There are many kinds of books on formal logic. Some have philosophers as their intended audience, some mathematicians, some computer scien tists. Although there is a common core to all such books, they will be very different in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise. Within computer science formal logic turns up in a number of areas, from pro gram verification to logic programming to artificial intelligence. This book is intended for computer scientists interested in automated theo rem proving in classical logic. To be more precise yet, it is essentially a theoretical treatment, not a how-to book, although how-to issues are not neglected. This does not mean, of course, that the book will be of no interest to philosophers or mathematicians. It does contain a thorough presentation of formal logic and many proof techniques, and as such it contains all the material one would expect to find in a course in formal logic covering completeness but, not incompleteness issues. The first item to be addressed is, What are we talking about and why are we interested in it? We are primarily talking about truth as used in mathematical discourse, and our interest in it is, or should be, self evident. Truth is a semantic concept, so we begin with models and their properties. These are used to define our subject.

The present text resulted from lectures given by the authors at the Rijks Universiteit at Utrecht. These lectures were part of a series on 'History of Contemporary Mathematics'. The need for such an enterprise was generally felt, since the curriculum at many universities is designed to suit an efficient treatment of advanced subjects rather than to reflect the development of notions and techniques. As it is very likely that this trend will continue, we decided to offer lectures of a less technical nature to provide students and interested listeners with a survey of the history of topics in our present-day mathematics. We consider it very useful for a mathematician to have an acquaintance with the history of the development of his subject, especially in the nineteenth century where the germs of many of modern disciplines can be found. Our attention has therefore been mainly directed to relatively young developments. In the lectures we tried to stay clear of both oversimplification and extreme technicality. The result is a text, that should not cause difficulties to a reader with a working knowledge of mathematics. The developments sketched in this book are fundamental for many areas in mathematics and the notions considered are crucial almost everywhere. The book may be most useful, in particular, for those teaching mathematics.

Mathematical Foundations of Computer Science, Volume I is the first of two volumes presenting topics from mathematics (mostly discrete mathematics) which have proven relevant and useful to computer science. This volume treats basic topics, mostly of a set-theoretical nature (sets, functions and relations, partially ordered sets, induction, enumerability, and diagonalization) and illustrates the usefulness of mathematical ideas by presenting applications to computer science. Readers will find useful applications in algorithms, databases, semantics of programming languages, formal languages, theory of computation, and program verification. The material is treated in a straightforward, systematic, and rigorous manner. The volume is organized by mathematical area, making the material easily accessible to the upper-undergraduate students in mathematics as well as in computer science and each chapter contains a large number of exercises. The volume can be used as a textbook, but it will also be useful to researchers and professionals who want a thorough presentation of the mathematical tools they need in a single source. In addition, the book can be used effectively as supplementary reading material in computer science courses, particularly those courses which involve the semantics of programming languages, formal languages and automata, and logic programming.

Drinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications is based on the author's original work establishing the correspondence between ell-adic rank r Galois representations and automorphic representations of GL(r) over a function field, in the local case, and, in the global case, under a restriction at a single place. It develops Drinfeld's theory of elliptic modules, their moduli schemes and covering schemes, the simple trace formula, the fixed point formula, as well as the congruence relations and a "simple" converse theorem, not yet published anywhere. This version, based on a recent course taught by the author at The Ohio State University, is updated with references to research that has extended and developed the original work. The use of the theory of elliptic modules in the present work makes it accessible to graduate students, and it will serve as a valuable resource to facilitate an entrance to this fascinating area of mathematics.

This is the first book-length treatment of hybrid logic and its proof-theory. Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model (where the points represent times, possible worlds, states in a computer, or something else). This is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about proof-theory for ordinary modal logic. Many modal-logical proof systems lack important properties and the relationships between proof systems for different modal logics are often unclear. In the present book we demonstrate that hybrid-logical proof-theory remedies these deficiencies by giving a spectrum of well-behaved proof systems (natural deduction, Gentzen, tableau, and axiom systems) for a spectrum of different hybrid logics (propositional, first-order, intensional first-order, and intuitionistic).

This book constitutes the proceedings of the 12th Biennial Meeting on Mathematics in Language, MOL 12, held in Nara, Japan, in September 2011.

Presented in this volume are 12 carefully selected papers, as well as the paper of the invited speaker Andreas Maletti. The papers cover such diverse topics as formal languages (string and tree transducers, grammar-independent syntactic structures, probabilistic and weighted context-free grammars, formalization of minimalist syntax), parsing and unification, lexical and compositional semantics, statistical language models, and theories of truth.

