Set representations of orders and a structural equivalent of saturation

Set representations are useful in the theory of knowledge spaces. A set representation of an order is an isomorphic mapping of its base set into the power set of some set ordered by set inclusion. Such a representation is basic if the union of the representing sets of the predecessors of an element contains strictly less elements than the representing set of this element, and it is parsimonious if the difference is exactly one element. This paper investigates the properties of the minimal number of elements which must be used in a parsimonious representation. This value is studied for several order operations. Moreover, orders which allow essentially only one parsimonious set representation are structurally characterized. These orders are called saturated. Finally, the way to apply these results to knowledge spaces are outlined.     

Representations and strategies for solving dynamic and static arithmetic word problems: The role of working memory capacities

This study aims at determining the structure of the representation constructed by adults to solve multiple-step arithmetic word problems. We show that this structure is isomorphic to the structure of the situation described in the text of the problem. In effect, dynamic problems, which describe sequential events, are more likely to be solved by sequential strategies than are static problems. In other words, the order of succession of subgoals reached by individuals for the resolution is determined by the order of succession of subgoals explicitly described in the text of the problem. However, an alternative mental representation is constructed by low span individuals when its cognitive cost is not too high and, therefore, certain static problems are solved by a sequential strategy as well. We conclude that the probability of the construction of an alternative representation to the one induced by the text of the problem depends on its cost and the relative amount of cognitive resources it releases.     

The isomorphic mapping hypothesis: evidence from Korean

This paper evaluates the relative merits of the trace deletion hypothesis, which attributes agrammatic comprehension difficulties to the loss of traces, and the isomorphic mapping hypothesis, which proposes that agrammatics have difficulty understanding sentences in which there is a nonisomorphic mapping between the syntactic representation and the corresponding event in the real world. (The two are isomorphic if the order of NPs reflects the place of entities in the event's 'action chain.' Since agents act on themes and transmit them to goals, the agent-theme-goal order is isomorphic with the corresponding event but the agent-goal-theme order is not.) The two hypotheses contrast in the predictions they make concerning goal-theme and theme-goal patterns in Korean: the TDH predicts degraded performance on the theme-goal structure (which is derived from the goal-theme structure); the IMH predicts difficulties on the goal theme structure, since it is nonisomorphic. Results of a comprehension study involving four Korean Broca's aphasics provide strong support for the IMH over the TDH.     

Isomorphism in Information Carrying Systems

Abstract: For the information theorist, the lawful generalizations that subsume instantiations of properties in the environment and instantiations of properties of perceptual representations determine the latter's content. Perceptual representations are also commonly thought to be isomorphic to what they represent, which presents the information theorist with a puzzle. What role could isomorphism play in perceptual representation when lawful generalizations determine content? I show that in order for the information that they carry to be available to cognition, perceptual representations must be isomorphic with respect to the constituent structure of the properties that they represent. Isomorphism therefore plays an important role in the information theorist's account of perceptual representation, even though it plays no role in determining content.     

The Distinction Between Quantitative and Qualitative Dimensions of Emotions: Clinical Implications

REBT theoreticians and practitioners describe two sets of emotions (and behaviors) as a reaction to adversity, whether these are functional or dysfunctional. This article deals with the ways in which REBT practitioners and theoreticians interpret these two sets of reactions, using either a quantitative or qualitative method. It favors the qualitative approach and illustrates it with a graphical representation of the two sets. The use of graphs turns out to be particularly useful for explaining certain phenomena to clients and for teaching novice practitioners. It also provides a framework for establishing an effective new thought or rational belief.     

Assembling a consistent set of sentences in relational probabilistic logic with stochastic independence

We examine the representation of judgements of stochastic independence in probabilistic logics. We focus on a relational logic where (i) judgements of stochastic independence are encoded by directed acyclic graphs, and (ii) probabilistic assessments are flexible in the sense that they are not required to specify a single probability measure. We discuss issues of knowledge representation and inference that arise from our particular combination of graphs, stochastic independence, logical formulas and probabilistic assessments.     

Reproducibility of functional network metrics and network structure: A comparison of task-related BOLD, resting ASL with BOLD contrast, and resting cerebral blood flow

Network analysis is an emerging approach to functional connectivity in which the brain is construed as a graph and its connectivity and information processing estimated by mathematical characterizations of graphs. There has been little to no work examining the reproducibility of network metrics derived from different types of functional magnetic resonance imaging (fMRI) data (e.g., resting vs. task related, or pulse sequences other than standard blood oxygen level dependent [BOLD] data) or of measures of network structure at levels other than summary statistics. Here, we take up these questions, comparing the reproducibility of graphs derived from resting arterial spin-labeling perfusion fMRI with those derived from BOLD scans collected while the participant was performing a task. We also examine the reproducibility of the anatomical connectivity implied by the graph by investigating test–retest consistency of the graphs’ edges. We compare two measures of graph-edge consistency both within versus between subjects and across data types. We find a dissociation in the reproducibility of network metrics, with metrics from resting data most reproducible at lower frequencies and metrics from task-related data most reproducible at higher frequencies; that same dissociation is not recapitulated, however, in network structure, for which the task-related data are most consistent at all frequencies. Implications for the practice of network analysis are discussed.     

