古代西方语义学的萌芽述评

“语义学”一词源于希腊文,其初是指一种与语音学相对的语言学的分支科学。后来“语义学”这个术语有了不同涵义。从研究对象和范畴来看,语义学可分为四种:语言学的语义学、哲学的语义学、普通语义学和逻辑语义学。作为一门独立学科的逻辑语义学只是到了十九世纪以后才出现,弗雷格作出了不可磨灭的贡献,罗素也作了极有意义的陈述,而真正的现代形式的逻辑语义学则从塔斯基算起,然而在西方,“如果我们说的语义学,就是(用一个最普遍的说法)指关于那些语言表达式和它们所指示的对象之间的关     

塔斯基的真理定义与物理论

塔斯基（Tarski）于1933年发表了他著名的真理定义,并相信该定义能够为其物理论的哲学立场服务;但费尔德（Field）批评说,塔斯基实际上所给出的真理定义并没能成功地达成这个目标。不过,费尔德同时也认为,一个部分奠基在塔斯基真理定义之上、并且是物理论者可以接受的化约性真理理论并非不可能。费尔德对于塔斯基真理定义的这些批评,在哲学家中曾经引起了许多意见不一的反应。本文的目的是想回答在这些讨论当中曾经被提出过的三个问题。首先,塔斯基实际上所给出的真理定义是不是一个物理论者可以接受的化约性定义？其次,费尔德所设想的那种可被物理论者所接受的化约性真理理论是否可能成功？最后,如果塔斯基实际上所给出的定义并不能符合物理论的化约目标,那么,一个物理论者是否便应该据此去反对塔斯基的真理定义？本文的最终结论是：这三个问题的正确答案都是否定的。     

论解释的形式基础和证据基础

自戴维森在 1967年发表他的著名论文《真理与意义》后, 学界把他关于真理与意义的理论称为戴维森纲领 (Davidson’sprogram)。戴维森纲领的卓越之处在于戴维森将塔斯基的约定 -T (conven tion T) 做了适当修改, 将真理理论相对于特定的语言及其使用者, 使约定 -T与意义联系起来, 为自然语言的恰当的语义学提供了一个清楚的、可检验的标准。戴维森基于“彻底解释” (radicalinter pretation) 概念提出的解释理论是戴维森纲领的重要内容, 其要旨在于说明理解 (understanding, mak ingintelligible) 一个言说者的言说行为所必须涉及的要素。…     

逻辑基础问题(下)

第三部分首先回应那些对在第二部分中所阐述的形式性/逻辑性标准的批评,然后为逻辑与数学之间的关系提出一个新的解释,以替换传统的逻辑主义解释。逻辑主义企图把数学归约到逻辑,但是它的缺陷之一在于无法为逻辑提供一个基础。通过把逻辑与数学都建基于形式之中,以及对二者的劳动分工与合作的解释,新基础解决了这个问题。本文进一步证明,新基础保证了逻辑的必然性、普遍性、主题中立性、强大规范性以及其他种种特性,并在最后讨论了逻辑中的错误和修正。     

动词体的球态语义学

本文介绍由塔斯基的立体几何导出的球态语义学,并将其应用于自然语言中的动词体现象。球态语义学特别适合应用于英语的进行体。这种方法有以下优点（i）它扩展了区间式语义,并同时避免了其缺陷,（ii）它解决了未完成体难题,（iii）它的解决方法无需诉诸最终结果策略。逻辑方法一般被认为难于处理自然语言的动词体问题。基于点的时间结构以及建立在该结构之上的经典普莱尔时态逻辑（[18]）太弱了。而基于区间的时态语义则缺乏足够的表达力,并且难以解释进行体（[4,8]）.本文给出一种新的基于球上整体－部分关系概念的模型和时态语义。这种球态语义学建基于塔斯基1927年引入的立体几何之上。与基于点和基于区间的语义不同,在球态语义学中很多动词体区分都由统一的逻辑方法刻画。在一个由封闭球构成的论域中,可达关系由相切性概念给出。相应地,我们可定义外切、内切、外径、内径以及同心等基本概念。与区间式语义不同,球是论域的初始概念,球态语义学不是在时间段而是在球中对事件赋值。因此,仅将时间区间作为初始概念而不承认其端点初性性的问题不复存在。英语中的进行体由球上的连续行动来刻画。行动是非终止的,只要球没有由外切相离。相应地,外切相离刻车动作完成。我们区分在均匀球和非均匀球中发生事件的整体-部分关系。非持续动作视为直径为零的同心球。球态语义学根据动作或执行完成的时刻来定义时间概念,其中不需要时间端点的概念。在保持与基于区间的时间模型类似的基础上,球态语义学暗示了一种关于可能世界的定性概念,并且它有利于解决时间的循环概念问题。     