Resolution Proof Systems: An Algebraic Theory presents a new algebraic framework for the design and analysis of resolution- based automated reasoning systems for a range of non-classical logics. It develops an algebraic theory of resolution proof systems focusing on the problems of proof theory, representation and efficiency of the deductive process. A new class of logical calculi, the class of resolution logics, emerges as a second theme of the book. The logical and computational aspects of the relationship between resolution logics and resolution proof systems is explored in the context of monotonic as well as nonmonotonic reasoning. This book is aimed primarily at researchers and graduate students in artificial intelligence, symbolic and computational logic. The material is suitable as a reference book for researchers and as a text book for graduate courses on the theoretical aspects of automated reasoning and computational logic.

Before his death in March, 1976, A. H. Lightstone delivered the manu script for this book to Plenum Press. Because he died before the editorial work on the manuscript was completed, I agreed (in the fall of 1976) to serve as a surrogate author and to see the project through to completion. I have changed the manuscript as little as possible, altering certain passages to correct oversights. But the alterations are minor; this is Lightstone's book. H. B. Enderton vii Preface This is a treatment of the predicate calculus in a form that serves as a foundation for nonstandard analysis. Classically, the predicates and variables of the predicate calculus are kept distinct, inasmuch as no variable is also a predicate; moreover, each predicate is assigned an order, a unique natural number that indicates the length of each tuple to which the predicate can be prefixed. These restrictions are dropped here, in order to develop a flexible, expressive language capable of exploiting the potential of nonstandard analysis. To assist the reader in grasping the basic ideas of logic, we begin in Part I by presenting the propositional calculus and statement systems. This provides a relatively simple setting in which to grapple with the some times foreign ideas of mathematical logic. These ideas are repeated in Part II, where the predicate calculus and semantical systems are studied."

Relevance logics came of age with the one and only International Conference on relevant logics in 1974. They did not however become accepted, or easy to promulgate. In March 1981 we received most of the typescript of IN MEMORIAM: ALAN ROSS ANDERSON Proceedings of the International Conference of Relevant Logic from the original editors, Kenneth W. Collier, Ann Gasper and Robert G. Wolf of Southern Illinois University. 1 They had, most unfortunately, failed to find a publisher - not, it appears, because of overall lack of merit of the essays, but because of the expense of producing the collection, lack of institutional subsidization, and doubts of publishers as to whether an expensive collection of essays on such an esoteric, not to say deviant, subject would sell. We thought that the collection of essays was still (even after more than six years in the publishing trade limbo) well worth publishing, that the subject would remain undeservedly esoteric in North America while work on it could not find publishers (it is not so esoteric in academic circles in Continental Europe, Latin America and the Antipodes) and, quite important, that we could get the collection published, and furthermore, by resorting to local means, published comparatively cheaply. It is indeed no ordinary collection. It contains work by pioneers of the main types of broadly relevant systems, and by several of the most innovative non-classical logicians of the present flourishing logical period. We have slowly re-edited and reorganised the collection and made it camera-ready.

Fuzzy Modelling: Paradigms and Practice provides an up-to-date and authoritative compendium of fuzzy models, identification algorithms and applications. Chapters in this book have been written by the leading scholars and researchers in their respective subject areas. Several of these chapters include both theoretical material and applications. The editor of this volume has organized and edited the chapters into a coherent and uniform framework. The objective of this book is to provide researchers and practitioners involved in the development of models for complex systems with an understanding of fuzzy modelling, and an appreciation of what makes these models unique. The chapters are organized into three major parts covering relational models, fuzzy neural networks and rule-based models. The material on relational models includes theory along with a large number of implemented case studies, including some on speech recognition, prediction, and ecological systems. The part on fuzzy neural networks covers some fundamentals, such as neurocomputing, fuzzy neurocomputing, etc., identifies the nature of the relationship that exists between fuzzy systems and neural networks, and includes extensive coverage of their architectures. The last part addresses the main design principles governing the development of rule-based models. Fuzzy Modelling: Paradigms and Practice provides a wealth of specific fuzzy modelling paradigms, algorithms and tools used in systems modelling. Also included is a panoply of case studies from various computer, engineering and science disciplines. This should be a primary reference work for researchers and practitioners developing models of complex systems.