Reductio ad contradictionem: An Algebraic Perspective

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms.     

Working memory capacity and fluid intelligence are strongly related constructs: comment on Ackerman, Beier, and Boyle (2005)

The authors agree with P. L. Ackerman, M. E. Beier, and M. O. Boyle (2005; see record 2004-22408-002) that working memory capacity (WMC) is not isomorphic with general fluid intelligence (Gf) or reasoning ability. However, the WMC and Gf/reasoning constructs are more strongly associated than Ackerman et al. (2005) indicate, particularly when considering the outcomes of latent-variable studies. The authors' reanalysis of 14 such data sets from 10 published studies, representing more than 3,100 young-adult subjects, suggests a strong correlation between WMC and Gf/reasoning factors (median r=.72), indicating that the WMC and Gf constructs share approximately 50% of their variance. This comment also clarifies the authors' "executive attention" view of WMC, it demonstrates that WMC has greater discriminant validity than Ackerman et al. (2005) implied, and it suggests some future directions and challenges for the scientific study of the convergence of WMC, attention control, and intelligence.     

Graphics in psychology: Pictures,data, and especially concepts

Over the last 50 years, psychologists have included numerous graphics in their journals and textbooks. The textbooks have contained mostly pictures, whereas the journals have contained mostly data graphs. Surprisingly, conceptual graphs (such as a network chart of the relations among types of memory) have been very infrequent. There are two main reasons for this absence of conceptual graphs: a lack of awareness of the important role that they have played in the history of our discipline, and a lack of knowledge about how to make them. Four types of conceptual graphs are presented here: mathematical graphs (e.g., Fechner’s law), geometric models (e.g., Henning’s smell prism), networks (e.g., Atkinson and Shiffrin’s STM-LTM model), and EulerVenn graphs (e.g., Shiffrin and Schneider’s model of control and automatic processes). Suggestions are offered forheuristics that can be used to facilitate the perception of conceptual relations in these graphs. It is argued that appropriate conceptual graphs can help researchers and theorists communicate the “big picture” relevant to their ideas, can help educators more effectively communicate with students, and may help clinical psychologists communicate their ideas to clients better.     

The monadic second-order logic of graphs XV: On a conjecture by D. Seese

A conjecture by D. Seese states that if a set of graphs has a decidable monadic second-order theory, then it is the image of a set of trees under a transformation of relational structures defined by monadic second-order formulas, or equivalently, has bounded clique-width. We prove that this conjecture is true if and only if it is true for the particular cases of bipartite undirected graphs, of directed graphs, of partial orders and of comparability graphs. We also prove that it is true for line graphs, for interval graphs and for partial orders of dimension 2. Our treatment of certain countably infinite graphs uses a representation of countable linear orders by binary trees that can be constructed by monadic second-order formulas. By using a counting argument, we show the intrinsic limits of the methods used here to handle this conjecture.     

The pond-and-duckweed problem; Three experiments on the misperception of exponential growth

The representation of duckweed multiplying itself in a pond is used as a research paradigm to study underestimation of exponential growth. The advantage of this paradigm is that the growth process is presented in a direct non-numerical way. The results show that the underestimation observed in earlier studies where growth was presented by means of tables or graphs, occurs in the pond-and-duckweed situation as well. By manipulating the way the process is presented it is possible to obtain some insight into the sampling strategies used by the subjects when they subjectively extrapolate the perceived processes. These experiments lead to the conclusion that subjects base their extrapolations on three or four samples only.     

Personhood,culture and family therapy

This paper argues that ideas about personhood and relationships vary cross-culturally and that although it is not always obvious or easy to discover, this variation is particularly salient to family therapy. This argument is supported by a dual definition of culture emphasizing both continuity of cultural themes (generative aspect) and the reconstitution, redefinition and change of such themes through interaction and communication (interactive aspect). The effectiveness of interventions which are isomorphic with ideas which families themselves hold about personhood is demonstrated through work with two families, one Punjabi and one Bangladeshi.     

