循环并不可恶--《恶性循环:非良基现象的数学》评介

上世纪初,罗素悖论和其他几个集合论悖论的出现引起了许多著名数学家的震惊。为了排除悖论,集合论学者们借助公理方法对康托尔的理论和方法进行了系统整理,建立起了多种严谨的集合论体系,当今应用广泛的ZF集合论就是其中最为著名的一种。直观上集合与类不分,一个性质决定一个类,ZF集合论不能承诺所有这些类都是集合,它用基础公理FA来保证所承诺的集合都是良基的。说一个类X为良基的,是指不能有一个由类组成的无穷序列X1、X2、…、Xn、… (不一定都不相同 )使得…∈Xn∈…∈X2∈X1∈X成立。这样的无穷序列被称为从X开始的一条无穷∈降…     

借“光”看宇宙,凭“道”说有无

如果哈姆雷特是一个宇宙学家 ,他大概会提出 :“有限 ,还是无限 ,这是一个问题。”这其实也是一个悖论 (ｐａｒａｄｏｘ ,似是而非的“吊诡”) :如果宇宙在时间空间上没有开端和边界 ,那么此前已经过去了无限 ,宇宙就被设想为完成了的总和 ,但无限不可能过去 ,总体也不可能完成 ;然而 ,如果宇宙有开端和边界 ,那么在开端之前、边界之外必须有一个空无的所在 ,但“无”中又怎么生出“有”来 (康德悖论 ) ?这个悖论还是“说谎者悖论” (被认为是罗素悖论的一个不严格表述 )的一个放大版 ,因为它同时具备了“自指” (ｓｅｌｆ ｒｅｆｅｒｅｎｃ…     

受欺负与青少年外化问题关系的“健康环境悖论”：敌意性归因的中介作用

“健康环境悖论”是指在总体受欺负水平较低的环境中, 受欺负的个体会表现出更多适应问题。本研究以来自47个班级的1764名5年级到8年级学生为被试(男生956人, 平均年龄14.46岁), 考察了班级平均受欺负水平在个体受欺负经历与外化问题间的调节作用及敌意性归因的中介作用。结果发现：(1)受欺负与外化问题的关系存在“健康环境悖论”现象, 即班级平均受欺负水平能调节个体受欺负经历与外化问题的关系, 在班级平均受欺负水平较低的班级中受欺负经历与外化问题的关联更强; (2)班级平均受欺负水平对受欺负与外化问题的调节作用通过敌意性归因的中介作用实现。本研究证实了受欺负与外化问题的健康环境悖论现象, 并首次揭示了敌意性归因的中介作用机制。     

语义悖论研究的新思路——情境语义学解悖方案探析

一、已有解悖方案的思路及不足自古老的说谎者悖论提出后,两千年来有关悖论的研究文献可谓汗牛充栋。特别是20世纪以来,罗素、塔尔斯基、克里普克等对悖论问题的关注,更进一步深化了人们的认识。但众所周知,无论罗素的类型论方案、塔尔斯基的语言层次理论,还是克里普克的真值间隙论方案,均没能合理地解决悖论问题。罗素的类型论方案试图通过对“自指”的限定来消解说谎者悖论,被认为是“用大炮打苍蝇,苍蝇虽然死了,但毁灭的东西也太多”!塔尔斯基试图通过给语言分层的方法来解决说谎者悖论,但“特设性”的指责却无法逃避。克里普克的方案最…     

认证悖论和绿蓝悖论

“悖论”(Paradox)一词是有岐义的。在关于演绎逻辑的讨论中,悖论是指一种特殊的逻辑矛盾,即:从命题A可推出非A.从非A又可推出A。但在关于归纳逻辑的讨论中,悖论不仅包括一切逻辑矛盾,而且包括那些与人们的直觉或常识相违的推论亦即怪论。这种广义的悖论在归纳逻辑中是为数不少的,其中最基本的并且至今悬而未决的悖论包括认证悖论和绿蓝悖论。下面我们分别介绍这两个归纳悖论。一、认证悖论认证悖论(the Paradoxes of Confirmation)是由著名逻辑学家和科学哲学家亨佩尔(CarlG.Hem. Pel)于1937年首先提出,以后他又在“认证逻辑研究”(即[1],此文最早发表于1945年)和     

自然语言中的自我指称及意义

这篇短文,意在驳斥那种认为所有自指命题从逻辑和语义的角度去分析都不合逻辑的理论。首先我认为一个命题可以以多种方式自指,并不能把所有的自指命题都看作同一类型。其次,在自然语言中,虽然如“这个命题是假的”这样的句子有其逻辑和语义的缺陷,但并不是所有的自指命题都是如此。在这方面,历史上有很多论述。罗素在其分类理论中。第一个指出所有自指命题都无意义,在其使数学基本原则规范化的研究中,分类理论的建立和发展特以用来处理自指悖论。其它逻辑学家在试图使集合理论规范化,精确阐述真假规则过程中也遇到了同样的悖论。显然,建立一种规范系统,确保此系统中形成和派生规则不产生逻辑悖论,是极其重要的。     

把握道德悖论需要注意的学理性问题——兼谈道德悖论研究的视界和价值与意义

不久前兴起的道德悖论研究虽然逐渐引起学界的重视,但对道德悖论的概念即"什么是道德悖论"的问题依然存在疑义,这是因为没有注意从学理上认同道德悖论的界定阅限、明晰道德价值实现结果之悖论与道德价值标准选择之悖论的关系、划清道德悖论与逻辑悖论的界限,以及理解道德悖论之现象与道德悖论之直党(思想)的区别.把握理解这几个学理的问题,是把握道德悖论概念、规范道德悖论研究并展现其应有价值和意义的方法论前提.     

