe that the item is in category A be p[A]. Then given p[A] and p[B] for a pair of constituent categories A and B, then, if the two beliefs are uncorrelated in the population, we would expect p[A&B], the probability that someone places the item in the conjunction "A which are also B", to be the product of the two probabilities p[A] and p[B]. There are however reasons why the beliefs may be correlated. First, global individual differences in the breadth of category boundaries would give rise to a positive correlation between believing that a given item is in Conceptual combination and negation 9 A and believing that it is in B. Second, variation in individual beliefs about the nature of the item being categorized could lead to either a positive or a negative correlation between p[A] and p[B] depending on how much the categories overlap or contrast. If the two category membership beliefs are fully negatively associated then the lower limit for p[A&B] should be zero or (p[A] + p[B] 1)/2 whichever is the greater. Alternatively if they are fully positively associated then the upper limit for p[A&B] should be the minimum of the two constituent probabilities. For example suppose that everyone holds the belief that all pets are animals. Then the probability that an item is classified as a Pet which is an Animal would be simply equal to the probability that it was classified as a Pet. Set logic applied to the incidence of beliefs in the population then predicts that p[A&B] should lie within the limits of zero or (p[A] + p[B] 1)/2 and the minimum of the two constituent probabilities. To the extent that this constraint is broken, there will be evidence against the view that the psychological representations of conjunctions are based on the logic of set intersection, as proposed by Osherson and Smith (1981). Furthermore, if the binary view of the separation between typicality effects and set membership espoused by Smith and Osherson (1984) is correct, then overextension, non-commutativity and concept dominance effects should be found in the mean rated typicalities, but should be much reduced, or even absent from the probability of categorization measure. Use of this design therefore helps to provide a test of Smith et al.'s theoretical position concerning the logical versus non-logical nature of concept conjunction. Conceptual combination and negation 10 The between subject design also helps settle a question about the validity of the earlier studies (Hampton, 1988b) which used within-subjects designs. Hampton (1988b) obtained categorization judgments in a fixed order, presenting constituent categories first, followed by conjunctions. If this order confound was in some way responsible for the findings of overextension in conjunctions, then overextension should be absent with a between subjects design. The second methodological change was introduced to provide a test of a modified extensional account of concept conjunctions proposed by Jones (1982), (see also Chater et al., 1990; Zadeh, 1982). The basis of this account is the idea that conjunctions are formed as a function of constituent class membership, but that conjunctive membership is rescaled, so that the best example of the set intersection becomes the prototype of the conjunctive set. Jones' (1982) conceptual stacks model proposed that people find the best fitting example of the two constituent sets, and then base their conjunction around this exemplar. Thus whatever instance happens to be the best example of a conjunction becomes maximally typical of the class. This proposal could also account for the emergent attributes and inheritance failure seen in Hampton (1987) if the conjunction's attributes are based on this maximal exemplar rather than on a combination of the two sets of constituent attributes. In order to test Jones' proposal, the materials for Experiment 1 deliberately included the best examples of the conjunctive concepts as generated by a pre-test. If his proposal is correct then at least some of these items should receive maximal typicality ratings for the conjunction (positive or negated) regardless of how typical they are in the constituent sets. Alternatively if Hampton's intensional model is correct, then it may be possible to find conjunctive classes with "empty centers", where the composite prototype formed from the two constituent prototypes does not correspond closely to any known instance, and so the "best known example" of the conjunction is still rated as relatively atypical. Experiment 1 Conceptual combination and negation 11 The primary aim of the first experiment was to explore the effects of negation in conceptual conjunctions. The following two questions were of particular importance concerning these negated conjunctions. First, how will membership in negated conjunctions be related to constituent membership? Will categorization probability for negated conjunctions (denoted as p[A&B]) follow the constraints of set logic? If not, then will negated conjunctions be overextended like other conjunctions? On the one hand, given that a conjunction is overextended with respect to its constituents, the corresponding negated conjunction may turn out to be underextended to a corresponding extent. On the other hand if overextension is the result of some decision process applying equally to all kinds of conjunction, then negated conjunctions may be equally overextended. Second, when mean membership/typicality values are considered will the negated conjunctions show effects of head versus modifier order and concept dominance parallel to those shown by their related conjunctions? The secondary aim was to test the generality of the earlier results with a between subjects design and with a more comprehensive selection of examples of the conjunctive categories. Two further hypotheses will then be tested; first whether categorization probability differs from mean typicality in the degree of non-logical effects observed, and second whether the most common exemplar in a conjunction is automatically rated as highly typical. Method Participants. Participants were 120 students at Stanford University who participated for course credit. They were randomly assigned, 20 to each of the 6 main conditions. A further 16 students participated in the pretest, also for course credit. Design and materials. The 6 pairs of overlapping concepts were taken from categories used in earlier research on this topic (Hampton, 1987, 1988b). The use of the same categories allowed the generality of the previous dominance effects to be tested. The concept pairs were Birds-Pets, Buildings-Dwellings, Furniture-Household Appliances, Conceptual combination and negation 12 Sports-Games, Tools-Weapons, and Vehicles-Machines. For each pair of categories, a new list of 20 items was created, containing exemplars of each concept, some in both concept categories, some in just one or the other, and others from the same general domain, but belonging to neither concept category. To ensure that the list contained the best examples of each conjunction, 16 participants performed an item generation task as a pretest. For each of the 6 conjunctions, participants listed as many examples as possible with no time constraint. (Order within conjunctions was balanced between two subgroups of participants). Between 25 and 45 different examples were generated. The 6 most frequently produced examples were then always included in the list of 20 selected items for each pair of categories. Items were typed in a randomly ordered vertical list headed by a category title. There were 2 lists to a page, and page order was randomized for each participant. Six versions of the booklet were created by varying the category title at the head of the item lists. Each version was given to a different group of 20 participants. Groups 1 and 2 each had one of the constituent concepts as category title (e.g. Bird for group 1 and Pet for group 2). Groups 3 and 4 each had a conjunction of the two concepts (e.g. group 3 had Birds which are also Pets, and group 4 had Pets which are also Birds). Groups 5 and 6 had the negated conjunction sets corresponding to groups 3 and 4 (e.g. group 5 had Birds which are not Pets, and group 6 had Pets which are not Birds). Procedure. The booklet was administered in a group testing session. A cover sheet contained instructions on how the items were to be rated. The rating scale was the same as used previously (Hampton, 1988a, 1988b). Participants first had to decide if an item belonged in the category, choosing a positive rating if it belonged, a negative rating if it did not, or a zero to indicate a borderline case. The positive and negative ratings each ranged from 1 to 3, corresponding either to the typicality of a category member (+1 = untypical, +3 = very typical), or to the relatedness of a category non-member (-1 = related, -3 = completely unrelated). If an item was unknown, participants could put a line through it. Those responses were then treated as missing data in the analyses. Conceptual combination and negation 13