集异璧－GEB，是数学家哥德尔、版画家艾舍尔、音乐家巴赫三个名字的前缀。《哥德尔、艾舍尔、巴赫书：集异璧之大成》是在英语世界中有极高评价的科普著作，曾获得普利策非小说奖。它通过对哥德尔的数理逻辑，艾舍尔的版画和巴赫的音乐三者的综合阐述，引人入胜地介绍了数理逻辑 学、可计算理 论、人工智能学、语言学、遗传学、音乐、绘画的理论等方面，构思精巧、含义深刻、视野广阔、富于哲学韵味。
中译本前后费时十余年，译者都是数学和哲学的专家，还得到原作者的直接参与，译文严谨通达，特别是在原作者的帮助下，把西方的文化典故和说法，尽可能转换为中国文化的典故和说法，使这部译本甚至可看作是一部新的创作，也是中外翻译史上的一个创举。

王浩（1921—1995），美籍华裔数学家、逻辑学家、计算机科学家、哲学家。1921年生于山东济南市。1943年毕业于西南联合大学数学系。1945年于清华大学研究生院哲学系毕业。曾师从金岳霖、王宪钧、沈有鼎等。1946年赴哈佛大学留学，师从蒯因（W. V. O. Quine），两年时间即获哈佛大学哲学博士学位。在哈佛短暂教学之后赴苏黎世与贝奈斯（Paul Bernays）一起工作。1954年—1956年，在牛津大学任第二届约翰•洛克讲座主讲，又任逻辑及数理哲学高级教职，主持数学基础讨论班。1961年—1967年，任哈佛大学教授。1967年—1991年，任洛克菲勒大学逻辑学教授。20世纪50年代初被选为美国科学院院士，后又被选为不列颠科学院外国院士。1983 年，被国际人工智能联合会授予第一届“数学定理机械证明里程碑奖”，以表彰他在数学定理机械证明研究领域中所作的开创性贡献。著有《数理逻辑概论》、《从数学到哲学》、《哥德尔》、《超越分析哲学》等专著。

Beyond Analytic Philosophy
Doing Justice to What We Know
Hao Wang
Preface
ix
Introduction
1
1. Russell and philosophy in this century 45
2. A digression on Wittgenstein's philosophy 75
3. From Vienna 1925 to America 1984 101
4. Quine's logical negativism 153
5. Metaphilosophical observations 191
References 215
Chronological table 231
Index
261

Douglas Hofstadter's long-awaited return to the themes of Gödel, Escher, Bach--an original and controversial view of the nature of consciousness and identity.
Can thought arise out of matter? Can self, a soul, a consciousness, an "I" arise out of mere matter? If it cannot, then how can you or I be here?
I Am a Strange Loop argues that the key to understanding selves and consciousness is the "strange loop"--a special kind of abstract feedback loop inhabiting our brains. The most central and complex symbol in your brain or mine is the one called "I." The "I" is the nexus in our brain, one of many symbols seeming to have free will and to have gained the paradoxical ability to push particles around, rather than the reverse.
How can a mysterious abstraction be real--or is our "I" merely a convenient fiction? Does an "I" exert genuine power over the particles in our brain, or is it helplessly pushed around by the laws of physics?
These are the mysteries tackled in I Am a Strange Loop, Douglas R. Hofstadter's first book-length journey into philosophy since Gödel, Escher, Bach. Compulsively readable and endlessly thought-provoking, this is the book Hofstadter's many readers have been waiting for.

