The Idea of an Exact Number: Children's Understanding of Cardinality and Equinumerosity

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one‐to‐one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood.     

Melting Lizards and Crying Mailboxes: Children's Preferential Recall of Minimally Counterintuitive Concepts

Previous research with adults suggests that a catalog of minimally counterintuitive concepts, which underlies supernatural or religious concepts, may constitute a cognitive optimum and is therefore cognitively encoded and culturally transmitted more successfully than either entirely intuitive concepts or maximally counterintuitive concepts. This study examines whether children's concept recall similarly is sensitive to the degree of conceptual counterintuitiveness (operationalized as a concept's number of ontological domain violations) for items presented in the context of a fictional narrative. Seven‐ to nine‐year‐old children who listened to a story including both intuitive and counterintuitive concepts recalled the counterintuitive concepts containing one (Experiment 1) or two (Experiment 2), but not three (Experiment 3), violations of intuitive ontological expectations significantly more and in greater detail than the intuitive concepts, both immediately after hearing the story and 1 week later. We conclude that one or two violations of expectation may be a cognitive optimum for children: They are more inferentially rich and therefore more memorable, whereas three or more violations diminish memorability for target concepts. These results suggest that the cognitive bias for minimally counterintuitive ideas is present and active early in human development, near the start of formal religious instruction. This finding supports a growing literature suggesting that diverse, early‐emerging, evolved psychological biases predispose humans to hold and perform religious beliefs and practices whose primary form and content is not derived from arbitrary custom or the social environment alone.     

Array training in a categorization task

Two components of categorization, within-category commonalities and between-category distinctiveness, were investigated in a categorization task. Subjects learned three prototype categories composed of moderately high distortions, by observing arrays containing patterns that belonged either to a common prototype category or to three different categories; a third group learned patterns presented one at a time, mirroring the standard paradigm. Following 6 learning blocks, subjects transferred to old patterns and new patterns at low-, medium-, and high-level distortions of the category prototype. The results showed that array training facilitated learning, especially when patterns in the array belonged to the same category. Transfer results showed a strong gradient effect across pattern distortion level for all conditions, with the highest performance obtained following array training on different category patterns and worst in the control condition. Interestingly, the old training patterns were classified worse than new low and no better than medium distortions. Neither this ordering nor the steepness of the gradient across prototype similarity for each condition could be predicted by the generalized context model. A prototype model better captured the steep gradient and ordinal pattern of results, although the overall fits were only slightly better than the exemplar model. The crucial role played by category commonalities and distinctiveness on categorical representations is addressed.     

A simplicity principle in unsupervised human categorization

We address the problem of predicting how people will spontaneously divide into groups a set of novel items. This is a process akin to perceptual organization. We therefore employ the simplicity principle from perceptual organization to propose a simplicity model of unconstrained spontaneous grouping. The simplicity model predicts that people would prefer the categories for a set of novel items that provide the simplest encoding of these items. Classification predictions are derived from the model without information either about the number of categories sought or information about the distributional properties of the objects to be classified. These features of the simplicity model distinguish it from other models in unsupervised categorization (where, for example, the number of categories sought is determined via a free parameter), and we discuss how these computational differences are related to differences in modeling objectives. The predictions of the simplicity model are validated in four experiments. We also discuss the significance of simplicity in cognitive modeling more generally.     

On Fodor's First Law of the Nonexistence of Cognitive Science

In his enormously influential The Modularity of Mind, Jerry Fodor (1983) proposed that the mind was divided into input modules and central processes. Much subsequent research focused on the modules and whether processes like speech perception or spatial vision are truly modular. Much less attention has been given to Fodor's writing on the central processes, what would today be called higher‐level cognition. In “Fodor's First Law of the Nonexistence of Cognitive Science,” he argued that central processes are “bad candidates for scientific study” and would resist attempts at empirical analysis. This essay evaluates his argument for this remarkable claim, concluding that although central processes may well be “messier” than input modules, this does not mean that they cannot be studied and understood. The article briefly reviews the scientific progress made in understanding central processes in the 35 years since the book was published, showing that Fodor's prediction is clearly falsified by massive advances in topics like decision making and analogy. The essay concludes that Fodor's Law was not based on a clear argument for why the complexities of central systems could not be studied but was likely based on intuitions and preferences that were common in psychology at the time.     

