Nonmonotonic extrapolation in function learning

This article reports the results of an experiment addressing extrapolation in function learning, in particular the issue of whether participants can extrapolate in a nonmonotonic manner. Existing models of function learning, including the extrapolation association model of function learning (EXAM; E. L. DeLosh, J. R. Busemeyer, & M. A. McDaniel, 1997), cannot account for this type of extrapolation pattern. We present the results of an experiment in which participants were shown a series of paired stimulus-response magnitudes where the relationship between these 2 dimensions conformed to a cyclic function. Participants were shown to extrapolate from these training data in a nonmonotonic way, contrary to predictions from EXAM. A new model of function learning is presented, which predicts responses more accurately than EXAM.     

Intuitive Hedonism

The hoary philosophical tradition of hedonism – the view that pleasure is the basic ethical or normative value – suggests that it is at least reasonably and roughly intuitive. But philosophers no longer treat hedonism that way. For the most part, they think that they know it to be obviously false on intuitive grounds, much more obviously false on such grounds than familiar competitors. I argue that this consensus is wrong. I defend the intuitive cogency of hedonism relative to the dominant desire-based and objectivist conceptions of well-being and the good. I argue that hedonism is still a contender, and indeed that our current understanding of commonsense intuition on balance supports it. An erratum to this article is available at .     

Intuitive knowledge

In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? A natural idea about perception is that a perception makes a belief amount to knowledge in part by making you sensorily aware of the concrete objects it is about. The analogous idea about intuition is that an intuition makes a belief amount to knowledge in part by making you intellectually aware of the abstract objects it is about. I expand both ideas into fuller accounts of perceptual and intuitive knowledge, explain the main challenge to this sort of account of intuitive knowledge (i.e. the challenge of making sense of intellectual awareness), and develop a response to it.     

Intuitive covariation estimation

Six experiments concerned people's ability to estimate the degree and sign of covariation represented in a bivariate distribution of stimuli with which they had just been presented as a series of pairs of stimuli. The stimuli were pairs of numbers, pairs of lines of variable lengths, or word-line pairs. In the latter case, subjects were asked to think of the words in terms of either pleasantness or familiarity; hence, the covariation relationship was between the normative pleasantness or familiarity value of the word and a line of variable length. In the sixth experiment, subjects were presented with two word-line pairs and were asked to access the covariation of both simultaneously. In most cases, the estimates reflected the sign and degree of covariation of the stimuli quite well. The estimates did not reflect accurately the stimulus covariation when the stimuli were numerical and the sign of the correlation of the stimuli was negative. A distinction is made between intuitive and strategic processes in the assessment of covariation.     

Method of serial extrapolation

A psychophysical procedure is proposed for measuring a subject's power of discerning “lawfulness,” “regularity,” “invariance” in a stimulus sequence. The task consists of supplying the next term in a given series. The method is tested by applying it to a visual series consisting of lines subdivided in geometric progression. The effect upon accuracy of judgement is investigated by varying the settings of the stimulus, in particular the rate of increase or gradient of the series and the number of segments given prior to extrapolation, and of presenting the next-but-one position in the series, thereby converting the task into one of interpolation. Where possible, quantitative predictions are made concerning these variables on the basis of certain explicit assumptions.

Results show that accuracy decreases, as predicted, as the gradient of the series increases, but does not seem to be affected by adding to the number of terms presented. Interpolation shows a reduction in variable error compared with extrapolation more or less as required by theory. Finally, discrepancies in the findings are discussed with special reference to their bearing on the validity of the method.     

Intuitive halving and doubling of figures

Summary In the first part of this paper we studied the factors which influenced older children and adults in solving the task of doubling and halving simple figures and solids. In the second we supplemented our results by a qualitative analysis of the behavior of younger children in the performance of similar tasks.

---

Zusammenfassung Im 1. Teil dieser Arbeit untersuchten wir die Faktoren, welche das Verhalten von ?lteren Kindern und Erwachsenen bestimmen, wenn sie die Aufgabe haben, zu einer gegebenen Figur eine doppelt oder halb so gro?e zu finden. Einige Versuche wurden statt an ebenen Figuren an einfachen K?rpern angestellt. Im 2. Teil erweiterten wir unsere Befunde durch eine qualitative Analyse der Vorg?nge, die unter ?hnlichen Bedingungen bei jüngeren Kindern auftreten.

---

---

(From the Research Laboratory of Smith College.)

---

With 4 Figures.     

