Classical Negation Strikes Back: Why Priest’s Attack on Classical Negation Can’t Succeed

Dialetheism is the view that some true sentences have a true negation as well. Defending dialetheism, Graham Priest argues that the correct account of negation should allow for true contradictions (sentences of the form \(\alpha \) and \(\lnot \alpha \)) without entailing triviality. A negation doing precisely that is said to have ‘surplus content’. Now, to defend that the correct account of negation does have surplus content, Priest advances arguments to hold that classical Boolean negation (which rules out surplus content) does not even make sense without begging the question against the dialetheist. We shall argue that Priest’s arguments may be turned upon themselves, and that he may also be accused of begging the question against the classical logician. We then advance an argument to the effect that Priest’s account of negation falls short of satisfying his own desiderata on a correct account of a negation: a theory of negation that attempts to represent contradictions cannot coherently allow surplus content, and vice-versa, a negation allowing for surplus content bans contradiction.     

Singular Propositions,Negation and the Square of Opposition

This paper contains two traditions of diagrammatic studies namely one, the Euler–Venn–Peirce diagram and the other, following tradition of Aristotle, the square of oppositions. We put together both the traditions to study representations of singular propositions (through a diagram system Venn-i, involving constants), their negations and the inter relationship between the two. Along with classical negation we have incorporated negation of another kind viz. absence (taking a cue from the notion of ‘abhãva’ existing in ancient Indian knowledge system). We have also considered the changes that take place in the context of open universe.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

Data mining feature selection for credit scoring models

The features used may have an important effect on the performance of credit scoring models. The process of choosing the best set of features for credit scoring models is usually unsystematic and dominated by somewhat arbitrary trial. This paper presents an empirical study of four machine learning feature selection methods. These methods provide an automatic data mining technique for reducing the feature space. The study illustrates how four feature selection methods—‘ReliefF’, ‘Correlation-based’, ‘Consistency-based’ and ‘Wrapper’ algorithms help to improve three aspects of the performance of scoring models: model simplicity, model speed and model accuracy. The experiments are conducted on real data sets using four classification algorithms—‘model tree (M5)’, ‘neural network (multi-layer perceptron with back-propagation)’, ‘logistic regression’, and ‘k-nearest-neighbours’.     

Estimating the demand distributions of single-period items having frequent stockouts

Very often, the service level of a single-period newsboy-type product is set at such a low level that: (i) stockouts occur in the majority of the periods, and (ii) a large right-hand side of the empirical demand distribution is never observable. This paper reports a practical approach for estimating the periodic-demand distribution of such a product. The approach has three components: (i) using the non-parametric ‘product limit’ method to estimate the fractiles of the observable left-hand side of the empirical distribution; (ii) using a subjective approach and an ‘extrapolation of hourly sales’ approach to ‘fill in’ the missing right-hand side of the empirical distribution; (iii) fitting the estimates obtained in the preceding two components to a Tocher curve — which can handle the diversity of shapes of a realistic demand distribution and is also computationally very convenient for subsequent calculations for production/inventory decisions. The entire approach is shown to be simpler but more powerful than existing alternatives for the problem.     

A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables

The bounded-dual-Lipschitz and Prohorov distances from the ‘empirical measure’ to the ‘average measure’ of independent random variables converges to zero almost surely if the sequence of average measures is tight. Three examples are also given.     

Variations on a theme: introducing new representations of fraction into two classrooms

Abstract

The interplay between generalisations and particular instances—examples—is an essential feature of mathematics teaching and learning. In this paper, we bring together our experiences of personal and classroom mathematics activity, and demonstrate that examples do not always fulfil their intended purpose (to point to generalisations). A distinction is drawn between ‘empirical’ and ‘structural’ generalisation, and the role of generic examples is discussed as a means of supporting the second of these qualities of generalisation.     

