The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution. Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schroder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Godel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later. Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.

At the heart of the justification for the reasoning used in modern mathematics lies the completeness theorem for predicate calculus. This unique textbook covers two entirely different ways of looking at such reasoning. Topics include:

• the representation of mathematical statements by formulas in a formal language;
• the interpretation of formulas as true or false in a mathematical structure;
• logical consequence of one formula from others;
• formal proof;
• the soundness and completeness theorems connecting logical consequence and formal proof;
• the axiomatization of some mathematical theories using a formal language;
• the compactness theorem and an introduction to model theory.

This book is designed for self-study by students, as well as for taught courses, using principles successfully developed by the Open University and used across the world. It includes exercises embedded within the text with full solutions to many of these. In addition there are a number of exercises without answers so that students studying under the guidance of a tutor may be assessed on the basis of what has been taught.

Some experience of axiom-based mathematics is required but no previous experience of logic. Propositional and Predicate Calculus gives students the basis for further study of mathematical logic and the use of formal languages in other subjects.

Derek Goldrei is Senior Lecturer and Staff Tutor at the Open University and part-time Lecturer in Mathematics at Mansfield College, Oxford, UK.

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses.

Rippling is a radically new technique for the automation of mathematical reasoning. It is widely applicable whenever a goal is to be proved from one or more syntactically similar givens. It was originally developed for inductive proofs, where the goal was the induction conclusion and the givens were the induction hypotheses. It has proved to be applicable to a much wider class of tasks, from summing series via analysis to general equational reasoning. The application to induction has especially important practical implications in the building of dependable IT systems, and provides solutions to issues such as the problem of combinatorial explosion. Rippling is the first of many new search control techniques based on formula annotation; some additional annotated reasoning techniques are also described here. This systematic and comprehensive introduction to rippling, and to the wider subject of automated inductive theorem proving, will be welcomed by researchers and graduate students alike.

The completion of the first draft of the human genome has led to an explosion of interest in genetics and molecular biology. The view of the genome as a network of interacting computational components is well-established, but researchers are now trying to reverse the analogy, by using living organisms to construct logic circuits. The potential applications for such technologies is huge, ranging from bio-sensors, through industrial applications to drug delivery and diagnostics. This book would be the first to deal with the implementation of this technology, describing several working experimental demonstrations using cells as components of logic circuits, building toward computers incorporating biological components in their functioning.

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Bob Hale and Crispin Wright draw together here the key writings in which they have worked out their distinctive approach to the fundamental questions: what is mathematics about, and how do we know it? The volume features much new material: introduction, postscript, bibliographies, and a new essay on a key problem. The Reason's Proper Study is the strongest presentation yet of the controversial neo-Fregean view that mathematical knowledge may be based a priori on logic and definitional abstraction principles. It will prove indispensable reading not just to philosophers of mathematics but to all who are interested in the fundamental metaphysical and epistemological issues which the programme raises.

Philosophical considerations, which are often ignored or treated casually, are given careful consideration in this introduction. Thomas Forster places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in an original analysis of well established topics. The presentation illustrates difficult points and includes many exercises. Little previous knowledge of logic is required and only a knowledge of standard undergraduate mathematics is assumed.

Information content and programming semantics are just two of the applications of the mathematical concepts of order, continuity and domains. This authoritative and comprehensive account of the subject will be an essential handbook for all those working in the area. An extensive index and bibliography make this an ideal sourcebook for all those working in domain theory.

Volume II, on formal (ZFC) set theory, incorporates a self-contained "chapter 0" on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques provides a solid foundation in set theory and a thorough context for the presentation of advanced topics (such as absoluteness, relative consistency results, two expositions of Godel's construstive universe, numerous ways of viewing recursion and Cohen forcing).

