This new book on mathematical logic by Jeremy Avigad gives a thorough introduction to the fundamental results and methods of the subject from the syntactic point of view, emphasizing logic as the study of formal languages and systems and their proper use. Topics include proof theory, model theory, the theory of computability, and axiomatic foundations, with special emphasis given to aspects of mathematical logic that are fundamental to computer science, including deductive systems, constructive logic, the simply typed lambda calculus, and type-theoretic foundations. Clear and engaging, with plentiful examples and exercises, it is an excellent introduction to the subject for graduate students and advanced undergraduates who are interested in logic in mathematics, computer science, and philosophy, and an invaluable reference for any practicing logician's bookshelf.

This book brings together contributions by leading researchers in computational complexity theory written in honor of Somenath Biswas on the occasion of his sixtieth birthday. They discuss current trends and exciting developments in this flourishing area of research and offer fresh perspectives on various aspects of complexity theory. The topics covered include arithmetic circuit complexity, lower bounds and polynomial identity testing, the isomorphism conjecture, space-bounded computation, graph isomorphism, resolution and proof complexity, entropy and randomness. Several chapters have a tutorial flavor. The aim is to make recent research in these topics accessible to graduate students and senior undergraduates in computer science and mathematics. It can also be useful as a resource for teaching advanced level courses in computational complexity.

This book provides an argumentation model for means end-reasoning, a distinctive type of reasoning used for problem-solving and decision-making. Means end-reasoning is modelled as goal-directed argumentation from an agent's goals and known circumstances, and from an action selected as a means, to a decision to carry out the action. Goal-based Reasoning for Argumentation provides an argumentation model of this kind of reasoning showing how it is employed in settings of intelligent deliberation where agents try to collectively arrive at a conclusion on what they should do to move forward in a set of circumstances. The book explains how this argumentation model can help build more realistic computational systems of deliberation and decision-making, and shows how such systems can be applied to solve problems posed by goal-based reasoning in numerous fields, from social psychology and sociology, to law, political science, anthropology, cognitive science, artificial intelligence, multi-agent systems, and robotics.

Conditional reasoning is reasoning that involves statements of the sort If A (Antecedent) then C (Consequent). This type of reasoning is ubiquitous; everyone engages in it. Indeed, the ability to do so may be considered a defining human characteristic. Without this ability, human cognition would be greatly impoverished. "What-if" thinking could not occur. There would be no retrospective efforts to understand history by imagining how it could have taken a different course. Decisions that take possible contingencies into account could not be made; there could be no attempts to influence the future by selecting actions on the basis of their expected effects. Despite the commonness and importance of conditional reasoning and the considerable attention it has received from scholars, it remains the subject of much continuing debate. Unsettled questions, both normative and empirical, continue to be asked. What constitutes normative conditional reasoning? How do people engage in it? Does what people do match what would be expected of a rational agent with the abilities and limitations of human beings? If not, how does it deviate and how might people's ability to engage in it be improved? This book reviews the work of prominent psychologists and philosophers on conditional reasoning. It describes empirical research on how people deal with conditional arguments and on how conditional statements are used and interpreted in everyday communication. It examines philosophical and theoretical treatments of the mental processes that support conditional reasoning. Its extensive coverage of the subject makes it an ideal resource for students, teachers, and researchers with a focus on cognition across disciplines.

The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

An important figure in the development of modern mathematical logic and abstract algebra, Augustus De Morgan (1806-71) was also a witty writer who made a hobby of collecting evidence of paradoxical and illogical thinking from historical sources as well as contemporary pamphlets and periodicals. Based on articles that had appeared in The Athenaeum during his lifetime, this work was edited by his widow and published in book form in 1872. It parades all varieties of crackpot, from circle-squarers to inventors of perpetual motion machines, all for the reader's entertainment and education. Filled with anecdotes, personal opinions and 'squibs' of every kind, the book remains enjoyable reading for those who are amused rather than appalled by the human condition. Also reissued in the Cambridge Library Collection are the Memoir of Augustus De Morgan (1882), prepared by his wife, and his ambitious Formal Logic (1847).

Pure inductive logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years plus the main contributions of the authors and their collaborators over the last decade to present a comprehensive account of the discipline within a single unified context. The exposition is structured around the traditional bases of rationality, such as avoiding Dutch Books, respecting symmetry and ignoring irrelevant information. The authors uncover further rationality concepts, both in the unary and in the newly emerging polyadic languages, such as conformity, spectrum exchangeability, similarity and language invariance. For logicians with a mathematical grounding, this book provides a complete self-contained course on the subject, taking the reader from the basics up to the most recent developments. It is also a useful reference for a wider audience from philosophy and computer science.

