First developed in the early 1980s by Lenstra, Lenstra, and Lov sz, the LLL algorithm was originally used to provide a polynomial-time algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.

Combinatory logic is one of the most versatile areas within logic that is tied to parts of philosophical, mathematical, and computational logic. Functioning as a comprehensive source for current developments of combinatory logic, this book is the only one of its kind to cover results of the last four decades. Using a reader-friendly style, the author presents the most up-to-date research studies. She includes an introduction to combinatory logic before progressing to its central theorems and proofs. The text makes intelligent and well-researched connections between combinatory logic and lambda calculi and presents models and applications to illustrate these connections.

This book is a collection of articles, some introductory, some extended surveys, and some containing previously unpublished research, on a range of topics linking infinite permutation group theory and model theory. Topics covered include: oligomorphic permutation groups and omega-categorical structures; totally categorical structures and covers; automorphism groups of recursively saturated structures; Jordan groups; Hrushovski's constructions of pseudoplanes; permutation groups of finite Morley rank; applications of permutation group theory to models of set theory without the axiom of choice. There are introductory chapters by the editors on general model theory and permutation theory, recursively saturated structures, and on groups of finite Morley rank. The book is almost self-contained, and should be useful to both a beginning postgraduate student meeting the subject for the first time, and to an active researcher from either of the two main fields looking for an overview of the subject.

This book presents several recent advances in natural language semantics and explores the boundaries between syntax and semantics over the last two decades. It is based on some of the most recent theories in logic, such as linear logic and ludics, first created by Jean-Yves Girard, and it also provides some sharp analyses of computational semantical representations, explaining advanced theories in theoretical computer sciences, such as the lambda-mu and Lambek-Grishin calculi which were applied by Philippe de Groote and Michael Moortgat. The author also looks at Aarne Ranta's 'proof as meaning' approach, which was first based on Martin-Loef's Type Theory.Meaning, Logic and Ludics surveys the many solutions which have been proposed for the syntax-semantics interface, taking into account the specifications of linguistic signs (continuous or discontinuous) and the fundamental mechanisms developed by linguists and notable Generativists. This pioneering publication also presents ludics (in a chapter co-authored with Myriam Quatrini), a framework which allows us to characterize meaning as an invariant with regard to interaction between processes. It is an excellent book for advanced students, and academics alike, in the field of computational linguistics.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic.

This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.

Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines. The book covers classical first-order logic and alternatives, including intuitionistic, free, and many-valued logic. It also considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, independent study as a course text, it covers more material than is typically covered in an introductory course. It also covers this material at greater length and in more depth with the purpose of making it accessible to those with no prior training in logic or formal systems. Online support material includes a detailed student solutions manual with a running commentary on all starred exercises, and a set of editable slide presentations for course lectures. Key Features Introduces an unusually broad range of topics, allowing instructors to craft courses to meet a range of various objectives Adopts a critical attitude to certain classical doctrines, exposing students to alternative ways to answer philosophical questions about logic Carefully considers the ways natural language both resists and lends itself to formalization Makes objectual semantics for quantified logic easy, with an incremental, rule-governed approach assisted by numerous simple exercises Makes important metatheoretical results accessible to introductory students through a discursive presentation of those results and by using simple case studies

This volume consists of papers selected from the presentations at the workshop and includes mainly recent developments in the fields of formal languages, automata theory and algebraic systems related to the theoretical computer science and informatics. It covers the areas such as automata and grammars, languages and codes, combinatorics on words, cryptosystems, logics and trees, Grobner bases, minimal clones, zero-divisor graphs, fine convergence of functions, and others.

The present monograph on matrix partial orders, the first on this topic, makes a unique presentation of many partial orders on matrices that have fascinated mathematicians for their beauty and applied scientists for their wide-ranging application potential. Except for the Loewner order, the partial orders considered are relatively new and came into being in the late 1970s. After a detailed introduction to generalized inverses and decompositions, the three basic partial orders - namely, the minus, the sharp and the star - and the corresponding one-sided orders are presented using various generalized inverses. The authors then give a unified theory of all these partial orders as well as study the parallel sums and shorted matrices, the latter being studied at great length. Partial orders of modified matrices are a new addition. Finally, applications are given in statistics and electrical network theory. Deceased

