Children's predictions of consistency in people's actions

Past research suggests that young children are often reluctant to generalize about people's behavior. Three experiments involving 102 4-5-year-olds, 84 7-8-year-olds, and 107 adults explored the conditions under which inductive inferences about people are made. There was an age-based increase in propensity to predict consistency in psychological/intentional causal relations. Children often predicted change; people would behave differently in the future than they did in the past. Younger children limited predictions of consistency to non-psychological contexts. Older children showed some appreciation of stable motivations (e.g. traits, preferences). The results are consistent with the hypothesis that children's theories of mind emphasize situational influences, with personal influences appearing in middle-childhood.     

Empiricism,probability, and knowledge of arithmetic: A preliminary defense

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgments of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity (Section 2.1), while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic (Section 2.2). Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the ω-rule (Section 3.1) or to the non-computability and non-continuity of probability assignments (Section 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments (Appendix A–Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4 Appendix B). In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar.     

Second order inductive logic and Wilmers' principle

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship with the first order principles of Regularity and Super Regularity.     

An observation on Carnapʼs Continuum and stochastic independencies

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson?s Sufficientness Principle holds, thus giving Carnap?s Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap?s Continuum for the language with just a single predicate.     

The law of large numbers in children's diversity-based reasoning

Adults increase the certainty of their inductive inferences by observing more diverse instances. However, most young children fail to do so. The present study tested the hypothesis that children's sensitivity to instance diversity is determined by three variables: ability to discriminate among instances (Discrimination); an intuition that large numbers of instances increase the strength of conclusion (Monotonicity); ability to detect subcategories and evaluate numerical differences between the subcategories, or Extraction. A total of 219 Chinese children aged 6 to 11 were tested for sensitivity to diversity by means of Discrimination, Monotonicity, and Extraction. The results indicated that children at all ages were able to discriminate instances and attend to set size. However, only 9- and 11-year-olds demonstrated Extraction and sensitivity to diversity. Furthermore, among all children diversity scores increased linearly with the level of Extraction. These results suggest that the law of large numbers plays a role in children's diversity-based reasoning.     

A computational analysis of general intelligence tests for evaluating cognitive development

The progression in several cognitive tests for the same subjects at different ages provides valuable information about their cognitive development. One question that has caught recent interest is whether the same approach can be used to assess the cognitive development of artificial systems. In particular, can we assess whether the ‘fluid’ or ‘crystallised’ intelligence of an artificial cognitive system is changing during its cognitive development as a result of acquiring more concepts? In this paper, we address several IQ tests problems (odd-one-out problems, Raven’s Progressive Matrices and Thurstone’s letter series) with a general learning system that is not particularly designed on purpose to solve intelligence tests. The goal is to better understand the role of the basic cognitive operational constructs (such as identity, difference, order, counting, logic, etc.) that are needed to solve these intelligence test problems and serve as a proof-of-concept for evaluation in other developmental problems. From here, we gain some insights into the characteristics and usefulness of these tests and how careful we need to be when applying human test problems to assess the abilities and cognitive development of robots and other artificial cognitive systems.     

Sample diversity and premise typicality in inductive reasoning: evidence for developmental change

Evaluating whether a limited sample of evidence provides a good basis for induction is a critical cognitive task. We hypothesized that whereas adults evaluate the inductive strength of samples containing multiple pieces of evidence by attending to the relations among the exemplars (e.g., sample diversity), six-year-olds would attend to the degree to which each individual exemplar in a sample independently appears informative (e.g., premise typicality). To test these hypotheses, participants were asked to select between diverse and non-diverse samples to help them learn about basic-level animal categories. Across various between-subject conditions (N=133), we varied the typicality present in the diverse and non-diverse samples. We found that adults reliably selected to examine diverse over non-diverse samples, regardless of exemplar typicality, six-year-olds preferred to examine samples containing typical exemplars, regardless of sample diversity, and nine-year-olds were somewhat in the midst of this developmental transition.     

Learning How to Generalize

Generalization is a fundamental problem solved by every cognitive system in essentially every domain. Although it is known that how people generalize varies in complex ways depending on the context or domain, it is an open question how people learn the appropriate way to generalize for a new context. To understand this capability, we cast the problem of learning how to generalize as a problem of learning the appropriate hypothesis space for generalization. We propose a normative mathematical framework for learning how to generalize by learning inductive biases for which properties are relevant for generalization in a domain from the statistical structure of features and concepts observed in that domain. More formally, the framework predicts that an ideal learner should learn to generalize by either taking the weighted average of the results of generalizing according to each hypothesis space, with weights given by how well each hypothesis space fits the previously observed concepts, or by using the most likely hypothesis space. We compare the predictions of this framework to human generalization behavior with three experiments in one perceptual (rectangles) and two conceptual (animals and numbers) domains. Across all three studies we find support for the framework's predictions, including individual‐level support for averaging in the third study.     

Developing knowledge of objects' motion properties in infancy

Three experiments with a novel variation of the inductive generalization procedure examined 18- and 22-month-olds' knowledge of objects' motion properties. Infants observed simple air and land movements modeled with an appropriate category member (e.g. dog) or an ambiguous block and were allowed to imitate with one or more of four exemplars. The experiments show that 18-month-olds' knowledge of land motions is grounded in causally relevant object parts, whereas 22-month-olds relate such motions more broadly to appropriate category members. Infants' basis for generalizing air motions suggested that at 22 months they have little knowledge about objects from that domain. The results are discussed in relation to the early development of the animate-inanimate distinction and the nature of the inductive generalization task.