The last twenty years have witnessed an accelerated development of pure and ap plied logic, particularly in response to the urgent needs of computer science. Many traditional logicians have developed interest in applications and in parallel a new generation of researchers in logic has arisen from the computer science community. A new attitude to applied logic has evolved, where researchers tailor a logic for their own use in the same way they define a computer language, and where auto mated deduction for the logic and its fragments is as important as the logic itself. In such a climate there is a need to emphasise algorithmic logic methodologies alongside any individual logics. Thus the tableaux method or the resolution method are as central to todays discipline of logic as classical logic or intuitionistic logic are. From this point of view, J. Goubault and I. Mackie's book on Proof Theory and Automated Deduction is most welcome. It covers major algorithmic methodolo gies as well as a variety of logical systems. It gives a wide overview for the ap plied consumer of logic while at the same time remains relatively elementary for the beginning student. A decade ago I put forward my view that a logical system should be presented as a point in a grid. One coordinate is its philosphy, motivation, its accepted theorems and its required non-theorems. The other coordinate is the algorithmic methodol ogy and execution chosen for its effective presentation. Together these two aspects constitute a 'logic'."

One of the most striking development of the last decades in the study of minimal surfaces, constant mean surfaces and harmonic maps is the discovery that many classical problems in differential geometry - including these examples - are actually integrable systems. This theory grew up mainly after the important discovery of the properties of the Korteweg-de Vries equation in the sixties. After C. Gardner, J. Greene, M. Kruskal et R. Miura 44] showed that this equation could be solved using the inverse scattering method and P. Lax 62] reinterpreted this method by his famous equation, many other deep observations have been made during the seventies, mainly by the Russian and the Japanese schools. In particular this theory was shown to be strongly connected with methods from algebraic geom etry (S. Novikov, V. B. Matveev, LM. Krichever. . . ), loop techniques (M. Adler, B. Kostant, W. W. Symes, M. J. Ablowitz . . . ) and Grassmannian manifolds in Hilbert spaces (M. Sato . . . ). Approximatively during the same period, the twist or theory of R. Penrose, built independentely, was applied successfully by R. Penrose and R. S. Ward for constructing self-dual Yang-Mills connections and four-dimensional self-dual manifolds using complex geometry methods. Then in the eighties it became clear that all these methods share the same roots and that other instances of integrable systems should exist, in particular in differential ge ometry. This led K."

Lattice (Boolean) functions are algebraic functions defined over an arbitrary lattice (Boolean algebra), while lattice (Boolean) equations are equations expressed in terms of lattice (Boolean) functions.This self-contained monograph surveys recent developments of Boolean functions and equations, as well as lattice functions and equations in more general classes of lattices; a special attention is paid to consistency conditions and reproductive general solutions.The contents include:- equational compactness in semilattices and Boolean algebras;- the theory of Post functions and equations (which is very close to that of Boolean functions and equations);- a revision of Boolean fundamentals;- closure operators on Boolean functions;- the decomposition of Boolean functions;- quadratic truth equations;- Boolean differential calculus;- Boolean geometry and other topics.There is also a chapter on equations in a very general sense. Applications refer to graph theory, automata theory, synthesis of circuits, fault detection, databases, marketing and others.

This unified approach to the foundations of mathematics in the theory of sets covers both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of "natural number" and "set". The book contains an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, and the analysis of proof by induction and definition by recursion. The book should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.

In The Moment of Proof, Benson attempts to convey to general readers the feeling of Eureka, the joy of discovery that mathematicians feel when they first encounter an elegant proof. The book is packed with intriguing puzzles: Loyd's Fifteen Puzzle, the Monty Hall Problem, the Prisoner's Dilemma, and more. Every fan of mathematical puzzles will be enthralled by this book.

This is the second volume in a series of well-respected works in temporal science and is by the same authors as the first. Volume one dealt primarily with basic concepts and methods, volume two discuses the more applicable aspects of temporal logics. The first four chapters continue the more theoretical presentations from volume one, covering automata, branching time and labelled deduction. The rest of the book is devoted to discussions of temporal databases, temporal execution and programming, actions and planning. With its inclusion of cutting-edge results and unifying methodologies, this book, and its companion are an indispensable reference for both the pure logician and the theoretical computer scientist.