It is necessary to practice methodological doubt, like Descartes, in - der to loosen the hold of mental habits; and it is necessary to cultivate logical imagination, in order to have a number of hypotheses at c- mand, and not to be the slave of the one which common sense has r- dered easy to imagine. These two processes, of doubting the familiar and imagining the unfamiliar, are corrective, and form the chief part of the mental training required for a philosopher. Bertrand Russell At every stage and in all circumstances knowledge is incomplete and provisional, conditioned and limited by the historical circumstances under which it was acquired, including the means and methods used for gaining it and the historically conditioned assumptions and categories used in the formulation of ideas and conclusions. Maurice Cornforth This monograph is the second in the series of meta-theoretic analysis of fuzzy paradigm and its contribution and possible contribution to formal reasoning in order to free the knowledge production process from the ridge frame of the classical paradigm that makes its application to soft and inexact sciences d- ficult or irrelevant. The work in the previous monograph was strictly devoted to problems of theory of knowledge and critique of classical, bounded and other rationalities in decision-choice processes regarding the principles of verification, falsification or corroboration in knowledge production. This monograph deals mostly with epistemic decision-choice models and theories and how they are related to both the classical and fuzzy paradigms.

Using a unique pedagogical approach, this text introduces mathematical logic by guiding students in implementing the underlying logical concepts and mathematical proofs via Python programming. This approach, tailored to the unique intuitions and strengths of the ever-growing population of programming-savvy students, brings mathematical logic into the comfort zone of these students and provides clarity that can only be achieved by a deep hands-on understanding and the satisfaction of having created working code. While the approach is unique, the text follows the same set of topics typically covered in a one-semester undergraduate course, including propositional logic and first-order predicate logic, culminating in a proof of Goedel's completeness theorem. A sneak peek to Goedel's incompleteness theorem is also provided. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. Familiarity with proofs and basic proficiency in Python is assumed.

Hex: The Full Story is for anyone - hobbyist, professional, student, teacher - who enjoys board games, game theory, discrete math, computing, or history. hex was discovered twice, in 1942 by Piet Hein and again in 1949 by John F. nash. How did this happen? Who created the puzzle for Hein's Danish newspaper column? How are Martin Gardner, David Gale, Claude Shannon, and Claude Berge involved? What is the secret to playing Hex well? The answers are inside... Features New documents on Hein's creation of Hex, the complete set of Danish puzzles, and the identity of their composer Chapters on Gale's game Bridg-it, the game Rex, computer Hex, open Hex problems, and more Dozens of new puzzles and solutions Study guide for Hex players Supplemenetary text for a course in game theory, discrete math, computer science, or science history

Formal languages are widely regarded as being above all mathematical objects and as producing a greater level of precision and technical complexity in logical investigations because of this. Yet defining formal languages exclusively in this way offers only a partial and limited explanation of the impact which their use (and the uses of formalisms more generally elsewhere) actually has. In this book, Catarina Dutilh Novaes adopts a much wider conception of formal languages so as to investigate more broadly what exactly is going on when theorists put these tools to use. She looks at the history and philosophy of formal languages and focuses on the cognitive impact of formal languages on human reasoning, drawing on their historical development, psychology, cognitive science and philosophy. Her wide-ranging study will be valuable for both students and researchers in philosophy, logic, psychology and cognitive and computer science.

Gerhard Gentzen has been described as logic's lost genius, whom Goedel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen's enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen's original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

Few mathematical results capture the imagination like Georg Cantor's groundbreaking work on infinity in the late nineteenth century. This opened the door to an intricate axiomatic theory of sets which was born in the decades that followed. Written for the motivated novice, this book provides an overview of key ideas in set theory, bridging the gap between technical accounts of mathematical foundations and popular accounts of logic. Readers will learn of the formal construction of the classical number systems, from the natural numbers to the real numbers and beyond, and see how set theory has evolved to analyse such deep questions as the status of the continuum hypothesis and the axiom of choice. Remarks and digressions introduce the reader to some of the philosophical aspects of the subject and to adjacent mathematical topics. The rich, annotated bibliography encourages the dedicated reader to delve into what is now a vast literature.

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians.

The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay's conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

Some of our earliest experiences of the conclusive force of an argument come from school mathematics: faced with a mathematical proof, we cannot deny the conclusion once the premises have been accepted. Behind such arguments lies a more general pattern of 'demonstrative arguments' that is studied in the science of logic. Logical reasoning is applied at all levels, from everyday life to advanced sciences, and a remarkable level of complexity is achieved in everyday logical reasoning, even if the principles behind it remain intuitive. Jan von Plato provides an accessible but rigorous introduction to an important aspect of contemporary logic: its deductive machinery. He shows that when the forms of logical reasoning are analysed, it turns out that a limited set of first principles can represent any logical argument. His book will be valuable for students of logic, mathematics and computer science.

