微分概念的历史发展及教学启示

微分的形式化定义是学生学习微分概念的主要困难.微分概念的历史发展表明,形式化的微分定义是微积分严格化的产物,朴素的微分定义更能体现微积分思想,而非标准分析给微分概念带来重生.在微积分学中应用非形式化的方法构建微分概念,以微分为主线(传统教材一般以导数为主线)进行微积分教学可以促进学生学习效果.     

计量逻辑学中的雪崩逻辑公式

将密码学中满足严格雪崩准则的布尔函数的概念引入到计量逻辑学之中,提出了雪崩逻辑公式的概念,并研究了雪崩逻辑公式的真度及其性质。证明了至少含有三个原子公式的雪崩逻辑公式的真度之集为H1={k/2n-12n-3≤k≤3×2n-3;n=3,4,…},在此基础上,通过引入函数ξ建立了n(n≥3)元雪崩布尔函数个数的表达式,给出了雪崩逻辑公式的构造方法。最后,研究了反射变换下k阶雪崩逻辑公式的性质。     

基于中介逻辑的时序逻辑系统

本文基于中介逻辑建立了一种中介时序逻辑系统ＭＴＬ（Ｍｅｄｉｕｍ Ｔｅｍｐｏｒａｌ Ｌｏｇｉｃ），文中着重讨论了ＭＴＬ的形式系统并给出了它的语义解释，证明了ＭＴＬ系统的可靠性．最后对ＭＴＬ系统和经典时序命题逻辑系统进行对比，指出经典时序命题逻辑系统是ＭＴＬ的子系统     

从事实逻辑到任务逻辑

在经典命题逻辑语言中引入附加算子几，较系统地研究近几年刚刚被提出的任务逻辑。这里把公式理解为“任务”，介绍了“任务逻辑”的语义理论，并从语构上定义形式系统L与之对应，证明该系统的可靠性、完备性及可判定性定理．最后建立系统L中的一系列基本定理。     

多值命题逻辑中逻辑理论的拓扑刻画

基于多值Lukasiewicz命题逻辑系统Ln中的计量逻辑学理论。研究了逻辑理论在逻辑意义下的性质与其在拓扑意义下的性质之间的联系，并给出了闭逻辑理论的拓扑性质描述及多值命题逻辑中逻辑理论的发散性的拓扑刻画。     

Logic Regression

Logic regression is an adaptive regression methodology that attempts to construct predictors as Boolean combinations of binary covariates. In many regression problems a model is developed that relates the main effects (the predictors or transformations thereof) to the response, while interactions are usually kept simple (two- to three-way interactions at most). Often, especially when all predictors are binary, the interaction between many predictors may be what causes the differences in response. This issue arises, for example, in the analysis of SNP microarray data or in some data mining problems. In the proposed methodology, given a set of binary predictors we create new predictors such as “X₁, X₂, X³, and X⁴ are true,” or “X⁵ or X⁶ but not X⁷ are true.” In more specific terms: we try to fit regression models of the form g(E[Y]) = b₀ + b₁ L₁ + · · · + b_n L_n , where L_j is any Boolean expression of the predictors. The L_j and b_j are estimated simultaneously using a simulated annealing algorithm. This article discusses how to fit logic regression models, how to carry out model selection for these models, and gives some examples.     

概率逻辑学基本定理在多值命题逻辑系统中的推广

通过引入概率测度空间,在n值Lukasiewicz命题逻辑系统中提出了满足Kolmogorov公理的命题公式的概率;证明了概率逻辑学基本定理,并将概率逻辑学基本定理推广到了更一般的形式,改进了对推理结论的不可靠度上界的估计;将概率逻辑学的基本方法引入计量逻辑学,建立了更一般的逻辑度量空间;通过概率逻辑学基本定理,证明了逻辑度量空间中概率MP,HS规则,它是真度MP,HS规则的推广.     