This volume contains the proceedings of the Second Joint IFSA-EC and EURO-WGFS Workshop on Progress in Fuzzy Sets in Europe held on April 6 -8, 1989 in Vienna, Austria. The workshop was organized by Prof. Dr. Wolfgang H. Janko from the University of Economics in Vienna under the auspices of IFSA-EC, the European chapter of the International Fuzzy Systems Association, and EURO-WGFS, the working group on Fuzzy Sets of the Association of Eu ropean Operational Research Societies. The workshop gathered more than 30 participants coming from Western European countries (Austria, Bel gium, England, Germany, Finland, France, Hungary, Italy, Scotland and Spain) Eastern European countries (Bulgaria, the German Federal Repu blic, Hungary and Poland) and non-European countries such as China and Japan. The 15 selected and refereed papers included in the volume are in prin ciple the author's own versions, with limited editorial changes and small corrections. They are arranged in alphabetical order. I wish to thank all the contributors for their valuable papers and an outstan ding cooperation in the editorial project. I also would like to express my sincere thanks to Professor Dr. H. J. Zimmermann for the cooperation in the refereeing procedure.

The History of the Book In August 1992 the author had the opportunity to give a course on resolution theorem proving at the Summer School for Logic, Language, and Information in Essex. The challenge of this course (a total of five two-hour lectures) con sisted in the selection of the topics to be presented. Clearly the first selection has already been made by calling the course "resolution theorem proving" instead of "automated deduction" . In the latter discipline a remarkable body of knowledge has been created during the last 35 years, which hardly can be presented exhaustively, deeply and uniformly at the same time. In this situ ation one has to make a choice between a survey and a detailed presentation with a more limited scope. The author decided for the second alternative, but does not suggest that the other is less valuable. Today resolution is only one among several calculi in computational logic and automated reasoning. How ever, this does not imply that resolution is no longer up to date or its potential exhausted. Indeed the loss of the "monopoly" is compensated by new appli cations and new points of view. It was the purpose of the course mentioned above to present such new developments of resolution theory. Thus besides the traditional topics of completeness of refinements and redundancy, aspects of termination (resolution decision procedures) and of complexity are treated on an equal basis."

The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras," which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time.

Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications.

This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naive" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science.

When I first participated in exploring theories of nonmonotonic reasoning in the late 1970s, I had no idea of the wealth of conceptual and mathematical results that would emerge from those halting first steps. This book by Wiktor Marek and Miroslaw Truszczynski is an elegant treatment of a large body of these results. It provides the first comprehensive treatment of two influen tial nonmonotonic logics - autoepistemic and default logic - and describes a number of surprising and deep unifying relationships between them. It also relates them to various modal logics studied in the philosophical logic litera ture, and provides a thorough treatment of their applications as foundations for logic programming semantics and for truth maintenance systems. It is particularly appropriate that Marek and Truszczynski should have authored this book, since so much of the research that went into these results is due to them. Both authors were trained in the Polish school of logic and they bring to their research and writing the logical insights and sophisticated mathematics that one would expect from such a background. I believe that this book is a splendid example of the intellectual maturity of the field of artificial intelligence, and that it will provide a model of scholarship for us all for many years to come. Ray Reiter Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 and The Canadian Institute for Advanced Research Table of Contents 1 1 Introduction ........."

Dr. KURT GODEL'S sixtieth birthday (April 28, 1966) and the thirty fifth anniversary of the publication of his theorems on undecidability were celebrated during the 75th Anniversary Meeting of the Ohio Ac ademy of Science at The Ohio State University, Columbus, on April 22, 1966. The celebration took the form of a Festschrift Symposium on a theme supported by the late Director of The Institute for Advanced Study at Princeton, New Jersey, Dr. J. ROBERT OPPENHEIMER: "Logic, and Its Relations to Mathematics, Natural Science, and Philosophy." The symposium also celebrated the founding of Section L (Mathematical Sciences) of the Ohio Academy of Science. Salutations to Dr. GODEL were followed by the reading of papers by S. F. BARKER, H. B. CURRY, H. RUBIN, G. E. SACKS, and G. TAKEUTI, and by the announcement of in-absentia papers contributed in honor of Dr. GODEL by A. LEVY, B. MELTZER, R. M. SOLOVAY, and E. WETTE. A short discussion of "The II Beyond Godel's I" concluded the session."

