Reasoning with quantifiers

In the semantics of natural language, quantification may have received more attention than any other subject, and one of the main topics in psychological studies on deductive reasoning is syllogistic inference, which is just a restricted form of reasoning with quantifiers. But thus far the semantical and psychological enterprises have remained disconnected. This paper aims to show how our understanding of syllogistic reasoning may benefit from semantical research on quantification. I present a very simple logic that pivots on the monotonicity properties of quantified statements--properties that are known to be crucial not only to quantification but to a much wider range of semantical phenomena. This logic is shown to account for the experimental evidence available in the literature as well as for the data from a new experiment with cardinal quantifiers ("at least n" and "at most n"), which cannot be explained by any other theory of syllogistic reasoning.     

Illusory inferences with quantifiers

The psychological study of reasoning with quantifiers has predominantly focused on inference patterns studied by Aristotle about two millennia ago. Modern logic has shown a wealth of inference patterns involving quantifiers that are far beyond the expressive power of Aristotelian syllogisms, and whose psychology should be explored. We bring to light a novel class of fallacious inference patterns, some of which are so attractive that they are tantamount to cognitive illusions. In tandem with recent insights from linguistics that quantifiers like “some” are treated as wh-questions, these illusory inferences are predicted by the erotetic theory of reasoning, which postulates that a process akin to question asking and answering is behind human inference making.     

Syllogisms with fractional quantifiers

Aristotle's syllogistic is extended to include denumerably many quantifiers such as more than 2/3 and exactly 2/3. Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given for claiming that syllogistic validity is relevant validity.     

Predicate provability logic with non-modalized quantifiers

Predicate modal formulas with non-modalized quantifiers (call them Q-formulas) are considered as schemata of arithmetical formulas, where is interpreted as the provability predicate of some fixed correct extension T of arithmetic. A method of constructing 1) non-provable in T and 2) false arithmetical examples for Q-formulas by Kripke-like countermodels of certain type is given. Assuming the means of T to be strong enough to solve the (undecidable) problem of derivability in QGL, the Q-fragment of the predicate version of the logic GL, we prove the recursive enumerability of the sets of Q-formulas all arithmetical examples of which are: 1) T-provable, 2) true. In. particular, the first one is shown to be exactly QGL and the second one to be exactly the Q-fragment of the predicate version of Solovay's logic S.     

Attentional focusing with quantifiers in production and comprehension

There is a very large number of quantifiers in English, so many that it seems impossible that the only information that they convey is about amounts. Building on the earlier work of Moxey and Sanford (1987), we report three experiments showing that positive and negative quantifiers focus on different subsets of the logical possibilities that quantifiers allow semantically. Experiments 1 and 2 feature a continuation task with quantifiers that span a full range of denotations (from near 0% to near 100%) and show that the effect is not restricted to quantifiers denoting small amounts. This enables a distinction to be made between generalization and complement set focus proper. The focus effects extend to comprehension, as shown by a self-paced reading study (Experiment 3). It is noted that the focus effects obtained are compatible with findings from earlier work by Just and Carpenter (1971), which used a verification paradigm, and in fact these effects constitute a direct test of inferences Just and Carpenter made about mechanisms of encoding negative quantifiers. A related but different explanation is put forward to explain the present data. The experiments show a quantifier function beyond the simple denotation of amount.     

A representation theorem for languages with generalized quantifiers through back-and-forth methods

We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.     

