Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis

A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism.     

Verbal Disagreements and Philosophical Scepticism

Many philosophers have suggested that disagreement is good grounds for scepticism. One response says that disagreement-motivated scepticism can be mitigated to some extent by the thesis that philosophical disputes are often verbal, not genuine. I consider the implications of this anti-sceptical strategy, arguing that it trades one kind of scepticism for others. I conclude with suggestions for further investigation of the epistemic significance of the nature of philosophical disagreement.     

概率性输入,噪音“与”门(PINA)模型

当前认知诊断测验的主要目的是对被试进行合理分类,进而采用类别变量去描述被试对某技能或知识(即认知属性)的掌握情况,但该粗糙的分类方法不能精细地区分不同被试之间的差异。对此,采用掌握概率这一连续变量去描述被试对某认知属性的掌握情况是一种值得尝试的做法。本文首先基于高阶潜在特质(简称"潜质")模型给出了认知属性掌握概率的量化定义,之后与多成分潜质模型相结合提出了概率性输入,噪音"与"门(PINA)模型;其次,采用MCMC算法实现了对PINA的参数估计,结果表明参数估计程序对各参数的估计返真性均较好;最后,以ECPE数据为例来说明PINA在实际测验分析中具有可行性。     

Reflexive theories of consciousness and unconscious perception

A core commitment of the reflexive theory of consciousness is that conscious states are themselves necessarily the contents of mental states. The strongest argument for this claim—the necessity of inner-content for consciousness—is the argument from unconscious perception. According to this argument, we find evidence for the necessity claim from cases of alleged unconscious perception, the most well-known and widely discussed of these being blindsight. However, the reflexive theory cannot partake in this argument and therefore, must rely on at least one of the other arguments for the necessity claim. These arguments are significantly less convincing than the argument from unconscious perception, and thus the reflexive theory is left in a dialectically weak position.     

实体和初级关系（FOR）匹配的分离与结合对故事类比通达的影响

采用移动窗口阅读技术深入探讨实体和初级关系（FOR）匹配的分离与结合对故事类比通达的影响,对表面特征和结构特征在类比通达中的作用进行检验。实验1探讨系统变化目标故事和线索故事之间实体匹配数目与实体和初级关系（FOR）匹配的分离与结合是否影响读线索故事时对目标故事的回想。实验2探讨系统变化目标故事和线索故事之间初级关系（FOR）匹配的数目与实体和初级关系（FOR）匹配的分离与结合是否影响读线索故事时对目标故事的回想。实验3探讨实体和初级关系（FOR）匹配的分离与结合对目标故事的通达是否与高级关系（HOR）的匹配有关。结果表明,实体匹配独立影响通达,实体和初级关系（FOR）分离与结合影响故事类比通达,高级的结构关系（HOR）是实体和初级关系（FOR）影响故事类比通达的必要条件。     

结构方程模型的应用及分析策略

差不多所有心理、教育、社会等概念,均难以直接准确测量,结构方程（SEM,Struc－tulalEquationModelling）提供一个处理测量误差的方法,采用多个指标去反映潜在变量,也令估计整个模型因子间关系,较传统回归方法更为准确合理。本文主要用一系列有关学习动机的虚拟例子,指出每个问题的主要分析策略,以展示SEM在教育及心理学可以应用的研究范畴。文内探讨的方法包括：验证性因素、高阶因子、路径及因果分析、多时段（multiwave）设计、单形模型（SimpleModel）、及多组比较等。     

Direct instruction: What it is and what it is becoming

This essay describes the principles of Direct Instruction design and delivery used to establish clear, unambiguous communication and maximize student responding. Direct Instruction research findings are summarized: achievement of low-income students in Follow Through, longitudinal results, and the effectiveness of Direct Instruction for students with handicaps. Finally, new directions for Direct Instruction, technological applications and instruction of higher-order skills, are discussed.     

Measurement bias and error correction in a two-stage estimation for multilevel IRT models

Among current state-of-the-art estimation methods for multilevel IRT models, the two-stage divide-and-conquer strategy has practical advantages, such as clearer definition of factors, convenience for secondary data analysis, convenience for model calibration and fit evaluation, and avoidance of improper solutions. However, various studies have shown that, under the two-stage framework, ignoring measurement error in the dependent variable in stage II leads to incorrect statistical inferences. To this end, we proposed a novel method to correct both measurement bias and measurement error of latent trait estimates from stage I in the stage II estimation. In this paper, the HO-IRT model is considered as the measurement model, and a linear mixed effects model on overall (i.e., higher-order) abilities is considered as the structural model. The performance of the proposed correction method is illustrated and compared via a simulation study and a real data example using the National Educational Longitudinal Survey data (NELS 88). Results indicate that structural parameters can be recovered better after correcting measurement biases and errors.     

A Generic Framework for Adaptive Vague Logics

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided.