First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas

The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).     

音乐训练对认知能力的影响

在当今社会中, 越来越多的儿童接受音乐训练, 试图通过音乐训练提高认知能力。针对认知活动中的语言能力、空间能力以及数学能力, 探究音乐训练与认知能力的关系, 试图为音乐教育提供实证依据。研究结果表明, 尽管音乐训练与认知能力的关系较为复杂, 但是, 音乐训练与个体的语言知觉和产生能力以及空间视觉化能力都存在正相关, 且在一定程度上可以提高这些认知能力。     

Animal cognition + optimal choice = behavior: A review of adaptive behavior and learning, 2nd Ed., by J. E. R. Staddon

Staddon discusses a vast array of topics in comparative psychology in this book. His view is that adaptive behavior in most cases is the result of optimal choice acting on an animal's knowledge about the world. Staddon refers to this as a functional teleonomic approach inasmuch as it attempts to understand an animal's behavior in terms of goals. He builds mathematical models based on this idea that are designed to reproduce specific sets of empirical observations, usually qualitatively. A natural consequence of Staddon's approach is that many models are developed, each of which applies to a specific set of observations. An alternative to functional teleonomy is a functional approach that builds on prior principles. In most cases, this approach favors a single‐theory account of behavior. Prior principles can be understood as functional stand‐ins for antecedent material causes, which means that these accounts are closer to mechanistic theories than are goal‐based teleonomic accounts. An ontological perspective, referred to as supervenient realism, is a means of understanding the relationship between functional theories and the material world. According to this perspective, the algorithmic operation of a successful functional theory may be understood to supervene on the material operation of the nervous system.     

MODELING RELATIONS

In the ordinary way of representing relations, the order of the relata plays a structural role, but in the states themselves such an order often does not seem to be intrinsically present. An alternative way to represent relations makes use of positions for the arguments. This is no problem for the love relation, but for relations like the adjacency relation and cyclic relations, different assignments of objects to the positions can give exactly the same states. This is a puzzling situation. The question is what is the internal structure of relations? Is the use of positions still justified, and if so, what is their ontological status? In this paper mathematical models for relations are developed that provide more insight into the structure of relations “out there” in the real world.     

Frame Based Formulas for Intermediate Logics

In this paper we define the notion of frame based formulas. We show that the well-known examples of formulas arising from a finite frame, such as the Jankov-de Jongh formulas, subframe formulas and cofinal subframe formulas, are all particular cases of the frame based formulas. We give a criterion for an intermediate logic to be axiomatizable by frame based formulas and use this criterion to obtain a simple proof that every locally tabular intermediate logic is axiomatizable by Jankov-de Jongh formulas. We also show that not every intermediate logic is axiomatizable by frame based formulas. Presented by Johan van Benthem     

The mathematics of geographic profiling

We begin by describing some of the mathematical foundations of the geographic profiling problem. We then present a new mathematical framework for the geographic profiling problem based on Bayesian statistical methods that makes explicit connections between assumptions on offender behaviour and the components of the mathematical model. It also can take into account local geographic features that either influence the selection of a crime site or influence the selection of an offender's anchor point. Copyright © 2009 John Wiley & Sons, Ltd.     

Subdirectly Irreducible Residuated Semilattices and Positive Universal Classes

CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)^k ≤ (x ⋀ e)^k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k). Dedicated to the memory of Willem Johannes Blok     

The roles of different executive functioning skills in young children’s mental computation and applied mathematical problem-solving

This study investigated the relationships of four executive functioning skills (including verbal working memory, spatial working memory, inhibitory control, and cognitive flexibility) with young children’s mental computation and applied mathematical problem-solving. Two hundred and twenty-five Chinese kindergarteners were tested with a battery of general cognitive, executive functioning and mathematics skills. Results showed that when children’s age, gender, non-verbal intelligence, and listening comprehension skills were controlled, verbal working memory and cognitive flexibility were significant correlates of mental computation, whereas verbal working memory, spatial working memory, and cognitive flexibility were significant correlates of applied mathematical problem-solving. Inhibitory control was not significantly associated with the two domains of mathematics under investigation. The findings highlight the differential roles of different executive functioning skills in early mathematical skills and offer practical implication for helping young children in learning complex mathematical skills.     

Conceptual engineering for mathematical concepts

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.