On January 22, 1990, the late John Bell held at CERN (European Laboratory for Particle Physics), Geneva a seminar organized by the Center of Quantum Philosophy, that at this time was an association of scientists interested in the interpretation of quantum mechanics. In this seminar Bell presented once again his famous theorem. Thereafter a discussion took place in which not only physical but also highly speculative epistemological and philosophical questions were vividly debated. The list of topics included: assumption of free will in Bell's theorem, the understanding of mind, the relationship between the mathematical and the physical world, the existence of unobservable causes and the limits of human knowledge in mathematics and physics. Encouraged by this stimulating discussion some of the participants decided to found an Institute for Interdisciplinary Studies (lIS) to promote philosoph ical and interdisciplinary reflection on the advances of science. Meanwhile the lIS has associated its activities with the Swiss foundation, Fondation du Leman, and the Dutch foundation, Stichting Instudo, registered in Geneva and Amsterdam, respectively. With its activities the lIS intends to strengthen the unity between the professional activities in science and the reflection on fun damental philosophical questions. In addition the interdisciplinary approach is expected to give a contribution to the progress of science and the socio economic development. At present three working groups are active within the lIS, i. e.: - the Center for Quantum Philosophy, - the Wealth Creation and Sustainable Development Group, - the Neural Science Group."

That philosophical themes could be studied in an exact manner by logical meanS was a delightful discovery to make. Until then, the only outlet for a philosophical interest known to me was the production of poetry or essays. These means of expression remain inconclusive, however, with a tendency towards profuseness. The logical discipline provides so me intellectual backbone, without excluding the literary modes. A master's thesis by Erik Krabbe introduced me to the subject of tense logic. The doctoral dissertation of Paul N eedham awaked me (as so many others) from my dogmatic slumbers concerning the latter's mono poly on the logical study of Time. Finally, a set of lecture notes by Frank Veltman showed me how classical model theory is just as relevant to that study as more exotic intensional techniques. Of the authors whose work inspired me most, I would mention Arthur Prior, for his irresistible blend of logic and philosophy, Krister Segerberg, for his technical opening up of a systematic theory, and Hans Kamp, for his mastery of all these things at once. Many colleagues have made helpful comments on the two previous versions of this text. I would like to thank especially my students Ed Brinksma, Jan van Eyck and Wilfried Meyer-Viol for their logical and cultural criticism. The drawings were contributed by the versatile Bauke Mulder. Finally, Professor H intikka's kind appreciation provided the stimulus to write this book."

This work introduces tools, from the field of category theory, that make it possible to tackle until now unsolvable representation problems (determination of the range of a given functor). The basic idea is: if a functor lifts many objects, then it also lifts many (poset-indexed) diagrams.

by Ivor Grattan-Guinness Until twenty years ago the outline history of logicism was well known. Frege had had the important ideas, until he was eclipsed by Wittgenstein. Russell was important in publicising the former and tutoring the latter, and also for working with Moore in the conversion of British philosophy from neo-Hegelianism to the new analytic tradition in the 1900s, but his own work on logic and especially logicism was very muddled. Around that time Russell, who was still alive, sold his manuscripts to McMaster University in Canada, and interest in his achievements in logic began to develop, especially after his death in 1970. Scholars found thousands of folios of unpublished holograph awaiting their attention, and also hundreds of pertinent letters (both in the Russell Archives and elsewhere in certain recipients' collections). Various facets of his work came to light for the first time, and others -which could have been gleaned from carefully reading of the published sources- gained new publicity from the evidence revealed in manuscripts. Even the technical passage work, which constitutes the unread majority of the Principia mathematica (1910-13) of Russell and Whitehead, began to receive a little respectful scrutiny. It turned out that Russell had done several pioneering things. While indeed often incoherent in reference and content, they comprised major forays into the new mathematical logic, of which he turned out to be a major founder: some are even of interest to modem studies.

Decision Criteria and Optimal Inventory Processes provides a theoretical and practical introduction to decision criteria and inventory processes. Inventory theory is presented by focusing on the analysis and processes underlying decision criteria. Included are many state-of-the-art criterion models as background material. These models are extended to the authors' newly developed fuzzy criterion models which constitute a general framework for the study of stochastic inventory models with special focus on the real world inventory theoretic reservoir operations problems. The applications of fuzzy criterion dynamic programming models are illustrated by reservoir operations including the integrated network of reservoir operation and the open inventory network problems. An interesting feature of this book is the special attention it pays to the analysis of some theoretical and applied aspects of fuzzy criteria and dynamic fuzzy criterion models, thus opening up a new way of injecting the much-needed type of non-cost, intuitive, and easy-to-use methods into multi-stage inventory processes. This is accomplished by constructing and optimizing the fuzzy criterion models developed for inventory processes. Practitioners in operations research, management science, and engineering will find numerous new ideas and strategies for modeling real world multi- stage inventory problems, and researchers and applied mathematicians will find this work a stimulating and useful reference.