This edited collection bridges the foundations and practice of constructive mathematics and focusses on the contrast between the theoretical developments, which have been most useful for computer science (eg constructive set and type theories), and more specific efforts on constructive analysis, algebra and topology. Aimed at academic logicians, mathematicians, philosophers and computer scientists Including, with contributions from leading researchers, it is up-to-date, highly topical and broad in scope. This is the latest volume in the Oxford Logic Guides, which also includes: 41. J.M. Dunn and G. Hardegree: Algebraic Methods in Philosophical Logic 42. H. Rott: Change, Choice and Inference: A study of belief revision and nonmonotoic reasoning 43. Johnstone: Sketches of an Elephant: A topos theory compendium, volume 1 44. Johnstone: Sketches of an Elephant: A topos theory compendium, volume 2 45. David J. Pym and Eike Ritter: Reductive Logic and Proof Search: Proof theory, semantics and control 46. D.M. Gabbay and L. Maksimova: Interpolation and Definability: Modal and Intuitionistic Logics 47. John L. Bell: Set Theory: Boolean-valued models and independence proofs, third edition

This book is about "diamond," a logic of paradox. In diamond, a statement can be true yet false; an "imaginary" state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown "modulator"; and Metamathematical Dilemma, which relates diamond to Gdelian meta-mathematics and dilemma games.

This book contains the material for a first course in pure model theory with applications to differentially closed fields. Topics covered in this book include saturated model criteria for model completeness and elimination of quantifiers; Morley rank and degree of element types; categoricity in power; two-cardinal theorems; existence and uniqueness of prime model extensions of substructures of models of totally transcendental theories; and homogeneity of models of ???1-categorical theories.

Logic is now widely recognized as one of the foundational disciplines of computing, and its applications reach almost every aspect of the subject, from software engineering and hardware to programming languages and AI. The Handbook of Logic in Computer Science is a multi-volume work covering all the major areas of application of logic to theoretical computer science. The handbook comprises six volumes, each containing five or six chapters giving an in-depth overview of one of the major topics in field. It is the result of many years of cooperative effort by some of the most eminent frontline researchers in the field, and will no doubt be the standard reference work in logic and theoretical computer science for years to come. Volume 3: Semantic Structures covers all the fundamental topics of semantics in logic and computation. The extensive chapters are the result of several years of coordinated research, and each have a thematic perspective. Together, they offer the reader the latest in research work, and the book will be indispensable to anyone seriously involved in the subject.

Machine Intelligence 14 contains material presented at the Anglo-Janpanese workshop of Novemver 1993 held at the Hitachi Research Laboratory. It marks the 70th birthday of Donald Michie, the founder of the series. The contents is divided into the following subjects: complex decision taking, inductive logic programming, applied machine learning, dynamic control, and computational learning theory. Applications include controlling a steel mill, discovery of protein structural constraints, and qualitative control for dynamic systems. This book is intended for researchers in Artificial Intelligence.

A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic
Logic of Mathematics combines a full-scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems:
* GAdel's theorems of completeness and incompleteness
* The independence of Goodstein's theorem from Peano arithmetic
* Tarski's theorem on real closed fields
* Matiyasevich's theorem on diophantine formulas
Logic of Mathematics also features:
* Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types
* Clear, concise explanations of all key concepts, from Boolean algebras to Skolem-LAwenheim constructions and other topics
* Carefully chosen exercises for each chapter, plus helpful solution hints
At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic-requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms.
Part I contains a thorough introduction to mathematical logic and model theory-including a full discussion of terms, formulas, and other fundamentals, plus detailed coverageof relational structures and Boolean algebras, GAdel's completeness theorem, models of Peano arithmetic, and much more.
Part II focuses on a number of advanced theorems that are central to the field, such as GAdel's first and second theorems of incompleteness, the independence proof of Goodstein's theorem from Peano arithmetic, Tarski's theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems.
With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use-the perfect textbook for advanced students of mathematics, computer science, and logic.

This superb collection of papers focuses on a fundamental question in logic and computation: What is a logical system? With contributions from leading researchers--including Ian Hacking, Robert Kowalski, Jim Lambek, Neil Tennent, Arnon Avron, L. Farinas del Cerro, Kosta Dosen, and Solomon Feferman--the book presents a wide range of views on how to answer such a question, reflecting current, mainstream approaches to logic and its applications. Written to appeal to a diverse audience of readers, What is a Logical System? will excite discussion among students, teachers, and researchers in mathematics, logic, computer science, philosophy, and linguistics.

Silly rabbit Your argument is ill-founded.

Have you read (or stumbled into) one too many irrational online debates? Ali Almossawi certainly had, so he wrote An Illustrated Book of Bad Arguments This handy guide is here to bring the internet age a much-needed dose of old-school logic (really old-school, a la Aristotle).