实践检验的两种功能──从现代数学的观点看

真理标准是真理性问题的重要组成部分。辩证唯物论在真理标准问题上认为,实践是检验真理的唯一标准,但并不排斥逻辑证明在认识过程中的作用。根据实践──认识──实践的唯物论认识论原理,这一观点不能说是错的。但是,现代数学的发展表明,它需要进一步具体说明逻辑检验（证明）与实践检验的关系,以及实践是如何检验数学的,才能真正坚持唯物论。本文将根据辩证唯物论原理和现代数学发展的新特点,提出：逻辑检验与实践检验属于两个不同认识层次的检验标准;实践检验的两种功能等观点。这些新见解是我的专著《数学的对象与性质》（即将…     

论现代数学背景下的康德几何观

<正>在对数学的哲学基础研究中,20世纪通常说来有三大流派:逻辑主义、形式主义和直觉主义。但在哥德尔定理出现以后,一致性问题给逻辑主义和形式主义带来了难以克服的困难;而直觉主义又很难对数学中的"抽象实体"作出合理的解释,并且数学中有些重要定理的证明完全超出了直觉主     

弱逻辑F的矢列演算

本文建立起弱逻辑F的G3-型矢列演算G3F,并证明弱化规则、收缩规则和切割规则在G3F中是可容许的。此外,本文还运用证明论的方法给出一个通过哥德尔–麦金西–塔斯基翻译将G3F嵌入到G3K的新证明。     

子集空间下的邻域扩张

子集空间逻辑是刻画知识及其在证据支持度提升时发生变化的一种具有拓扑逻辑风格的简单架构的模态认知逻辑。与关系语义学不同,子集空间逻辑的语义学借助"邻域"而非"可通达关系"来表达不确定性区间。邻域的缩小体现不确定性的减少;在认知语境下,这种不确定性的减少就表现为知识的增长。子集空间逻辑主要刻画邻域的缩小。与邻域缩小相对应的邻域扩张同样具有理论研究的价值,然而却没有在子集空间逻辑中得到刻画。本文在子集空间逻辑的框架下探讨邻域的扩张。主要成果是给出带有邻域扩张算子的子集空间逻辑,为其引入关系语义学和公理系统,并证明该系统的完全性。     

相对主义的成真条件语义学

本文在扼要考察相对主义的基本思路和常见批评之后,概括了语义学相对主义的基本动机和技术手段。我们将麦克法兰式的激进相对主义语义学框架与卡普兰式二维语义方案进行比较之后,通过麦克法兰的论述解释如何从哲学上为相对主义的真概念提供实质性的说明。最后我们亦简要分析和回应了语义学相对主义可能面临的挑战。     

Logical structuralism and Benacerraf’s problem

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.     

The defeater version of Benacerraf’s problem for a priori knowledge

Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue that a new, more challenging, version of Benacerraf’s problem can be constructed. The new version—what I call the Defeater Version—of Benacerraf’s problem makes use of a no-defeater condition on knowledge and justification. I conclude by arguing that the best way to avoid the problem is to construct a theory of how a priori judgments reliably track the facts. I also suggest four different kinds of theories worth pursuing.     

The Innateness Hypothesis and Mathematical Concepts

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge?     

On the creative role of axiomatics. The discovery of lattices by Schröder,Dedekind, Birkhoff,and others

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schröder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative aspect of axiomatics for mathematical practice is brought to the fore.     