Feature Statistics Modulate the Activation of Meaning During Spoken Word Processing

Understanding spoken words involves a rapid mapping from speech to conceptual representations. One distributed feature‐based conceptual account assumes that the statistical characteristics of concepts’ features—the number of concepts they occur in (distinctiveness/sharedness) and likelihood of co‐occurrence (correlational strength)—determine conceptual activation. To test these claims, we investigated the role of distinctiveness/sharedness and correlational strength in speech‐to‐meaning mapping, using a lexical decision task and computational simulations. Responses were faster for concepts with higher sharedness, suggesting that shared features are facilitatory in tasks like lexical decision that require access to them. Correlational strength facilitated responses for slower participants, suggesting a time‐sensitive co‐occurrence‐driven settling mechanism. The computational simulation showed similar effects, with early effects of shared features and later effects of correlational strength. These results support a general‐to‐specific account of conceptual processing, whereby early activation of shared features is followed by the gradual emergence of a specific target representation.     

A standard conceptual framework for the study of subjective time

Research on the mental representation of time (‘subjective time’) has provided broad insights into the nature of time perception and temporal processing. As the field comprises different scientific disciplines, such as psychology, philosophy, and neuroscience, studies differ with regard to the basic terms and concepts used. For this reason, research on subjective time lacks a coherent conceptual system. We argue that research in the field of subjective time should aim at establishing such a system, i.e., a more standardized terminology, in order to strengthen its theoretical basis and to support an efficient communication of results. Based on key empirical findings and concepts that are commonly (but inconsistently) used in the literature, we argue for a conceptual framework for the study of subjective time that differentiates between three types of mental representations of time: basic temporal processing, time perception in terms of passage, and time perception in terms of duration.     

The paradox of the universal triangle: Concepts,language, and prototypes

For over 300 years, the humble triangle has served as the paradigmatic example of the problem of abstraction. How can we have the idea of a general triangle even though every experience with triangles is with specific ones? Classical cognitive science seemed to provide an answer in symbolic representation. With its easily enumerated necessary and sufficient conditions, the triangle would appear to be an ideal candidate for being represented in a symbolic form. I show that it is not. Across a variety of tasks—drawing, speeded recognition, unspeeded visual judgments, and inference—representations of triangles appear to be graded and context dependent. I show that using the category name “triangle” activates a more prototypical representation than using an arguably coextensive cue, “three-sided polygon”. For example, when asked to draw “triangles” people draw more typical triangles than when asked to draw “three-sided polygons”. Altogether, the results support the view that (even formal) concepts have a graded and flexible structure, which takes on a more prototypical and stable form when activated by category labels.     

Categories of Large Numbers in Line Estimation

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time are expected near the boundaries between categories. In contrast to previous work (Landy, Silbert, & Goldin, 2013) that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers.     

The empirical case for role-governed categories

Most theories of categorization posit feature-based representations. Markman and Stilwell (2001) argued that many natural categories name roles in relational systems and therefore they are role-governed categories. There is little extant empirical evidence to support the existence of role-governed categories. Three experiments examine predictions for ways that role-governed categories should differ from feature-based categories. Experiment 1 shows that our knowledge of role-governed categories, in contrast to feature-based categories, is largely about properties extrinsic to category members. Experiment 2 shows that role-governed categories have more prominent ideals than feature-based categories. Experiment 3 demonstrates that novel role-governed categories are licensed by the instantiation of novel relational structures. We then discuss broader implications for the study of categories and concepts.