The conceptual basis of function learning and extrapolation: Comparison of rule-based and associative-based models

The purpose of this article is to provide a foundation for a more formal, systematic, and integrative approach to function learning that parallels the existing progress in category learning. First, we note limitations of existing formal theories. Next, we develop several potential formal models of function learning, which include expansion of classic rule-based approaches and associative-based models. We specify for the first time psychologically based learning mechanisms for the rule models. We then present new, rigorous tests of these competing models that take into account order of difficulty for learning different function forms and extrapolation performance. Critically, detailed learning performance was also used to conduct the model evaluations. The results favor a hybrid model that combines associative learning of trained input—prediction pairs with a rule-based output response for extrapolation (EXAM).     

Intuitive Covariation Assessment of the Illusory Correlation

Most of the research concerned with the illusory correlation is modeled after the seminal work of D. L. Hamilton and R. K. Gifford (1976). However, S. A. Haslam and C. McGarty (1994) have voiced concerns over the dependent measures used within this paradigm. Therefore, in this study, the authors tested a new dependent variable that has high face validity. This measure was modeled after the work of J. R. McGahan and R. Wight (1989) and consisted of a set of propositional statements representing either the illusory correlation, the contingency opposite the illusory correlation, or the noncontingency. A second purpose of this study was to validate other studies that have used dependent measures modeled after the work of J. R. McGahan and R. Wight (1989). Demonstrating that this measure can be used to detect a well-documented phenomenon (i.e., the illusory correlation) would strengthen the results and conclusions from other studies. To this end, results from 2 experiments indicate that this measure does provide a valid alternative to those measures that are commonly used in illusory correlation studies. The results thereby give credence to other studies that have used similar dependent measures.     

Intuitive covariation assessment of the illusory correlation

Most of the research concerned with the illusory correlation is modeled after the seminal work of D. L. Hamilton and R. K. Gifford (1976). However, S. A. Haslam and C. McGarty (1994) have voiced concerns over the dependent measures used within this paradigm. Therefore, in this study, the authors tested a new dependent variable that has high face validity. This measure was modeled after the work of J. R. McGahan and R. Wight (1989) and consisted of a set of propositional statements representing either the illusory correlation, the contingency opposite the illusory correlation, or the noncontingency. A second purpose of this study was to validate other studies that have used dependent measures modeled after the work of J. R. McGahan and R. Wight (1989). Demonstrating that this measure can be used to detect a well-documented phenomenon (i.e., the illusory correlation) would strengthen the results and conclusions from other studies. To this end, results from 2 experiments indicate that this measure does provide a valid alternative to those measures that are commonly used in illusory correlation studies. The results thereby give credence to other studies that have used similar dependent measures.     

Intuitive probabilistic inference in capuchin monkeys

The ability to reason about probabilities has ecological relevance for many species. Recent research has shown that both preverbal infants and non-human great apes can make predictions about single-item samples randomly drawn from populations by reasoning about proportions. To further explore the evolutionary origins of this ability, we conducted the first investigation of probabilistic inference in a monkey species (capuchins; Sapajus spp.). Across four experiments, capuchins (N = 19) were presented with two populations of food items that differed in their relative distribution of preferred and non-preferred items, such that one population was more likely to yield a preferred item. In each trial, capuchins had to select between hidden single-item samples randomly drawn from each population. In Experiment 1 each population was homogeneous so reasoning about proportions was not required; Experiments 2–3 replicated previous probabilistic reasoning research with infants and apes; and Experiment 4 was a novel condition untested in other species, providing an important extension to previous work. Results revealed that at least some capuchins were able to make probabilistic inferences via reasoning about proportions as opposed to simpler quantity heuristics. Performance was relatively poor in Experiment 4, so the possibility remains that capuchins may use quantity-based heuristics in some situations, though further work is required to confirm this. Interestingly, performance was not at ceiling in Experiment 1, which did not involve reasoning about proportions, but did involve sampling. This suggests that the sampling task posed demands in addition to reasoning about proportions, possibly related to inhibitory control, working memory, and/or knowledge of object permanence.     

Efficient extrapolation of the view with a dynamic and predictive stimulus

In the real-world, the retinal projection of an object changes as we move, or as a moving object passes in front of us. We have to recognise objects, despite such retinal-projection changes. Many studies have shown that the time required to identify objects after a change in the retinal projection is longer than when there is no retinal-projection change. This recognition cost is referred to as the view-dependent effect. Previous researchers have studied the view-dependent effect while disregarding the predictability of retinal-projection changes. Here, we demonstrate that there is no view-dependent effect when the predictability is introduced, in the case where participants track moving objects by head-turning or eye-movement in a virtual environment. Violation of the predictability, such as an unpredictable retinal-projection change or a movement of the first stimulus that was inconsistent with a subsequent retinal-projection change, caused a view-dependent effect. Moreover, we found that extraretinal information such as head-turning or eye-movement was unnecessary for view-independent recognition. These results indicate that humans can extrapolate to the tested view from the studied view in a view-independent way when retinal-projection change is predictable from the visual stimulus.