Classification of concatenation measurement structures according to scale type

A relational structure is said to be of scale type (M,N) iff M is the largest degree of homogeneity and N the least degree of uniqueness (Narens, 1981a, Narens, 1981b) of its automorphism group.Roberts (in Proceedings of the first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conference on graph theory and its applications, New York: Wiley, 1984) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures 〈X, ?, ° 〉, where ? is a total ordering and ° is a monotonic operation. Section 2 establishes that for concatenation structures with M>0 and N<∞ the only scale types are (1,1), (1,2), and (2,2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type (M, M) (Theorem 3.1) and of Alper (Journal of Mathematical Psychology, 1985, 29, 73–81) for scale type (1, 2) (Theorem 3.2). For M>0, concatenation structures are all isomorphic to numerical ones for which the operation can be written $\text{x°y = yf(}\text{x}\text{y}\text{)}$, where f is strictly increasing and $\text{f(x)}\text{x}$ is strictly decreasing (unit structures). The equation f(x^?)=f(x)^? is satisfied for all x as follows: for and only for ? = 1 in the (1,1) case; for and only for ?=kⁿ, k > 0 fixed, and n ranging over the integers, in the (1, 2) case; and for all ?>0 in the (2, 2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation catenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other results, are mostly formulated in terms of the set of right translations of the induced operation. In Section 5 we consider the existence of representations of concatenation structures. The case of positive ones was dealt with earlier (Narens & Luce (Journal of Pure & Applied Algebra27, 1983, 197–233). For idempotent ones, closure, density, solvability, and Archimedean are shown to be sufficient (Theorem 5.1). The rest of the section is concerned with incomplete results having to do with the representation of cases with M>0. A variety of special conditions, many suggested by the conjoint equivalent of a concatenation structure, are studied in Section 6. The major result (Theorem 6.4) is that most of these concepts are equivalent to bisymmetry for idempotent structures that are closed, dense, solvable, and Dedekind complete. This result is important in Section 7, which is devoted to a general theory of scale type (2, 2) for the utility of gambles. The representation is a generalization of the usual SEU model which embodies a distinctly bounded form of rationality; by the results of Section 6 it reduces to the fully rational SEU model when rationality is extended beyond the simplest equivalences. Theorem 7.3 establishes that under plausible smoothness conditions, the ratio scale case does not introduce anything different from the (2, 2) case. It is shown that this theory is closely related to, but somewhat more general, than Kahneman and Tversky's (Econometrica47, 1979, 263–291) prospect theory.     

Disclosure of Personal Problems, Self-Esteem, and the Facilitativeness of Friends and Lovers

The relationship between disclosure of personal problems, self-esteem, and Rogers' facilitative qualities of unconditional acceptance, empathy, and congruence in a casual friend, the closest friend of the same and opposite sex, and the romantic partner, was investigated in 104 females and 19 males (pooled together). They completed the Rosenberg Self-Esteem Scale, a shortened Barrett-Lennard Relationship Inventory, and a two-item index of disclosure of personal problems. Self-esteem was most strongly positively correlated with the facilitative qualities in the romantic partner, followed by the casual friend. Disclosure of personal problems was also positively related to these facilitative qualities. When the facilitativeness of the romantic relationship was controlled through partial correlation, the positive correlation between self-esteem and disclosure of personal problems was reduced to near zero, while that between self-esteem and the facilitativeness of the casual friend was little affected.     

