Linear Measurement Models-Axiomatizations and Axiomatizability

This paper examines, in the scope of representational measurement theory, different axiomatizations and axiomatizability of linear and bilinear representations of ordinal data contexts in real vector spaces. The representation theorems proved in this paper are modifications and generalizations of Scott's characterization of finite linear measurement models. The advantage of these representation theorems is that they use only finitely many axioms, the number of which depends on the size of the given ordinal data context. Concerning the axiomatizability, it is proved by model-theoretic methods that finite linear measurement models cannot be axiomatized by a finite set of first order axioms. Copyright 2000 Academic Press.     

Free trees and bidirectional trees as representations of psychological distance

Two types of graph-theoretic representations of psychological distances or dissimilarities are proposed: weighted free trees and weighted bidirectional trees. A weighted free tree is a generalization of the sort of graph representation used in hierarchical clustering. A weighted bidirectional tree is a further generalization which allows for asymmetric dissimilarities. The properties of these structures are discussed and numerical methods are presented that can be used to derive a representation for any given set of dissimilarities. The applicability of these structures is illustrated by using them to represent the data from experiments on the similarities among animal terms and on memory for sentences. An analysis based upon free trees is compared and contrasted with analyses using hierarchical clustering and multidimensional scaling.     

A parametric procedure for ultrametric tree estimation from conditional rank order proximity data

The psychometric and classification literatures have illustrated the fact that a wide class of discrete or network models (e.g., hierarchical or ultrametric trees) for the analysis of ordinal proximity data are plagued by potential degenerate solutions if estimated using traditional nonmetric procedures (i.e., procedures which optimize a STRESS-based criteria of fit and whose solutions are invariant under a monotone transformation of the input data). This paper proposes a new parametric, maximum likelihood based procedure for estimating ultrametric trees for the analysis of conditional rank order proximity data. We present the technical aspects of the model and the estimation algorithm. Some preliminary Monte Carlo results are discussed. A consumer psychology application is provided examining the similarity of fifteen types of snack/breakfast items. Finally, some directions for future research are provided.     

Short-term recognition memory for serial order and timing

Recent evidence suggests that a common temporal representation underlies memory for serial order of items in a sequence, and the timing of items in a sequence. This stands in contrast to other data suggesting a reliance on only ordinal information in short-term memory tasks. An experiment is reported here in which participants were post cued to perform a comparison between a probe and study list of items irregularly spaced in time, on the basis of order or temporal information.Participants' performance on theserial recognition task was notaffected by the temporal proximity of items, although participants were able to use temporal information to perform a temporal recognition task. Application of a temporal matching model of serial and temporal recognition suggests that although participants were able to remember the timing of items, this memory for timing was unlikely to determine serial recognition performance. The results suggest a dissociation between ordinal and temporal information in short-term memory.     

Ordinality and novel sequence learning in jackdaws

A hallmark of higher cognition is the flexible use of information. This requires an abstract representation of the information. In sequence learning, ordinal position knowledge is seen as a more versatile representation when compared to chaining. Here, we assessed which of these mental representations is the most natural and most dominant in jackdaws. Two jackdaws (Corvus monedula) were trained on 14 separate three-item sequences (triplets), made up of abstract images. On each trial, the three items of one triplet were presented in fixed order. The images represented either the first, second or third ordinal position. Test stimuli consisted of the three images and a distractor image that was chosen randomly from the remaining sequences. We rewarded pecking in the correct order to the images belonging to the same sequence. The most common error the birds made was to peck at a distractor item from the same ordinal position. To look at how versatile the jackdaws' ordinal knowledge was, we replaced a familiar item with a novel item in some sequences. We then created novel sequences with these items, which the birds completed correctly. It appears, then, that jackdaws have a concept of ordinal position.     

Numerical ordering ability mediates the relation between number-sense and arithmetic competence

What predicts human mathematical competence? While detailed models of number representation in the brain have been developed, it remains to be seen exactly how basic number representations link to higher math abilities. We propose that representation of ordinal associations between numerical symbols is one important factor that underpins this link. We show that individual variability in symbolic number-ordering ability strongly predicts performance on complex mental-arithmetic tasks even when controlling for several competing factors, including approximate number acuity. Crucially, symbolic number-ordering ability fully mediates the previously reported relation between approximate number acuity and more complex mathematical skills, suggesting that symbolic number-ordering may be a stepping stone from approximate number representation to mathematical competence. These results are important for understanding how evolution has interacted with culture to generate complex representations of abstract numerical relationships. Moreover, the finding that symbolic number-ordering ability links approximate number acuity and complex math skills carries implications for designing math-education curricula and identifying reliable markers of math performance during schooling.     