Cross-linguistic relations between quantifiers and numerals in language acquisition: Evidence from Japanese

A study of 104 Japanese-speaking 2- to 5-year-olds tested the relation between numeral and quantifier acquisition. A first study assessed Japanese children’s comprehension of quantifiers, numerals, and classifiers. Relative to English-speaking counterparts, Japanese children were delayed in numeral comprehension at 2 years of age but showed no difference at 3 and 4 years of age. Also, Japanese 2-year-olds had better comprehension of quantifiers, indicating that their delay was specific to numerals. A second study examined the speech of Japanese and English caregivers to explore the syntactic cues that might affect integer acquisition. Quantifiers and numerals occurred in similar syntactic positions and overlapped to a greater degree in English than in Japanese. Also, Japanese nouns were often dropped, and both quantifiers and numerals exhibited variable positions relative to the nouns they modified. We conclude that syntactic cues in English facilitate bootstrapping numeral meanings from quantifier meanings and that such cues are weaker in classifier languages such as Japanese.     

Bayesian analysis of structural equation models with mixed exponential family and ordered categorical data

Structural equation models are very popular for studying relationships among observed and latent variables. However, the existing theory and computer packages are developed mainly under the assumption of normality, and hence cannot be satisfactorily applied to non‐normal and ordered categorical data that are common in behavioural, social and psychological research. In this paper, we develop a Bayesian approach to the analysis of structural equation models in which the manifest variables are ordered categorical and/or from an exponential family. In this framework, models with a mixture of binomial, ordered categorical and normal variables can be analysed. Bayesian estimates of the unknown parameters are obtained by a computational procedure that combines the Gibbs sampler and the Metropolis–Hastings algorithm. Some goodness‐of‐fit statistics are proposed to evaluate the fit of the posited model. The methodology is illustrated by results obtained from a simulation study and analysis of a real data set about non‐adherence of hypertension patients in a medical treatment scheme.     

Progressing from programmatic to discovery research: a case example with the overjustification effect

Scientific research progresses along planned (programmatic research) and unplanned (discovery research) paths. In the current investigation, we attempted to conduct a single-case evaluation of the overjustification effect (i.e., programmatic research). Results of the initial analysis were contrary to the overjustification hypothesis in that removal of the reward contingency produced an increase in responding. Based on this unexpected finding, we conducted subsequent analyses to further evaluate the mechanisms underlying these results (i.e., discovery research). Results of the additional analyses suggested that the reward contingency functioned as punishment (because the participant preferred the task to the rewards) and that withdrawal of the contingency produced punishment contrast.     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

Function identification from noisy data with recursive error bounds

New success criteria of inductive inference in computational learning theory are introduced which model learning total (not necessarily recursive) functions with (possibly everywhere) imprecise theories from (possibly always) inaccurate data. It is proved that for any level of error allowable by the new success criteria, there exists a class of recursive functions such that not all f are identifiable via the criterion at that level of error. Also, necessary and sufficient conditions on the error level are given for when more classes of functions may be identified.     

Theory and data on developmental changes in novelty preference

The present study was designed to investigate the predictive value of an informational analysis of vicarious consequences. Two levels of amount of vicarious consequences (complete, partial) were varied with three types of vicarious consequences (reward, punishment, reward-punishment). Each of six groups of preschool age boys and girls observed an adult male model receive one of the combinations of amount and type of vicarious consequences for his choices from a commodity preference series. A seventh group of boys and girls observed the model receive no consequences for his choices. After observing the model, each subject was asked to make his own choices from the commodity preference series and to recall the choices which the model made. In contradiction to the predictions of an informational analysis of vicarious consequences, the amount of vicarious consequences did not differentially affect the recall of modeled behavior while the type of vicarious consequences did have a differential effect.     

Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs

Using a Monte Carlo simulation and the Kenward–Roger (KR) correction for degrees of freedom, in this article we analyzed the application of the linear mixed model (LMM) to a mixed repeated measures design. The LMM was first used to select the covariance structure with three types of data distribution: normal, exponential, and log-normal. This showed that, with homogeneous between-groups covariance and when the distribution was normal, the covariance structure with the best fit was the unstructured population matrix. However, with heterogeneous between-groups covariance and when the pairing between covariance matrices and group sizes was null, the best fit was shown by the between-subjects heterogeneous unstructured population matrix, which was the case for all of the distributions analyzed. By contrast, with positive or negative pairings, the within-subjects and between-subjects heterogeneous first-order autoregressive structure produced the best fit. In the second stage of the study, the robustness of the LMM was tested. This showed that the KR method provided adequate control of Type I error rates for the time effect with normally distributed data. However, as skewness increased—as occurs, for example, in the log-normal distribution—the robustness of KR was null, especially when the assumption of sphericity was violated. As regards the influence of kurtosis, the analysis showed that the degree of robustness increased in line with the amount of kurtosis.     