Children's academic attainment is linked to the global organization of the white matter connectome

Literacy and numeracy are important skills that are typically learned during childhood, a time that coincides with considerable shifts in large‐scale brain organization. However, most studies emphasize focal brain contributions to literacy and numeracy development by employing case‐control designs and voxel‐by‐voxel statistical comparisons. This approach has been valuable, but may underestimate the contribution of overall brain network organization. The current study includes children (N = 133 children; 86 male; mean age = 9.42, SD = 1.715; age range = 5.92–13.75y) with a broad range of abilities, and uses whole‐brain structural connectomics based on diffusion‐weighted MRI data. The results indicate that academic attainment is associated with differences in structural brain organization, something not seen when focusing on the integrity of specific regions. Furthermore, simulated disruption of highly‐connected brain regions known as hubs suggests that the role of these regions for maintaining the architecture of the network may be more important than specific aspects of processing. Our findings indicate that distributed brain systems contribute to the etiology of difficulties with academic learning, which cannot be captured using a more traditional voxel‐wise statistical approach.     

Transition structure versus commitment in sequential subjective probability revision

When Ss revise subjective probabilities, in the light of new evidence, a common finding is that they are conservative with respect to Bayes' theorem; revisions are too small. One kind of hypothesis to account for this is ‘model specific’, assuming a breakdown in an otherwise potentially Bayesian process. The other kind assumes that statistically irrelevant, task-specific information is processed. An example of the latter is the commitment hypothesis, assuming a commitment building up to the indications of early evidence, causing Ss to lag behind Bayes' theorem in their later judgements. Evidence is presented suggesting that Ss are not necessarily sensitive to mere sub-sets of a sequence, but that this form of suboptimality may result from overall sequence structure; specifically from a bias against long runs of like evidence. This would fit with findings from other areas of research, and would suggest that there is a general form of suboptimality operating which is relevant to all sequential processing tasks.     

Affect and memory in young children

The state-dependent theory of the relationship between affective states and memory holds that recall will be best when the affective state at recall matches that during learning. Sequential happy, neutral, and sad affective states that were either consistent (e.g., Happy-Happy) or inconsistent (e.g., Sad-Neutral) were experimentally induced in preschool children prior to encoding and then again prior to retrieval (free and cued recall, recognition memory). Facial ratings indicated that the inductions were effective in inducing affect. Nevertheless, emotional states did not influence children's ability to recall items under free or cued conditions, and recognition memory was essentially perfect for all subjects. Thus, there was no evidence for state-dependent learning or for a positive loop between subjects' positive affect at retrieval and memory for positively rated information. Results are discussed in terms of the generally inconsistent findings in the literature on the role of affect in children's memory and factors that may limit affective state-dependent learning in children.This research was supported by Research Grant No. 11776 from the National Institute of Child Health and Human Development to Marion Perlmutter, by Grant BNS 78-01108 from the National Science Foundation to John C. Masters, and by Program Project Grant No. 0527 to the Institute of Child Development. Wayne Duncan is now at the University of Denver, and Christine Todd is now at the University of Illinois, Urbana. Marion Perlmutter is now at the University of Michigan. We would like to thank Keith Elliott and LuAnne Tczap for their work as experimenters; Jule Kogan, Carol Revermann, and Sonya Hernandez for their help in coding data; and Jayne Grady-Reitan for her administrative assistance throughout the study.     

Moral Evil,Freedom and the Goodness of God: Why Kant Abandoned Theodicy

Kant proclaimed that all theodicies must fail in ‘On the Miscarriage of All Philosophical Trials in Theodicy’, but it is mysterious why he did so since he had developed a theodicy of his own during the critical period. In this paper, I offer an explanation of why Kant thought theodicies necessarily fail. In his theodicy, as well as in some of his works in ethics, Kant explained moral evil as resulting from unavoidable limitations in human beings. God could not create finite beings without such limitations and so could not have created humans that were not prone to committing immoral acts. However, the work of Carl Christian Eberhard Schmid showed Kant that given his own beliefs about freedom and the nature of responsibility one could not account for moral evil in this way without tacitly denying that human beings were responsible for their actions. This result is significant not only because it explains an otherwise puzzling shift in Kant's philosophy of religion, but also because it shows that the theodicy essay provides powerful evidence that Kant's thinking about moral evil and freedom underwent fundamental shifts between early works such as the Groundwork and later works like the Religion within the Limits of Mere Reason.