How Mathematics Isn't Logic

If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting : nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The '=' of mathematics is an objectual relation between numbers; the '=' of logic marks a syntactic relation of coreferring terms.     

论中医养生内涵与中国传统文化

以古代文献为依据,研究中医养生内涵与中国传统文化形态闻的密切关联.认为道家文化是中医养生理论的思想渊源,儒家的养德、致中和、卫生思想促进和丰富了中医养生,佛家的思想对中医养生文化起到补充作用.相互间不断发生碰撞,彼此交融渗透,最终发展成独具特色的中医养生体系.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.     

中国传统"革命"思想的两个维度及毛泽东的"革命"逻辑

传统中国蕴含着两种“革命”思想的维度:一是圣人救世的“汤武革命”,一是基于压迫而进行反抗的农民起义。这两个维度对于既系统地接受过中国传统的规范性文化教育,同时又深受非规范性文化熏染的毛泽东来说都产生了极大的影响,并形成了他“圣人救世”与“压迫-反抗”的两种“革命”逻辑。这两种具有不同文化气质的“革命”逻辑在毛泽东思想中得以结合的根本基础是他尚变求动的宇宙观和人生观。     

How to Apply Mathematics

This paper presents a novel account of applied mathematics. It shows how we can distinguish the physical content from the mathematical form of a scientific theory even in cases where the mathematics applied is indispensable and cannot be eliminated by paraphrase.     

Counterfactual Logic and the Necessity of Mathematics

Journal of Philosophical Logic - This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious...     

Logic and Mathematics in the Seventeenth Century

According to the received view (Bocheński, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good reasons, it overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Bocheński characterizes as ‘classical logic’, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic.     

Against the Judgment-Dependence of Mathematics and Logic

Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases.     

为什么形而上学需要逻辑学，而数学则不需要

本文对皮尔士有关数学、形而上学和逻辑学之间关系的理论进行了辩护.与常识观点不同,皮尔士认为,以学院逻辑为主的逻辑学是一门实证科学,它的任务在于对思想进行有效控制,以便获得真理.数学不需要这样的逻辑学,因为数学不是一种实证科学,它所关注的不是实证事实或真理,而是精确、清晰和连贯.数学是对经验事实的理想化,因而它首先需要的是想象力、专注力和概括力.在皮尔士看来,科学的形而上学应该是一种探讨经验事实的实证科学,它需要逻辑学尤其是学院逻辑来加以控制,因而形而上学需要逻辑学.