支撑模糊集合转换的模糊逻辑是伪逻辑

模糊逻辑系统用模糊集合描述模糊信息、用模糊集合转换处理模糊信息.阐述支撑模糊集合转换的模糊逻辑为什么是伪逻辑的原因.指出定义在论域一个空间上的模糊集合,因为破坏了集合中元素的"不可分割性",所以模糊集合描述的模糊信息不能用数学计算通过模糊集合转换处理.实际应用中的模糊信息定义在论域与状态空间两个空间上,其正确表达方式是满足"归一化"条件的隶属度向量;处理的正确途径是,研究基于状态转移矩阵的隶属度转换;支撑隶属度转换的是近似推理逻辑,目的是使构建的隶属度转换模型是当前条件下人们可能构建的"最优"近似模型.     

Abstract Logics, Logic Maps, and Logic Homomorphisms

What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. This research was supported by the grant CNPq/FAPESB 350092/2006-0.     

模糊综合评判加权平均模型的误导性阐释

模糊综合评判在不能界定隶属度转换是线性转换条件下,硬性用加权平均线性模型实现隶属度转换.通过揭示指标隶属度中通常包含确定目标隶属度不起作用的非线性冗余值,建立区分权滤波的冗余理论,用冗余理论界定隶属度转换不是线性转换,并构建近似推理逻辑支撑的隶属度转换模型.阐述所建模型为什么是当前条件下人们能构建的最优近似模型.当指标隶属度只取1或0两个值时模型将退化成模糊综合评判的"加权平均"模型.     

A Bounded Translation of Intuitionistic Propositional Logic into Basic Propositional Logic

In this paper we prove a bounded translation of intuitionistic propositional logic into basic propositional logic. Our new theorem, compared with the translation theorem in [1], has the advantage that it gives an effective bound on the translation, depending on the complexity of formulas.     

The Place of Operational Research in the Development of Modern Society

Jacques Lesourne is Professor of Economics at the Saint-Etienne School of Mining and Professor of Company Economics at the Demonstration School of the National Institute of Statistics and Economic Research. He has lately had a book published by Dunod, entitled Technique Economique et Gestion Industrielle.     

分布逻辑的理论与分析

以往的逻辑理论讨论的是理论上命题之间的逻辑关系,作为一种基本准则,可以放在许多情况下使用,但是一般的逻辑理论相对抽象,命题形式过于简单,这样有时不能很准确很方便地处理实际问题,在传统数学中,分布是一种重要的思想,基于这种思想建立了分布逻辑理论,同时对命题值在分布结构上的真假以及变化进行了讨论。     

The Logic of Analogy

How should a scientist argue when the data are insufficient to allow him to reason by classical or statistical models? After all, in most real world situations - in business or in war - that is the unhappy norm. In such cases the ordinary man instinctively argues by analogy, as Leibniz long ago showed; indeed if time presses, there is no alternative. The trouble, however, is that if we then include such arguments in our scientific reasoning, then, as we all know, this can lead to false conclusions. To escape from this dilemma, is there any alternative logical basis from which we can start our reasoning? What is proposed here is that instead of the well tried three valued logic of true, false or probable, we should adopt the three valued logic of true, false or possible. A rational system for analogue arguments can then be developed by these means, and with it the advantages brought by the use of symbols and so on. Such a method, however, includes many necessary changes as to how to structure our problems and how to apply new criteria; and it is some of these changes that are outlined in this note. For instance, it outlines the meaning of ‘causal relationships’ in analogue arguments, as well as how to define ‘rational choice’ in terms of analogue propositions. The advantage throughout is that this allows us to argue with less rather than more data.     

Hypothetical Logic of Proofs

The logic of proofs is a refinement of modal logic introduced by Artemov in 1995 in which the modality ?A is revisited as ?t?A where t is an expression that bears witness to the validity of A. It enjoys arithmetical soundness and completeness and is capable of reflecting its own proofs (?A implies ? ?t?A, for some t). We develop the Hypothetical Logic of Proofs, a reformulation of LP based on judgemental reasoning.