Stewart Shapiro presents a distinctive original view of the foundations of mathematics, arguing that second-order logic has a central role to play in laying these foundations. He gives an accessible account of second-order and higher-order logic, paying special attention to philosophical and historical issues. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic.

'In this excellent treatise Shapiro defends the use of second-order languages and logic as frameworks for mathematics. His coverage of the wide range of logical and philosophical . . . is thorough, clear, and persuasive.' Michael D. Resnik, History and Philosophy of Logic

This is a classic introduction to set theory, suitable for students with no previous knowledge of the subject. Providing complete, up-to-date coverage, the book is based in large part on courses given over many years by Professor Hajnal. The first part introduces all the standard notions of the subject; the second part concentrates on combinatorial set theory. Exercises are included throughout and a new section of hints has been added to assist the reader.

This is a classic introduction to set theory in three parts. The first part gives a general introduction to set theory, suitable for undergraduates; complete proofs are given and no background in logic is required. Exercises are included, and the more difficult ones are supplied with hints. An appendix to the first part gives a more formal foundation to axiomatic set theory, supplementing the intuitive introduction given in the first part. The final part gives an introduction to modern tools of combinatorial set theory. This part contains enough material for a graduate course of one or two semesters. The subjects discussed include stationary sets, delta systems, partition relations, set mappings, measurable and real-valued measurable cardinals. Two sections give an introduction to modern results on exponentiation of singular cardinals, and certain deeper aspects of the topics are developed in advanced problems.

Together, Models and Computability and its sister volume Sets and Proofs will provide readers with a comprehensive guide to the current state of mathematical logic. All the authors are leaders in their fields and are drawn from the invited speakers at 'Logic Colloquium '97' (the major international meeting of the Association of Symbolic Logic). It is expected that the breadth and timeliness of these two volumes will prove an invaluable and unique resource for specialists, post-graduate researchers, and the informed and interested nonspecialist.

Now the most used texbook for introductory cryptography courses in both mathematics and computer science, the Third Edition builds upon previous editions by offering several new sections, topics, and exercises. The authors present the core principles of modern cryptography, with emphasis on formal definitions, rigorous proofs of security.

The study of stable groups connects model theory, algebraic geometry and group theory. It analyzes groups that possess a certain general dependence relation and tries to derive structural properties from this. These may be group-theoretic (nilpotency or solubility of a given group), algebro-geometric (identification of a group as an algebraic group), or model-theoretic (description of the definable sets). In this book, the general theory of stable groups is developed from the beginning (including a chapter on preliminaries in group theory and model theory), concentrating on the model- and group-theoretic aspects. It brings together the various extensions of the original finite rank theory under a unified perspective and provides a coherent exposition of the current knowledge in the field.

This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.

Type theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. By concentrating on the principles that make the theory work in practice, the author covers all the key ideas without getting involved in the complications of more advanced systems. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm that lies at the heart of every such system. Also featured are two other interesting algorithms that until now have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making it the first book at this level that can be used as an introduction to type theory for computer scientists.

This volume surveys recent interactions between model theory and other branches of mathematics, notably group theory. Beginning with an introductory chapter describing relevant background material, the book contains contributions from many leading international figures in this area. Topics described include automorphism groups of algebraically closed fields, the model theory of pseudo-finite fields and applications to the subgroup structure of finite Chevalley groups. Model theory of modules, and aspects of model theory of various classes of groups, including free groups, are also discussed. The book also contains the first comprehensive survey of finite covers. Many new proofs and simplifications of recent results are presented and the articles contain many open problems. This book will be a suitable guide for graduate students and a useful reference for researchers working in model theory and algebra.