悖论·逻辑矛盾·辩证矛盾——悖论漫谈之一

何谓悖论?就一个形式系统而言,出现“如果p则非p;如果非p则p”(即“pp)这样的循环的“怪圈”,这就是一直困扰着人们的悖论的逻辑形式。 长期以来,人们曾把悖论看成是一种文字游戏,但是自本世纪以来,西方已兴起了对悖论的科学的探讨。这种“兴起”与现代科学的发展相关。现代科学既高度分化又高度综合的整体化趋势,使得普通逻辑的分析性思维一下子显得不够用了,许多科学家和哲学家转而去     

芝诺悖论今昔谈

爱利亚的芝诺为了捍卫老师巴门尼德关于“存在”不动、是一的学说,提出了著名的运动悖论和多悖论,以表明运动和多是不可能的。尽管他的结论在常人看来很荒谬,但他居然给出了乍看起来颇令人信服的论证,故人们常常称这些论证构成了悖论或佯谬,不过,若细细推敲,其结论未必荒谬,其论证未必令人信服,故中性地称这些论证为芝诺论辩(Argument)最为合适。芝诺的运动论辩全部得自亚里士多德在《物理学》中的转述,有四个:     

说谎者悖论的一个新解决方案的纲要(英文)

首先,本文在第二节中提出了谎者悖论所包含的一些问题,比如加强的说谎者悖论,Curry悖论,Yablo悖论,偶然的说谎者悖论,以及塔尔斯基的真不可定义性定理。其次,在第三节简单介绍了塔尔斯基的分层理论,并且指出了他方案的一个最大不足之处就是把一个真谓词分成了很多个个真谓词,这和我们的直观很不符合。之后在第四节介绍了我的一个解悖方案,并讨论了它的一些性质,比如可靠性,完全性,以及我的方案和塔尔斯基方案的区别。第五节说明了第二节中提到的一些问题在我的这个方案下如何处理。     

A Berry and A Russell Without Self-Reference

In this paper I present two new paradoxes, a definability paradox (related to the paradoxes of Berry, Richard and König), and a paradox about extensions (related to Russell’s paradox). However, unlike the familiar definability paradoxes and Russell’s paradox, these new paradoxes involve no self-reference or circularity.     

An entirely non-self-referential Yabloesque paradox

Graham Priest (Analysis 57:236–242, 1997) has argued that Yablo’s paradox (Analysis 53:251–252, 1993) involves a kind of ‘hidden’ circularity, since it involves a predicate whose satisfaction conditions can only be given in terms of that very predicate. Even if we accept Priest’s claim that Yablo’s paradox is self-referential in this sense—that the satisfaction conditions for the sentences making up the paradox involve a circular predicate—it turns out that there are paradoxical variations of Yablo’s paradox that are not circular in this sense, since they involve satisfaction conditions that are not recursively specifiable, and hence not recognizable in the sense required for Priest’s argument. In this paper I provide a general recipe for constructing infinitely many (in fact, continuum-many) such noncircular Yabloesque paradoxes, and conclude by drawing some more general lessons regarding our ability to identify conditions that are necessary and sufficient for paradoxically more generally.     

Liar-type Paradoxes and the Incompleteness Phenomena

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.     

The Elimination of Self-Reference: Generalized Yablo-Series and the Theory of Truth

Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is self-referential but which, taken together, are paradoxical. Yablo’s paradox consists of a countable series of linearly ordered sentences s(0), s(1), s(2),... , where each s(i) says: For each k > i, s(k) is false (or equivalently: For no k > i is s(k) true). We generalize Yablo’s results along two dimensions. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k > i, s(k) is true, where Q is a generalized quantifier (e.g., no, every, infinitely many, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo’s results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference. Various translation procedures that eliminate self-reference from a non-quantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo’s paradox and the translations we offer do not involve self-reference.     

Paradoxical hypodoxes

Synthese - Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’)....     

Rosser-Type Undecidable Sentences Based on Yablo’s Paradox

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type formalizations of Yablo’s paradox for each consistent but not Σ₁-sound theory is dependent on the choice of a standard proof predicate.     

Two Notions of Circularity

Crispin Wright’s epistemic response to McKinsey’s paradox is to argue that introspective knowledge of the first premise fails to transmit across the semantic externalist entailment in the second premise to the conclusion that one has such untoward knowledge of the external world. This paper argues first that Stewart Cohen and Jonathan Vogel’s bootstrapping arguments suffer from a novel kind of epistemic circularity, which triggers failure of transmission but allows for the possibility of basic perceptual knowledge. It is then argued that McKinsey’s paradox falls out as a special case of this template for transmission failure. The circularity in play is semantic: the paradox illicitly imports semantically relevant properties of knowledge-individuating sources into the contents of the knowledge states that those sources individuate by instantiating those properties. Importantly, this diagnosis permits the possibility of basic introspective knowledge as propounded by Tyler Burge and other semantic externalists.     

A neglected response to the paradoxes of confirmation

Hempel's paradox of the ravens, and his take on it, are meant to be understood as being restricted to situations where we have no additional background information. According to him, in the absence of any such information, observations of FGs confirm the hypothesis that all Fs are G. In this paper I argue against this principle by way of considering two other paradoxes of confirmation, Goodman’s “grue” paradox and the “tacking” (or “irrelevant conjunct”) paradox. What these paradoxes reveal, I argue, is that a presumption of causal realism is required to ground any confirmation; but once we grant causal realism, we have no reason to accept the central principles giving rise to the paradoxes.     

Hobson’s Conception of Definable Numbers

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.