PLF.sets的实证研究及其高质量仿真应用

为高效率与高质量仿真城市人口迁移与演变趋势, 基于Zadeh模糊集的基础知识, 引入哲学否定与计算机逻辑中对立否定、中介否定和矛盾否定的概念。在反复研究和论证了三种逻辑否定之间的内在关系、基本特征和融合条件后, 创新性地提出了PLF.sets泛逻辑性模糊集, 形成可直接参与计算的多种逻辑变量及其演变形式的评价因数集。PLF.sets在饱和验证其运算规则和逻辑转化可行性的前提下, 通过已知局部隶属函数g(x)构建出逼近性能良好的整体隶属度函数f(x)。它能够在确定的值域U[a, b]内, 使得任意的x(x∈U)映射于f(x)值。并在高度仿真和逼近原始系统规律的同时, 逻辑简化系统函数产生的复杂性计算。PLF.Sets于《管理信息系统》教学质量评价的实证研究结果充分证明, PLF.sets方法可以化繁为简, 简化运算方法和步骤;显示出高效性、准确性、普适性和简洁性的特点。PLF.sets方法在对北京城市人口迁移与演变趋势仿真的应用表明:(1)北京城市人口流动变化净增加值的定位均是“多”的等级;(2)北京城市人口流动网络及其关联度特征的定性等级均是“高”级别。最终高质量地评估了正向迁移城市的人口流动格局、人口迁移与演变趋势。     

A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.     

Process and content: two ways of using SSM

Soft systems methodology (SSM) includes several ways of gaining a rich appreciation of the problem situation addressed. ‘Analysis One’, exploration of the intervention itself, is the subject here, since it is sparsely covered in the literature. The analysis is conducted in terms of three roles: ‘client’, ‘problem solver’ and ‘problem owner’. Whoever is in the role of ‘problem solver’ is free to define a list of possible ‘problem owners’, which brings many perspectives to bear on the situation. It was realized that ‘client’ and ‘problem solver’ should themselves feature in the ‘problem owner’ list. The ‘problem’ owned by the ‘problem solver’ is that of undertaking the intervention. This led to a realization that SSM is relevant to both the content of a perceived situation (SSMc) and the process of dealing with that content (SSMp). This development is described and illustrated by work in the National Health Service. The focus of the SSM use was to define the intellectual process for a service specification project which NHS professionals would themselves carry out.     

Causal study of low stakeholder engagement in healthcare simulation projects

Stakeholder engagement plays a fundamental role in the success of ‘operational research’ initiatives including simulation projects. However, there is little empirical evidence of real engagement in the context of healthcare simulation. This paper principally examines this issue and aims to provide insights into the possible causes. The paper reports on the results of a literature review and 10 field studies within the UK healthcare settings, supplemented with the authors’ experience in order to arrive at an initial list of the causes, which will then be tested through a survey of expert opinions. Twelve primary and 26 secondary causal factors, which received statistically significant level of agreement from the experts, are presented in a fish-bone diagram. The findings indicate that communication gap between simulation and stakeholder groups is the top primary factor contributing the most to the poor stakeholder engagement in healthcare simulation projects, followed by ‘poor management support’, ‘clinician’s high workload’ and ‘failure in producing tangible and quick results’.     

Can we have confidence in generic structures?

The system dynamics concept of ‘generic structure’ is dividable into three sub-types. This paper analyses the validity of these three, using both practical and theoretical perspectives. Firstly, a new set of measures is developed for generating validity—‘confidence’—amongst a group using generic structures in a practical modelling situation. It is concluded that different confidence criteria are implicitly employed; there is an argument for trading-off model precision and analytical quality for simplicity and ease of use and future research is needed to combine these ‘process’ and ‘content’ aspects of confidence. From a theoretical stance it is shown that with two of the sub-types a scientific notion of confidence is achievable whereas the third (‘archetypes’) involves merely metaphorical thinking. It is concluded that the theoretical status of archetypes requires further development, whilst ensuring that its benefits are retained.     

Natural Deduction for Post’s Logics and their Duals

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued (\(k\geqslant 3\)) logics as well as for all Dual Post’s k-valued logics.     

Evaluating foreign direct investments: new challenges for strategic planners

Business and hence strategic planning is becoming increasingly international. Within that internationalisation, foreign direct investment (FDI) has become increasingly important, especially in E/SE Asia. A logical step then is to seek appropriate methods for modelling and evaluating those FDIs. Excluding the core financial appraisal the literature is sparse. This paper describes an empirical examination of these issues based on a sample from the UK's top 250 listed companies. The results of responses to a questionnaire are described with elaboration gleaned from eight detailed, follow up interviews. The picture to emerge is one of systematic treatment of the basic financial calculations but fairly rudimentary modelling of non-financial aspects. The treatment of risk was most typically through amendment of cash flows for core business risks and adjustment of the hurdle rate for ‘wilder’ risks. There is discussion of some of the subtler points of ‘wild’ risk's treatment and some outline suggestions for models to tackle other non-financial aspects of overall project evaluation. It is also suggested that the thinking developed may be applicable to a wider class of investment proposals.     