Here are cogent explanations of the straw man fallacy, the slippery slope argument, the ad hominem attack, and other common attempts at reasoning that actually fall short plus a beautifully drawn menagerie of animals who (adorably) commit every logical faux pas. Rabbit thinks a strange light in the sky must be a UFO because no one can prove otherwise (the appeal to ignorance). And Lion doesn t believe that gas emissions harm the planet because, if that were true, he wouldn t like the result (the argument from consequences).

Once you learn to recognize these abuses of reason, they start to crop up everywhere from congressional debate to YouTube comments which makes this geek-chic book a must for anyone in the habit of holding opinions. It s the antidote to fuzzy thinking, with furry animals "

The Handbook of Logic in Computer Science is a multi-volume work covering all major areas of application of logic to theoretical computer science.

The requirement to reason logically forms the basis of all mathematics, and hence mathematical logic is one of the most fundamental topics that students will study. Assuming no prior knowledge of the topic, this book provides an accessible introduction for advanced undergraduate students.

This book is not a conventional history of mathematics as such, a museum of documents and scientific curiosities. Instead, it identifies this vital science with the thought of those who constructed it and in its relation to the changing cultural context in which it evolved. Particular emphasis is placed on the philosophic and logical systems, from Aristotle onward, that provide the basis for the fusion of mathematics and logic in contemporary thought. Ettore Carruccio covers the evolution of mathematics from the most ancient times to our own day. In simple and non-technical language, he observes the changes that have taken place in the conception of rational theory, until we reach the lively, delicate and often disconcerting problems of modern logical analysis. The book contains an unusual wealth of detail (including specimen demonstrations) on such subjects as the critique of Euclid's fifth postulate, the rise of non-Euclidean geometry, the introduction of theories of infinite sets, the construction of abstract geometry, and-in a notably intelligible discussion-the development of modern symbolic logic and meta-mathematics. Scientific problems in general and mathematical problems in particular show their full meaning only when they are considered in the light of their own history. This book accordingly takes the reader to the heart of mathematical questions, in a way that teacher, student and layman alike will find absorbing and illuminating. The history of mathematics is a field that continues to fascinate people interested in the course of creativity, and logical inference u quite part and in addition to those with direct mathematical interests.

This book is a compilation of papers presented at the 2002 European Summer Meeting of the Association for Symbolic Logic and the associated Colloquium Logicum 2002 conference. It includes tutorials and research articles from some of the world's preeminent logicians. The topics presented span all areas of mathematical logic, with a particular emphasis on Computability Theory and Proof Theory.

This proceedings volume contains research papers in mathematical logic, especially in model theory and its applications to algebra and formal theories of arithmetic. Other papers address interpretability theory, computable analysis, modal logic, and the history of mathematical logic in Iran. The conference was held in Tehran, Iran, in October 2003, with the expressed purpose of bringing together researchers with connections to Iranian logicians and promoting further research in mathematical logic in Iran.

Kurt Goedel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Goedel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Goedel's Nachlass. These long-awaited final two volumes contain Goedel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Goedel's Nachlass. All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Goedel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Goedel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

The papers presented at the Symposium focused mainly on two fields of interest. First, there were papers dealing with the theoretical background of fuzzy logic and with applications of fuzzy reasoning to the problems of artificial intelligence, robotics and expert systems. Second, quite a large number of papers were devoted to fuzzy approaches to modelling of decision-making situations under uncertainty and vagueness and their applications to the evaluation of alternatives, system control and optimization.Apart from that, there were also some interesting contributions from other areas, like fuzzy classifications and the use of fuzzy approaches in quantum physics.This volume contains the most valuable and interesting papers presented at the Symposium and will be of use to all those researchers interested in fuzzy set theory and its applications.

Kurt Gödel is regarded as one of the most outstanding logician of the twentieth century, famous for his work on logic and number theory. This third volume of a comprehensive edition of Godel's works comprises a selection of previously unpublished manuscripts and lectures. It includes introductory notes that provide extensive explanations and historical commentary on each of the papers. This book is accessible to a wide audience without sacrificing historical or scientific accuracy and will be an essential part of the working library of both professionals and students.

Logic languages are free from the ambiguities of natural languages, and are therefore specially suited for use in computing. Model theory is the branch of mathematical logic which concerns the relationship between mathematical structures and logic languages, and has become increasingly important in areas such as computing, philosophy and linguistics. As the reasoning process takes place at a very abstract level, model theory applies to a wide variety of structures. It is also possible to define new structures and classify existing ones by establishing links between them. These links can be very useful since they allow us to transfer our knowledge between related structures. This book provides a clear and readable introduction to the subject, and is suitable for both mathematicians and students from outside the subject. It includes some historically relevant information before each major topic is introduced, making it a useful reference for non-experts. The motivation of the subject is constantly explained, and proofs are also explained in detail.