The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features

A method is discussed which extends principal components analysis to the situation where the variables may be measured at a variety of scale levels (nominal, ordinal or interval), and where they may be either continuous or discrete. There are no restrictions on the mix of measurement characteristics and there may be any pattern of missing observations. The method scales the observations on each variable within the restrictions imposed by the variable's measurement characteristics, so that the deviation from the principal components model for a specified number of components is minimized in the least squares sense. An alternating least squares algorithm is discussed. An illustrative example is given.Copies of this paper and of the associated PRINCIPALS program may be obtained by writing to Forrest W. Young, Psychometric Laboratory, Davie Hall 013-A, Chapel Hill, NC 27514.     

Testing conclusions from functional imaging of working memory with data from acute stroke

Functional imaging studies indicate that the left hemisphere mediates verbal working memory, while the right hemisphere mediates both verbal and spatial working memory. We evaluated acute stroke patients with working memory tests and imaging to identify whether unilateral dysfunction causes deficits in spatial and/or verbal working memory deficits. While left cortical stroke patients had verbal working memory impairments (p<0.003), right cortical stroke patients had both verbal p<0.007) and spatial working memory (p<0.03) impairments, confirming functional imaging results. Patients with transient ischemic stroke and patients with non-cortical stroke did not have significant deficits in working memory in either modality.     

Modelling human problem solving with data from an online game

Since the beginning of cognitive science, researchers have tried to understand human strategies in order to develop efficient and adequate computational methods. In the domain of problem solving, the travelling salesperson problem has been used for the investigation and modelling of human solutions. We propose to extend this effort with an online game, in which instances of the travelling salesperson problem have to be solved in the context of a game experience. We report on our effort to design and run such a game, present the data contained in the resulting openly available data set and provide an outlook on the use of games in general for cognitive science research. In addition, we present three geometrical models mapping the starting point preferences in the problems presented in the game as the result of an evaluation of the data set.     

Theory and data interactions of the scientific mind: evidence from the molecular and the cognitive laboratory.

A number of researchers and scholars have stressed the importance of disconfirmation in the quest for the development of scientific knowledge (e.g., Popper, 1959). Paradoxically, studies examining human reasoning in the laboratory have typically found that people display a confirmation bias in that they are more likely to seek out and attend to data consistent rather than data inconsistent with their initial theory (Wason, 1968). We examine the strategies that scientists and students use to evaluate data that are either consistent or inconsistent with their expectations. First, we present findings from scientists reasoning "live" in their laboratory meetings. We show that scientists often show an initial reluctance to consider inconsistent data as "real." However, this initial reluctance is often overcome with repeated observations of the inconsistent data such that they modify their theories to account for the new data. We further examine these issues in a controlled scientific causal thinking simulation specifically developed to examine the reasoning strategies we observed in the natural scientific environment. Like the scientists, we found that participants in our simulation initially displayed a propensity to discount data inconsistent with a theory provided. However, with repeated observations of the inconsistent data, the students, like the scientists, began to see the once anomalous data as "real" and the initial bias to discount that data was significantly diminished.     

The Underlying Theory: Drawn from Experience with Individuals and Groups

This article is based on a presentation Carl made at the University of California, Irvine, on November 8, 1986. Carl focuses on the conditions that facilitate change—both in individuals and groups. Carl connects his theory of personal change with the process for achieving peace.