An information theoretic measure for the evaluation of ordinal scale data

This article describes a new measure of dispersion as an indication of consensus and dissention. Building on the generally accepted Shannon entropy, this measure utilizes a probability distribution and the ordered ranking of categories in an ordinal scale distribution to yield a value confined to the unit interval. Unlike other measures that need to be normalized, this measure is always in the interval 0 to 1. The measure is typically applied to the Likert scale to determine degrees of agreement among ordinal-ranked categories when one is dealing with data collection and analysis, although other scales are possible. Using this measure, investigators can easily determine the proximity of ordinal data to consensus (agreement) or dissention. Consensus and dissention are defined relative to the degree of proximity of values constituting a frequency distribution on the ordinal scale measure. The authors identify a set of criteria that a measure must satisfy in order to be an acceptable indicator of consensus and show how the consensus measure satisfies all the criteria.     

The construction of spatial situation models during reading

Psycholinguistic research faces a major challenge in describing the mental representations readers construct from a text. It is now widely accepted that readers end with a representation of the situation described in the text. However, it is unclear whether this representation allows the activation of elements in accordance with their situation proximity. To answer this question, two experiments were conducted. Participants read texts, sentence by sentence, which gave them instructions about how to arrange items in a layout; they then performed a recognition task. By manipulating the spatial proximity between prime and probe items, this task allowed the measurement of a spatial priming effect. In the first experiment, a larger priming effect was observed for closer items on the spatial layout. The second experiment replicated these findings and showed that the priming effects are better explained by categorical distance than by Euclidean distance.     

Representational constraints on children's suggestibility

ABSTRACT— In a multistage experiment, twelve 4- and 9-year-old children participated in a triad rating task. Their ratings were mapped with multidimensional scaling, from which euclidean distances were computed to operationalize semantic distance between items in target pairs. These children and age-mates then participated in an experiment that employed these target pairs in a story, which was followed by a misinformation manipulation. Analyses linked individual and developmental differences in suggestibility to children's representations of the target items. Semantic proximity was a strong predictor of differences in suggestibility: The closer a suggested distractor was to the original item's representation, the greater was the distractor's suggestive influence. The triad participants' semantic proximity subsequently served as the basis for correctly predicting memory performance in the larger group. Semantic proximity enabled a priori counterintuitive predictions of reverse age-related trends to be confirmed whenever the distance between representations of items in a target pair was greater for younger than for older children.     

Text representations as reflected in patterns of cognitive distance

Subjects performed a task that involved learning a text and then indicating, for each of a series of pairs of words, whether they belonged to the same sentence of that text. When the principle of argument overlap or causal coherence was the sole integrative factor in a text, negative response times and error rates reflected distances in the network representation predicted by the principle. When the two principles predicted conflicting representations for a text, response times and error rates reflected the predictions of the principle of argument overlap. The results were interpreted as suggesting that network connections predicted by the two principles are both present in the memory representations of texts and that causal connections are recorded at a more abstract level of representation than are argument overlap connections.     

Orienting numbers in mental space: Horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.     

Orienting numbers in mental space: horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.     

Social representations of drugs,causal judgment and social perception

The aim of this study was to analyse the influence of social representation on both social perceptions and social judgments. In the first stage of this study, 47 drug users and 80 ‘normal’ subjects were asked to respond to a questionnaire about representations of drugs. Three weeks later we contacted the same subjects. They were asked to answer some questions about a fictitious story in which an actor labelled as ‘a drug user’ or ‘a person’ disputed with a trader. Three different social representations of drugs were found. It was shown that these social representations were anchored in different social groups which were defined by their proximity to the world of drugs. Subjects who were themselves drug users shared an accepting or an ambivalent social representation of drugs but they also made the most negative judgements about the causes of a fictitious dispute between a trader and a drug addict. Moreover, these subjects had the most negative perception of the drug addict. Furthermore, some factors which increase the salience of social representations were studied. The effect on social perception and causal attributions of the interaction between social representations, the context and personal involvement in drugs was also shown. Some relations between the theory of social representations and the theories about asymmetrical intergroup relationships are exposed.     