Polynomial induction and length minimization in intuitionistic bounded arithmetic

It is shown that the feasibly constructive arithmetic theory IPV does not prove (double negation of) LMIN(NP), unless the polynomial hierarchy CPV‐provably collapses. It is proved that PV plus (double negation of) LMIN(NP) intuitionistically proves PIND(coNP). It is observed that PV + PIND(NP ∪ coNP) does not intuitionistically prove NPB, a scheme which states that the extended Frege systems are not polynomially bounded. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Markov chain models for delinquency: Transition matrix estimation and forecasting

A Markov chain is a natural probability model for accounts receivable. For example, accounts that are ‘current’ this month have a probability of moving next month into ‘current’, ‘delinquent’ or ‘paid‐off’ states. If the transition matrix of the Markov chain were known, forecasts could be formed for future months for each state. This paper applies a Markov chain model to subprime loans that appear neither homogeneous nor stationary. Innovative estimation methods for the transition matrix are proposed. Bayes and empirical Bayes estimators are derived where the population is divided into segments or subpopulations whose transition matrices differ in some, but not all entries. Loan‐level models for key transition matrix entries can be constructed where loan‐level covariates capture the non‐stationarity of the transition matrix. Prediction is illustrated on a $7 billion portfolio of subprime fixed first mortgages and the forecasts show good agreement with actual balances in the delinquency states. Copyright © 2010 John Wiley & Sons, Ltd.     

Axioms for a fuzzy measure of inequality

It has been argued that the concept of inequality is inherently imprecise. A difficulty with standard inequality measures is that they generally make no allowances for this, and when they do, it is by dropping the ‘completeness’ axiom in ranking social states (e.g. the Lorenz criterion). It is argued here that the erring axiom is not ‘completeness’ but ‘exactness’ which, being implicit, tends to escape notice. A fuzzy measure of inequality, along with a set of necessary and sufficient axioms, is established. The new measure has several attractive properttes: It allows for tentative judgements and doubts. It is easy to interpret and compute, and the Gini ranking turns out to be a nearest exact approximation of it.     

Re-sourcing teachers’ work and interactions: a collective perspective on resources, their use and transformation

This paper reviews the literature on the theme of mathematics teachers’ work and interactions with resources, taking a particular perspective, the so-called ‘collective perspective’ on resources, their use and transformation. The review is presented under three headings: (1) theoretical frameworks commonly used in this area of research; (2) teachers’ interactions with resources in terms of their design and use; and (3) teachers’ interactions with resources in terms of teacher learning and professional development. From the literature, and the collection of papers in this issue, we argue that the collective dimensions play an important role in mathematics teachers’ work with resources and in their professional learning/development. Further empirical investigations are likely to be needed on: how teachers may work in collectives and with resources, and in which ways ‘productive’ collectives may form and work together; which roles particular resources can play in these delicate constellations and how particular resources may support teachers in their work and learning; and which kinds of resources offer opportunities for community building.     

Poverty equilibrium—An introductory systemic analysis

This paper represents an attempt to clarify the field of the studies concerning the poor stagnant economy. Using systemic analysis the original theories are reformulated in such a way as to highlight their similarities and dynamic structure. The works of Ragnar Nurkse, Gunnar Myrdal, John K. Galbraith and Harvey Leibenstein are presented as components of a general theory of stagnation at low incomes. The paper ends with an attempt to formulate a ‘modern version’ of the theory, using the ‘chaotic dynamics approach’ to depict the situation. The main purpose of this paper is to ‘formulate’ the concept or, in other words, to clarify mathematically the hypothesis in order to test it. The building of the empirical tests and of economic models to explain it is treated in other papers.     

The epistemology of complex systems

The human knowledge structure is acquired by a very complex process of learning. ‘Folk knowledge’, acquired in ordinary daily life, may be distinguished from ‘Scholarly knowledge’. Folk knowledge has images which range from about 10⁵ to 10⁻⁵ of the human size; scholarly knowledge produces images which range from about 10²⁰ to 10⁻²⁰ of the human size. Knowledge results from the interaction of internal and external messages. Internal messages produce perceptions of identities—propositions which cannot be untrue, and also perceptions of ‘near-identities’—propositions which have a high probability of being true. Beyond these are the empirical propositions, derived from ordered observation of the records of the past and the present, and from experiment. These are always subjet both to error, and the reduction of error. Each field of knowledge has to find its appropriate method, depending on the nature of the system about which knowledge is sought. These systems may be classified as predictable (with stable parameters) and partially predictable (systems involving information, or subject to parametric change.) It is a mistake to transfer methods appropriate in one field to another, where they may be inappropriate.