Metamathematics is mathematics used to study mathematics', or it involves the application of a philosophy of mathematics. The first part of this general description appears tautological, or is perhaps open to Bertrand Russell's and Alfred Whitehead's types of antimonies (e.g., "the of all sets is not a set"), as described in their famous "Principia Mathematica." An alternative, non-circular definition is as follows: Metamathematics is the study of metatheories of standard theories in mathematics, or about mathematical--not purely logical'-- theories. Thus, in Encyclop]dia Britannica, metatheory is defined as a ," MT, the subject matter of which is another theory, T . A finding proved in the former (MT) that deals with the latter (T) is known as a metatheorem " (cited from Metatheory-Encyclop]dia Britannica Online). Thus, a major part of metamathematics deals with: metatheorems, that is " about theorems," meta-propositions about propositions, metatheories about mathematical proofs (that of course utilize logic, but also are based upon fundamental mathematics concepts), and so on. Meta-mathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. Bertrand Russell's and Alfred Whitehead's type of paradoxes is yet another important example of possible contradictions due to such failures in the 'old' set theory.

Both in science and in practical affairs we reason by combining facts only inconclusively supported by evidence. Building on an abstract understanding of this process of combination, this book constructs a new theory of epistemic probability. The theory draws on the work of A. P. Dempster but diverges from Depster's viewpoint by identifying his 'lower probabilities' as epistemic probabilities and taking his rule for combining 'upper and lower probabilities' as fundamental. This book opens with a critique of the well-known Bayesian theory of epistemic probability. It then proceeds to develop an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called 'monotone of order of infinity.' and Dempster's rule for combining such set functions. This rule, together with the idea of 'weights of evidence,' leads to both an extensive new theory and a better understanding of the Bayesian theory. This book concludes with a brief treatment of statistical inference and a discussion of the limitations of epistemic probability. Appendices contain mathematical proofs, which are relatively elementary and seldom depend on mathematics more advanced that the binomial theorem.

From the Reviews: "...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." - "Philosophy and Phenomenological Research".

Uncertainty is a fundamental and unavoidable feature of daily life; in order to deal with uncertaintly intelligently, we need to be able to represent it and reason about it. In this book, Joseph Halpern examines formal ways of representing uncertainty and considers various logics for reasoning about it. While the ideas presented are formalized in terms of definitions and theorems, the emphasis is on the philosophy of representing and reasoning about uncertainty; the material is accessible and relevant to researchers and students in many fields, including computer science, artificial intelligence, economics (particularly game theory), mathematics, philosophy, and statistics.Halpern begins by surveying possible formal systems for representing uncertainty, including probability measures, possibility measures, and plausibility measures. He considers the updating of beliefs based on changing information and the relation to Bayes' theorem; this leads to a discussion of qualitative, quantitative, and plausibilistic Bayesian networks. He considers not only the uncertainty of a single agent but also uncertainty in a multi-agent framework. Halpern then considers the formal logical systems for reasoning about uncertainty. He discusses knowledge and belief; default reasoning and the semantics of default; reasoning about counterfactuals, and combining probability and counterfactuals; belief revision; first-order modal logic; and statistics and beliefs. He includes a series of exercises at the end of each chapter.

Computers are ubiquitous yet to many they remain objects of irreducible mystery. This text looks at the question of how today's computers can perform such a variety of tasks if computing is just glorified arithmetic. The author illustrates how the answer lies in the fact that computers are essentially engines of logic and that their hardware and software embody concepts developed over centuries by logicians. "Engines of Logic" gives the reader a clear explanation of how and why computers work.

Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics. The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Godel's Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the author's website.

This book draws on ideas from philosophical logic, computational logic, multi-agent systems, and game theory to offer a comprehensive account of logic and games viewed in two complementary ways. It examines the logic of games: the development of sophisticated modern dynamic logics that model information flow, communication, and interactive structures in games. It also examines logic as games: the idea that logical activities of reasoning and many related tasks can be viewed in the form of games.
In doing so, the book takes up the “intelligent interaction" of agents engaging in competitive or cooperative activities and examines the patterns of strategic behavior that arise. It develops modern logical systems that can analyze information-driven changes in players’ knowledge and beliefs, and introduces the “Theory of Play" that emerges from the combination of logic and game theory. This results in a new view of logic itself as an interactive rational activity based on reasoning, perception, and communication that has particular relevance for games.
Logic in Games, based on a course taught by the author at Stanford University, the University of Amsterdam, and elsewhere, can be used in advanced seminars and as a resource for researchers.