The Structural Memory: A network model for human perception of serial objects

Summary The Structural Memory, a network model for human perception of serial objects, such as series, is presented. Our theoretical assumptions originate from the Structural Information (Theory (SIT) (Buffart & Leeuwenberg, 1983; Leeuwenberg, 1969), a theory concerning the perceptible structures in an object and the human preference for one of these structures. A symbolic notation of such a perceptible structure in an object is called a representation. For a given maximum number of symbols we can generate all representations automatically. From this procedure we define G(eneration)-relations between the representations. We also define S(tructure)-relations based upon the structures described by the representations. The representations and relations can be seen as respectively the nodes and links in a network. This network is the basis for the Structural Memory. We therefore assign an activation value to each representation in the network, expressing the strength of the preference for the described structure at a certain moment. By means of a process model we are able to make predictions for the strength of the preference for a perceptible structure in an object. The process is only based upon the network structure, because of a relation found between the preference measure, used in SIT, for a perceptible structure in an object and the number of G- and S-relations of the corresponding representation in the network. It is shown that with two of these process models some experiments by Van Leeuwen and Buffart can be simulated.     

Additive similarity trees

Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer program, ADDTREE, for the construction of additive trees is described and applied to several sets of data. A comparison of these results to the results of multidimensional scaling illustrates some empirical and theoretical advantages of tree representations over spatial representations of proximity data.We thank Nancy Henley and Vered Kraus for providing us with data, and Jan deLeeuw for calling our attention to relevant literature. The work of the first author was supported in part by the Psychology Unit of the Israel Defense Forces.     

Social representations and intergroup conflicts: Who's smoking here?

We studied the influence of explicitating a social conflict between smokers and nonsmokers on the social representations held about tobacco. Two representations were found: a psychological' representation which was defined by psychological explanations of the causes of beginning to smoke, a characterization of smokers as psychologically disturbed, and negative feelings towards smokers; a second defensive representation which attributes smoking to social factors, and hold both a positive stereotype of smokers and positive feelings towards smokers. The percentage of smokers who adhere to the defensive representation increased significantly when the conflict was made salient. Non-smokers were not influenced by the explicitation of the social conflict. The structure of each representation was analyzed. It was found that in the psychological' representation the structuring element was the negative stereotype of smokers. By contrast, the structuring element in the defensive representation was identification with smokers. The implicatlons of this data for theories of both intergroup relations and social representations are discussed.     

Aligning Logical and Psychological Contributions to the Understanding of Human Reasoning

Modern logic is an abstract study of consequence rather than a mechanistic study of reasoning. This abstract view has much to offer psychological studies of the representations which implement human reasoning. This paper reviews recent results showing the abstract equivalence of all the main psychological competence models of syllogistic reasoning. Their apparently contrasting representations are computationally equivalent for the kind of data presented. The true empirical contrasts between theories lie in other details of implementation — in whether integrated representations are constructed and in the sequence of cases considered. These contrasts are independent of sentential, graphical or mental model representation.Logic is also a guide to representation requirements. Linear syllogism models are representable by sequences of tokens, whereas categorial syllogism models generally require attribute binding. However, one consequence of the present semantic analysis is that all categorial syllogism competence models converge on an algorithm which avoids representation of more than a single sequence.The paper goes on to consider what evidence could distinguish the representations implementing reasoning in working memory. The example of dual task paradigms is considered and some conceptual questions are raised about the identification of the modalities of working memory subsystems with linguistic and spatial representations. Carefully distinguishing memory medium from memory interpretation leads to the conclusion that it is the interpretation of the contents of memory which determines whether it holds spatial or linguistic information. It is argued that the critical dimension for understanding reasoning processes is the expressiveness of interpretation of representations.     

Linguistic Distributional Knowledge and Sensorimotor Grounding both Contribute to Semantic Category Production

The human conceptual system comprises simulated information of sensorimotor experience and linguistic distributional information of how words are used in language. Moreover, the linguistic shortcut hypothesis predicts that people will use computationally cheaper linguistic distributional information where it is sufficient to inform a task response. In a pre-registered category production study, we asked participants to verbally name members of concrete and abstract categories and tested whether performance could be predicted by a novel measure of sensorimotor similarity (based on an 11-dimensional representation of sensorimotor strength) and linguistic proximity (based on word co-occurrence derived from a large corpus). As predicted, both measures predicted the order and frequency of category production but, critically, linguistic proximity had an effect above and beyond sensorimotor similarity. A follow-up study using typicality ratings as an additional predictor found that typicality was often the strongest predictor of category production variables, but it did not subsume sensorimotor and linguistic effects. Finally, we created a novel, fully grounded computational model of conceptual activation during category production, which best approximated typical human performance when conceptual activation was allowed to spread indirectly between concepts, and when candidate category members came from both sensorimotor and linguistic distributional representations. Critically, model performance was indistinguishable from typical human performance. Results support the linguistic shortcut hypothesis in semantic processing and provide strong evidence that both linguistic and grounded representations are inherent to the functioning of the conceptual system. All materials, data, and code are available at https://osf.io/vaq56/ .