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

..."The book, written by one of the main researchers on the field, gives a complete account of the theory of r.e. degrees. .... The definitions, results and proofs are always clearly motivated and explained before the formal presentation; the proofs are described with remarkable clarity and conciseness. The book is highly recommended to everyone interested in logic. It also provides a useful background to computer scientists, in particular to theoretical computer scientists." Acta Scientiarum Mathematicarum, Ungarn 1988 ..."The main purpose of this book is to introduce the reader to the main results and to the intricacies of the current theory for the recurseively enumerable sets and degrees. The author has managed to give a coherent exposition of a rather complex and messy area of logic, and with this book degree-theory is far more accessible to students and logicians in other fields than it used to be." Zentralblatt fur Mathematik, 623.1988

This volume takes its name from a popular series of intensive mathematics workshops hosted at institutions in Appalachia and surrounding areas. At these meetings, internationally prominent set theorists give one-day lectures that focus on important new directions, methods, tools and results so that non-experts can begin to master these and incorporate them into their own research. Each chapter in this volume was written by the workshop leaders in collaboration with select student participants, and together they represent most of the meetings from the period 2006-2012. Topics covered include forcing and large cardinals, descriptive set theory, and applications of set theoretic ideas in group theory and analysis, making this volume essential reading for a wide range of researchers and graduate students.

This second volume of a collection of papers offers new perspectives and challenges in the study of logic. It is presented in honor of the fiftieth birthday of Jean-Yves Beziau. The papers touch upon a wide range of topics including paraconsistent logic, quantum logic, geometry of oppositions, categorical logic, computational logic, fundamental logic notions (identity, rule, quantification) and history of logic (Leibniz, Peirce, Hilbert). The volume gathers personal recollections about Jean-Yves Beziau and an autobiography, followed by 25 papers written by internationally distinguished logicians, mathematicians, computer scientists, linguists and philosophers, including Irving Anellis, Dov Gabbay, Ivor Grattan-Guinness, Istvan Nemeti, Henri Prade. These essays will be of interest to all students and researchers interested in the nature and future of logic.

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

As society comes to rely increasingly on software for its welfare and prosperity there is an urgent need to create systems in which it can trust. Experience has shown that confidence can only come from a more profound understanding of the issues, which in turn can come only if it is based on logically sound foundations.
This volume contains contributions from leading researchers in the critical disciplines of computing and information science, mathematics, logic, and complexity. All contributions are self-contained, aiming at comprehensibility as well as comprehensiveness. The volume also contains introductory hints to technical issues, concise surveys, introductions, and various fresh results and new perspectives.

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.

Mountaineers use pitons to protect themselves from falls. The lead climber wears a harness to which a rope is tied. As the climber ascends, the rope is paid out by a partner on the ground. As described thus far, the climber receives no protection from the rope or the partner. However, the climber generally carries several spike-like pitons and stops when possible to drive one into a small crack or crevice in the rock face. After climbing just above the piton, the climber clips the rope to the piton, using slings and carabiners. A subsequent fall would result in the climber hanging from the piton if the piton stays in the rock, the slings and carabiners do not fail, the rope does not break, the partner is holding the rope taut and secure, and the climber had not climbed too high above the piton before falling. The climber's safety clearly depends on all of the components of the system. But the piton is distinguished because it connects the natural to the artificial. In 1987 I designed an assembly-level language for Warren Hunt's FM8501 verified microprocessor. I wanted the language to be conveniently used as the object code produced by verified compilers. Thus, I envisioned the language as the first software link in a trusted chain from verified hardware to verified applications programs. Thinking of the hardware as the "rock" I named the language "Piton."

Gert H. Muller The growth of the number of publications in almost all scientific areas,* as in the area of (mathematical) logic, is taken as a sign of our scientifically minded culture, but it also has a terrifying aspect. In addition, given the rapidly growing sophistica- tion, specialization and hence subdivision of logic, researchers, students and teachers may have a hard time getting an overview ofthe existing literature, partic- ularly if they do not have an extensive library available in their neighbourhood: they simply do not even know what to ask for! More specifically, if someone vaguely knows that something vaguely connected with his interests exists some- where in the literature, he may not be able to find it even by searching through the publications scattered in the review journals. Answering this challenge was and is the central motivation for compiling this Bibliography. The Bibliography comprises (presently) the following six volumes (listed with the corresponding Editors): I. Classical Logic W. Rautenberg II. Non-c1assical Logics W. Rautenberg IH. Model Theory H. -D. Ebbinghaus IV. Recursion Theory P. G. Hinman V. Set Theory A. R. Blass VI. ProofTheory; Constructive Mathematics J. E. Kister; D. van Dalen & A. S. Troelstra.