Rule-based fuzzy modeling has been recognised as a powerful technique for the modeling of partly-known nonlinear systems. Fuzzy models can effectively integrate information from different sources, such as physical laws, empirical models, measurements and heuristics. Application areas of fuzzy models include prediction, decision support, system analysis, control design, etc. Fuzzy Modeling for Control addresses fuzzy modeling from the systems and control engineering points of view. It focuses on the selection of appropriate model structures, on the acquisition of dynamic fuzzy models from process measurements (fuzzy identification), and on the design of nonlinear controllers based on fuzzy models. To automatically generate fuzzy models from measurements, a comprehensive methodology is developed which employs fuzzy clustering techniques to partition the available data into subsets characterized by locally linear behaviour. The relationships between the presented identification method and linear regression are exploited, allowing for the combination of fuzzy logic techniques with standard system identification tools. Attention is paid to the trade-off between the accuracy and transparency of the obtained fuzzy models. Control design based on a fuzzy model of a nonlinear dynamic process is addressed, using the concepts of model-based predictive control and internal model control with an inverted fuzzy model. To this end, methods to exactly invert specific types of fuzzy models are presented. In the context of predictive control, branch-and-bound optimization is applied. The main features of the presented techniques are illustrated by means of simple examples. In addition, three real-world applications are described. Finally, software tools for building fuzzy models from measurements are available from the author.

This book has a fundamental relationship to the International Seminar on Fuzzy Set Theory held each September in Linz, Austria. First, this volume is an extended account of the eleventh Seminar of 1989. Second, and more importantly, it is the culmination of the tradition of the preceding ten Seminars. The purpose of the Linz Seminar, since its inception, was and is to foster the development of the mathematical aspects of fuzzy sets. In the earlier years, this was accomplished by bringing together for a week small grou ps of mathematicians in various fields in an intimate, focused environment which promoted much informal, critical discussion in addition to formal presentations. Beginning with the tenth Seminar, the intimate setting was retained, but each Seminar narrowed in theme; and participation was broadened to include both younger scholars within, and established mathematicians outside, the mathematical mainstream of fuzzy sets theory. Most of the material of this book was developed over the years in close association with the Seminar or influenced by what transpired at Linz. For much of the content, it played a crucial role in either stimulating this material or in providing feedback and the necessary screening of ideas. Thus we may fairly say that the book, and the eleventh Seminar to which it is directly related, are in many respects a culmination of the previous Seminars.

Fuzzy knowledge and fuzzy systems affect our lives today as systems enter the world of commerce. Fuzzy systems are incorporated in domestic appliances (washing machine, air conditioning, microwave, telephone) and in transport systems (a pilotless helicopter has recently completed a test flight). Future applications are expected to have dramatic implications for the demand for labor, among other things. It was with such thoughts in mind that this first international survey of future applications of fuzzy logic has been undertaken. The results are likely to be predictive for a decade beyond the millenium. The predictive element is combined with a bibliography which serves as an historical anchor as well as being both extensive and extremely useful. Analysis and Evaluation of Fuzzy Systems is thus a milestone in the development of fuzzy logic and applications of three representative subsystems: Fuzzy Control, Fuzzy Pattern Recognition and Fuzzy Communications.

Logic plays a central conceptual role in modern mathematics. However, mathematical logic has grown into one of the most recondite areas of mathematics. As a result, most of modern logic is inaccessible to all but the specialist. This new book is a resource that provides a quick introduction and review of the key topics in logic for the computer scientist, engineer, or mathematician. Handbook of Logic and Proof Techniques for Computer Science presents the elements of modern logic, including many current topics, to the reader having only basic mathematical literacy. Computer scientists will find specific examples and important ideas such as axiomatics, recursion theory, decidability, independence, completeness, consistency, model theory, and P/NP completeness. The book contains definitions, examples and discussion of all of the key ideas in basic logic, but also makes a special effort to cut through the mathematical formalism, difficult notation, and esoteric terminology that is typical of modern mathematical logic.T This handbook delivers cogent and self-contained introductions to critical advanced topics, including: * Godel's completeness and incompleteness theorems * Methods of proof, cardinal and ordinal numbers, the continuum hypothesis, the axiom of choice, model theory, and number systems and their construction * Extensive treatment of complexity theory and programming applications * Applications to algorithms in Boolean algebra * Discussion of set theory and applications of logic The book is an excellent resource for the working mathematical scientist. The graduate student or professional in computer science and engineering or the systems scientist who needs to have a quick sketch of a key idea from logic will find it here in this self-contained, accessible, and easy-to-use reference.