The role of practice and habituation in Socrates’ theory of ethical development

ABSTRACT

The goal of this paper is to challenge the standard view that Socrates of the early Platonic dialogues is an intellectualist with respect to virtue. Through a detailed analysis of the educational theory laid out in the early dialogues, it will be argued that Socrates believes that the best way to cultivate virtues in his interlocutors is not to convince them of ethical truths by way of reason and argument alone, but to encourage them to participate in the practice of virtue. Habit and practice are essential to the cultivation of virtue because they mould the desires and dispositions of the agent and promote a kind of knowledge that cannot be achieved discursively – craft-knowledge. Only when agents have achieved craft-knowledge can they be counted on to act virtuously on every occasion; and craft-knowledge can only be achieved by way of practice and habituation.     

Understanding understanding mathematics

In this paper we look at some of the ingredients and processes involved in the understanding of mathematics. We analyze elements of mathematical knowledge, organize them in a coherent way and take note of certain classes of items that share noteworthy roles in understanding. We thus build a conceptual framework in which to talk about mathematical knowledge. We then use this representation to describe the acquisition of understanding. We also report on classroom experience with these ideas.     

Epistemology Without History is Blind

In the spirit of James and Dewey, I ask what one might want from a theory of knowledge. Much Anglophone epistemology is centered on questions that were once highly pertinent, but are no longer central to broader human and scientific concerns. The first sense in which epistemology without history is blind lies in the tendency of philosophers to ignore the history of philosophical problems. A second sense consists in the perennial attraction of approaches to knowledge that divorce knowing subjects from their societies and from the tradition of socially assembling a body of transmitted knowledge. When epistemology fails to use the history of inquiry as a laboratory in which methodological claims can be tested, there is a third way in which it becomes blind. Finally, lack of attention to the growth of knowledge in various domains leaves us with puzzles about the character of the knowledge we have. I illustrate this last theme by showing how reflections on the history of mathematics can expand our options for understanding mathematical knowledge.     

Defining and measuring conceptual knowledge in mathematics

A long tradition of research on mathematical thinking has focused on procedural knowledge, or knowledge of how to solve problems and enact procedures. In recent years, however, there has been a shift toward focusing, not only on solving problems, but also on conceptual knowledge. In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking literature, and (2) how conceptual knowledge is defined, operationalized, and measured in three mathematical domains: equivalence, cardinality, and inversion. We uncovered three general issues. First, few investigators provide explicit definitions of conceptual knowledge. Second, the definitions that are provided are often vague or poorly operationalized. Finally, the tasks used to measure conceptual knowledge do not always align with theoretical claims about mathematical understanding. Together, these three issues make it challenging to understand the development of conceptual knowledge, its relationship to procedural knowledge, and how it can best be taught to students. In light of these issues, we propose a general framework that divides conceptual knowledge into two facets: knowledge of general principles and knowledge of the principles underlying procedures.     

Knowledge and the regularity theory of information

Fred Dretske's Knowledge and the Flow of Information is an extended attempt to develop a philosophically useful theory of information. Dretske adapts central ideas from Shannon and Weaver's mathematical theory of communication, and applies them to some traditional problems in epistemology. In doing so, he succeeds in building for philosophers a much-needed bridge to important work in cognitive science. The payoff for epistemologists is that Dretske promises a way out of a long-standing impasse — the Gettier problem. He offers an alternative model of knowledge as information-based belief, which purports to avoid the problems justificatory accounts face. This essay looks closely at Dretske's theory. I argue that while the information-theoretic framework is attractive, it does not provide an adequate account of knowledge. And there seems to be no way of tightening the theory without introducing some version of a theory of justification — the very notion Dretske's theory was designed to avoid.     

Scientific Philosophy,Mathematical Philosophy,and All That

This article suggests that scientific philosophy, especially mathematical philosophy, might be one important way of doing philosophy in the future. Along the way, the article distinguishes between different types of scientific philosophy; it mentions some of the scientific methods that can serve philosophers; it aims to undermine some worries about mathematical philosophy; and it tries to make clear why in certain cases the application of mathematical methods is necessary for philosophical progress.