LÉVY HIERARCHY IN WEAK SET THEORIES

We investigate the interactions of formula complexity in weak set theories with the axioms available there. In particular, we show that swapping bounded and unbounded quantification preserves formula complexity in presence of the axiom of foundation weakened to an arbitrary set base, while it does not if the axiom of foundation is further weakened to a proper class base. More attention is being paid to the necessary axioms employed in the positive results, than to the combinatorial strength of the positive results themselves.     

Propositions, Sets, and Worlds

If we agree with Michael Jubien that propositions do not exist, while accepting the existence of abstract sets in a realist mathematical ontology, then the combined effect of these ontological commitments has surprising implications for the metaphysics of modal logic, the ontology of logically possible worlds, and the controversy over modal realism versus actualism. Logically possible worlds as maximally consistent proposition sets exist if sets generally exist, but are equivalently expressed as maximally consistent conjunctions of the same propositions in corresponding sets. A conjunction of propositions, even if infinite in extent, is nevertheless itself a proposition. If sets and hence proposition sets exist but propositions do not exist, then whether or not modal realism is true depends on which of two apparently equivalent methods of identifying, representing, or characterizing logically possible worlds we choose to adopt. I consider a number of reactions to the problem, concluding that the best solution may be to reject the conventional model set theoretical concept of logically possible worlds as maximally consistent proposition sets, and distinguishing between the actual world alone as maximally consistent and interpreting all nonactual merely logically possible worlds as submaximal. I am grateful to the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS), Royal Netherlands Academy of Arts and Sciences (KNAW), for supporting this among related research projects in philosophical logic and philosophy of mathematics during my Resident Research Fellowship in 2005-2006.     

Single and double alternation learning in rats: The role of set size and correction

In many experiments, rats have evidenced extreme difficulty mastering alternation patterns. In three experiments, we explored rats’ ability to learn double alternation patterns and possible reasons behind their past difficulties with such patterns. In Experiment 1, rats learned single and double alternation patterns. In the second and third experiment, we explored the role of correction and interference from stimulus set size on rats’ learning of double alternation patterns. Our results showed that rats easily learned both single and double alternation patterns and that stimulus set size had no noticeable effect on learning double alternation patterns. Findings additionally evidenced that correction played an important role in rats’ learning double alternation patterns.     

Another Paradox In Naive Set-Theory

Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox. Presented by Robert Goldblatt     

The Buss–Perry Aggression Questionnaire: Some unfinished business

This article gauges the extent to which items and sum scores of the Buss–Perry Aggression Questionnaire (BPAQ) are contaminated with social desirability response set (SDR), and the extent to which that contamination results in biased findings. Special validity coefficients are introduced for that purpose. Although the SDR confound was quite substantial across all items (with one exception), it is argued on the basis of the size of the introduced validity coefficients and other, logical considerations that paradoxically the confound had minimally distorting effects in estimating the BPAQ models selected for probing the extent of bias.     

数量化维度特征搜索的影响因素探索

有关数量化维度特征搜索的研究表明 ,目标搜索斜率 (反应时间对呈现项目数量函数的斜率 )是由目标与干扰子之间的差异决定的 ,然而以往研究中其差异的值取决于具体的实验条件。根据韦伯定律 ,笔者推测 ,目标搜索斜率应由目标与干扰子之间的相对差异 (C)决定 ,C为目标值 (T)与干扰子值 (D)的差值与该干扰子值的比值。当C值对应相等时 ,不同几何形状刺激的目标搜索斜率应无差异。为此笔者分别采用圆 (实验一 )和三角形 (实验二 )进行实验研究。实验结果支持了上述假设 ,并发现目标大于干扰子时的目标搜索斜率对C值的函数是单调递减函数     

Internal approach to external sets and universes

In this article we show how the universe of HST, Hrbaek set theory (a nonstandard set theory of external type, which includes, in particular, the ZFC Replacement and Separation schemata for all formulas in the language containing the membership and standardness predicates, and Saturation for standard size families of internal sets, but does not include the Power set axiom) admits a system of subuniverses which keep the Replacement, model Power set and Choice (in fact all of ZFC, with the exception of the Regularity axiom, which indeed is replaced by the Regularity over the internal subuniverse), and also keep as much of Saturation as it is necessary.This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.Partially supported by AMS grants in 1993 – 1995 and DFG grants in 1994 – 1995. Robert Goldblatt     

Conditional discrimination in mentally retarded adults: the development of generalized skills.

The development of generalized conditional discrimination skills was examined in adults with retardation. Two subjects with histories of failure to acquire arbitrary matching under trial-and-error procedures were successful under procedures that trained one or more prerequisite skills. The successive discrimination between the sample stimuli was established by training the subjects to name the stimuli. The simultaneous discrimination between the comparison stimuli was established using either (a) standard simple discrimination training with reversals or (b) a procedure in which each of the two sample-comparison relations in the conditional discrimination was presented in blocks of trials, with the size of the blocks decreasing gradually until sample presentation was randomized. The amount of prerequisite training required varied across subjects and across successive conditional discriminations. After acquiring either two or three conditional discriminations with component training, both subjects learned new conditional discriminations under trial-and-error procedures. In general, each successive conditional discrimination was acquired more rapidly. Tests showed that conditional responding had become a generalized skill. Symmetry was shown for almost all trained relations. Symmetry trial samples were ultimately named the same as the stimuli to which they were related in training.     

Some properties of two measures of multivariate association

Two kinds of measures of multivariate association, based on Wilks' and the Bartlett-Nanda-Pillai trace criterionV, respectively, are compared in terms of properties of the univariateR ² which they generalize. A unified set of derivations of the properties is provided which are self-contained and not restricted to decompositions in canonical variates. One conclusion is that asymmetric index based on allows generalization of the multiplicative decomposition ofR ² in terms of squared partial correlations, but not the additive decomposition in terms of squared semipartial correlations, while the reverse is true for anasymmetric index based onV.We are indebted to Jos M. F. ten Berge for some fruitful discussions.