Problem solving,recall, and isomorphic transfer among third-grade and sixth-grade children

Analogical reasoning processes were studied in third- and sixth-grade children. One group at each grade level received two analogs of the “farmer's dilemma”, a scheduling problem that requires seven moves for solution, during acquisition. Other groups received one analog or only a transfer task. On each trial the child heard a list of statements representing the exact series of moves necessary to solve the problem, was immediately asked to recall the list, materials representing the problem were then produced, and the child was asked to show how to solve it. A trial was terminated and a new one begun when an error was made on the physical task. Following criterial performance all children were transferred to an isomorphic analog and simply instructed to solve it. Third-graders in the two-analog condition required more trials in acquisition than the other groups, which did not differ from each other. Transfer was better following two analogs, but was unaffected by grade level. Grade level affected recall accuracy, but recall accuracy was not a good predictor of transfer performance. Trial-by-trial analyses suggested that propositions representing the solution are acquired piecemeal, but consolidation of the generalizable problem representation is abrupt.     

Psychoanalysis and Trauma

This article articulates some of the problems that surround the use of the term trauma in psychoanalytic theory and suggests that the key element for a theory of pathogenesis and mental functioning is not the either/or of external versus internal causation or trauma versus drive. Rather, it is an understanding of whether, or to what extent, the raw data of existential experience is or is not transformed into psychological experience. From this perspective, trauma is whatever outstrips and disrupts the psyche’s capacity for representation or mentalization. Absent the potential for mental representation, these events and phenomena are historical only from an external, third-person perspective. Until they are mentalized, they remain locked within an ahistorical, repetitive process as potentials for action, somatization, and projection.     

A graph theoretic approach to similarity relations

Summary A similarity relation is reflexive and symmetric. A graph may be regarded as a geometric representation of an irreflexive symmetric relation; anr-graph then coincides with a similarity relation. For the realization of psychophysical phenomena, the points of a graph stand for the stimuli and a line for indistinguishability of a stimulus pair. Not all graphs, however, are likely to arise as similarity relations in a real situation. A special class of graphs, called interval graphs, does have this realistic property. These graphs are described and characterized.The preparation of this article was supported by Grant NSF-G-17771 from the National Science Foundation.     

Self-presentation, Representation and the Self

Chisholm held that some states of ourselves are self-presenting and provide a stopping place in the quest for justification. The justification we have for accepting that we are in those states is transparent to us in a way that enables us to answer questions about justification. Representation enables us to apprehend such self-presenting states through themselves in a representational loop. It is a loop of exemplarization wherein the state is used as an exemplar to represent the kind of state it is. The result is that the representation of the state provides the subject with a kind of representation that loops back onto itself escaping the bondage of stratified mentality. This form of representation by exemplarization is shown to resolve problems and paradoxes concerning subjectivity, consciousness and the self raised by the writings of Hume, Kierkegaard, Ferrier, Sartre and Frank Jackson.     

Knowledge space theory, formal concept analysis, and computerized psychological assessment

In the present study, the use of knowledge space theory (KST), jointly with formal concept analysis (FCA), is proposed for developing a formal representation of the relations between the items of a questionnaire and a set of psychodiagnostic criteria. This formal representation can be used to develop an efficient adaptive tool for psychological assessment. Rusch and Wille (1996) have shown some interesting connections between KST and FCA; these connections are applied in the construction of knowledge structures, starting from a formal context representing the relations between items and criteria. The proposed general methodology was applied, as an example, to the Maudsley Obsessional-Compulsive Questionnaire. We used a data set provided by a sample of patients with a diagnosis of obsessive-compulsive disorder to validate the obtained structures. The parameters of the basic local independence model (BLIM) were estimated for the obtained knowledge structures. The fit of each model was tested by parametric bootstrap because of the sparseness of the derived data matrix. The results are discussed in light of both psychological and methodological relapses. In particular, we propose a reinterpretation of the BLIM parameters that seems suitable for testing reliability and construct validity; furthermore, it is pointed out how the obtained structures could represent the starting point for the development of a computerized assessment tool.     

Similarity, kernels, and the triangle inequality

Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS, similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and closeness of stimuli in a geometric representation may be intuitively plausible, Tversky and Gati [Tversky, A., & Gati, I. (1982). Similarity, separability, and the triangle inequality. Psychological Review, 89(2), 123-154] have reported data which are inconsistent with the usual geometric representations that are based on segmental additivity. We show that similarity measures based on Shepard’s universal law of generalization [Shepard, R. N. (1987). Toward a universal law of generalization for psychologica science. Science, 237(4820), 1317-1323] lead to an inner product representation in a reproducing kernel Hilbert space. In such a space stimuli are represented by their similarity to all other stimuli. This representation, based on Shepard’s law, has a natural metric that does not have additive segments whilst still retaining the intuitive notion of connecting similarity and distance between stimuli. Furthermore, this representation has the psychologically appealing property that the distance between stimuli is bounded.     

Categoricity Theorems and Conceptions of Set

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to the pure sets of the other.This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine.