Machine generated contents note: Foreword Maurice Nivat; Introduction; 1. Overview; 2. Graph algebras and widths of graphs; 3. Equational and recognizable sets in many-sorted algebras; 4. Equational and recognizable sets of graphs; 5. Monadic second-order logic; 6. Algorithmic applications; 7. Monadic second-order transductions; 8. Transductions of terms and words J. Engelfriet; 9. Relational structures; 10. Conclusion and open problems; References; Index.
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. (shrink)
Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert’s metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.
This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or (...) creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)
Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic ...
This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4–9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
Proceedings of the Tenth Brazilian Conference on Mathematical Logic. Coleção CLE, volume 14, 1995. Centro De Lógica, Epistemologia e História da Ciência, Unicamp, Campinas, SP, Brazil.
This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in (...) the main body of the text is rigorous, but, a section of 'historical remarks' traces the evolution of the ideas presented in each chapter. Sources of the original accounts of these developments are listed in the bibliography. (shrink)
A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics (...) -- Soundness and completeness -- Why is first order logic "First Order"? (shrink)
This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...) striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)
An excellent introduction to mathematical logic, this book provides readers with a sound knowledge of the most important approaches to the subject, stressing the use of logical methods in attacking nontrivial problems. It covers the logic of classes, of propositions, of propositional functions, and the general syntax of language, with a brief introduction that also illustrates applications to so-called undecidability and incompleteness theorems. Other topics include the simple proof of the completeness of the theory of combinations, Church's theorem (...) on the recursive unsolvability of the decision problem for the restricted function calculus, and the demonstrable properties of a formal system as a criterion for its acceptability. 1950 ed. (shrink)
LN , so f lies in the elementary submodel M'. Clearly co 9 M' . It follows that 6 = {f(n): n em} is included in M'. Hence the ordinals of M' form an initial ...
Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...) II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)
This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is (...) kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis. (shrink)
Springer link: http://www.springer.com/philosophy/logic+and+philosophy+of+language/book/978-94-007-6090-5 -/- This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining (...) it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. -/- Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)
Mathematics often seems incomprehensible, a melee of strange symbols thrown down on a page. But while formulae, theorems, and proofs can involve highly complex concepts, the math becomes transparent when viewed as part of a bigger picture. What Is a Number? provides that picture. Robert Tubbs examines how mathematical concepts like number, geometric truth, infinity, and proof have been employed by artists, theologians, philosophers, writers, and cosmologists from ancient times to the modern era. Looking at a broad range of (...) topics -- from Pythagoras's exploration of the connection between harmonious sounds and mathematical ratios to the understanding of time in both Western and pre-Columbian thought -- Tubbs ties together seemingly disparate ideas to demonstrate the relationship between the sometimes elusive thought of artists and philosophers and the concrete logic of mathematicians. He complements his textual arguments with diagrams and illustrations. This historic and thematic study refutes the received wisdom that mathematical concepts are esoteric and divorced from other intellectual pursuits -- revealing them instead as dynamic and intrinsic to almost every human endeavor. (shrink)
Mathematical Logic for Computer Science is a mathematics textbook with theorems and proofs, but the choice of topics has been guided by the needs of computer science students. The method of semantic tableaux provides an elegant way to teach logic that is both theoretically sound and yet sufficiently elementary for undergraduates. To provide a balanced treatment of logic, tableaux are related to deductive proof systems.The logical systems presented are:- Propositional calculus (including binary decision diagrams);- Predicate calculus;- Resolution;- Hoare logic;- (...) Z;- Temporal logic.Answers to exercises (for instructors only) as well as Prolog source code for algorithms may be found via the Springer London web site: http://www.springer.com/978-1-85233-319-5 Mordechai Ben-Ari is an associate professor in the Department of Science Teaching of the Weizmann Institute of Science. He is the author of numerous textbooks on concurrency, programming languages and logic, and has developed software tools for teaching concurrency. In 2004, Ben-Ari received the ACM/SIGCSE Award for Outstanding Contributions to Computer Science Education. (shrink)
The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of recent developments. Highlights of this volume from the 2005 meeting include three papers on different aspects of connections between model theory and algebra; a survey of recent major advances in combinatorial set theory; a tutorial on proof theory and modal logic; and a description of (...) Bernay's philosophy of mathematics. (shrink)
Filling the need for an accessible, carefully structured introductory text in symbolic logic, Modern Logic has many features designed to improve students' comprehension of the subject, including a proof system that is the same as the award-winning computer program MacLogic, and a special appendix that shows how to use MacLogic as a teaching aid. There are graded exercises at the end of each chapter--more than 900 in all--with selected answers at the end of the book. Unlike competing texts, Modern (...) Logic gives equal weight to semantics and proof theory and explains their relationship, and develops in detail techniques for symbolizing natural language in first-order logic. After a general introduction featuring the notion of logical form, the book offers sections on classical sentential logic, monadic predicate logic, and full first-order logic with identity. A concluding section deals with extensions of and alternatives to classical logic, including modal logic, intuitionistic logic, and fuzzy logic. For students of philosophy, mathematics, computer science, or linguistics, Modern Logic provides a thorough understanding of basic concepts and a sound basis for more advanced work. (shrink)
Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay oN phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, (...) Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincare; and Frege. (shrink)
This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...
Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in (...) the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians. (shrink)
Learn how to develop your reasoning skills and how to write well-reasoned proofs Learning to Reason shows you how to use the basic elements of mathematical language to develop highly sophisticated, logical reasoning skills. You'll get clear, concise, easy-to-follow instructions on the process of writing proofs, including the necessary reasoning techniques and syntax for constructing well-written arguments. Through in-depth coverage of logic, sets, and relations, Learning to Reason offers a meaningful, integrated view of modern mathematics, cuts through confusing terms (...) and ideas, and provides a much-needed bridge to advanced work in mathematics as well as computer science. Original, inspiring, and designed for maximum comprehension, this remarkable book: Clearly explains how to write compound sentences in equivalent forms and use them in valid arguments Presents simple techniques on how to structure your thinking and writing to form well-reasoned proofs Reinforces these techniques through a survey of sets-the building blocks of mathematics Examines the fundamental types of relations, which is "where the action is" in mathematics Provides relevant examples and class-tested exercises designed to maximize the learning experience Includes a mind-building game/exercise space at www.wiley.com/products/subject/mathematics/. (shrink)
Chihara here develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics from the viewpoint of the constructibility theory developed. This innovative analysis will appeal to mathematicians and philosophers of logic, mathematics, and science.
A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises (...) designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. (shrink)
Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in that (...) it is much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (shrink)
This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search including proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences.
Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in (...) proof and the possibility of eliminating impredictive reasoning from proof. Students and lecturers of philosophy, philosophy of logic, and philosophy of mathematics will find this to be essential reading. A companion volume entitled Proof and Logic in Mathematics is also available from Routledge. (shrink)
This introduction to rigorous mathematical logic is simple enough in both presentation and context for students of a wide range of ages and abilities. Starting with symbolizing sentences and sentential connectives, it proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables. Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Throughout the book, the (...) authors emphasize the pervasive and important problem of translating English sentences into logical or mathematical symbolism. 1964 edition. Index. (shrink)
Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, (...) a traditional stumbling block for students on the way to the Godel incompleteness theorems. (shrink)
Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.
This collection of new essays offers a 'state-of-the-art' conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the centre of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures, published here for the first time.
This lively introduction to mathematical logic, easily accessible to non-mathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursivefunctions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.
Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a (...) derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optinal sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science. (shrink)
Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, (...) a traditional stumbling block for students on the way to the Godel incompleteness theorems. (shrink)
Appropriate for upper-level undergraduates and graduate students, this volume includes a variety of Boole's writings on logical subjects, along with papers on related questions of probability. His earlier work, The Mathematical Analysis of Logic, appears here, together with an account of the notes Boole made on his own interleaved copy. In addition, the appendices contain relevant papers by contemporaries with whom the author engaged in discussion, making it possible to trace interesting developments in Boolean reasoning-particularly in regard to his (...) extended treatment of the relation between formal logic and the theory of probabilities. 1952 ed. (shrink)
A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.
Suitable for advanced undergraduates and graduate students from diverse fields and varying backgrounds, this self-contained course in mathematical logic features numerous exercises that vary in difficulty. The author is a Professor of Mathematics at the University of Wisconsin.
Originally published in 1910, Principia Mathematica led to the development of mathematical logic and computers and thus to information sciences. It became a model for modern analytic philosophy and remains an important work. In the late 1960s the Bertrand Russell Archives at McMaster University in Canada obtained Russell's papers, letters and library. These archives contained the manuscripts for the new Introduction and three Appendices that Russell added to the second edition in 1925. Also included was another manuscript, 'The Hierarchy (...) of Propositions and Functions', which was divided up and re-used to create the final changes for the second edition. These documents provide fascinating insight, including Russell's attempts to work out the theorems in the flawed Appendix B, 'On Induction'. An extensive introduction describes the stages of the manuscript material on the way to print and analyzes the proposed changes in the context of the development of symbolic logic after 1910. (shrink)
The theme of this book is formed by a pair of concepts: the concept of formal language as carrier of the precise expression of meaning, facts and problems, and the concept of algorithm or calculus, i.e. a formally operating procedure for the solution of precisely described questions and problems. The book is a unified introduction to the modern theory of these concepts, to the way in which they developed first in mathematical logic and computability theory and later in automata (...) theory, and to the theory of formal languages and complexity theory. Apart from considering the fundamental themes and classical aspects of these areas, the subject matter has been selected to give priority throughout to the new aspects of traditional questions, results and methods which have developed from the needs or knowledge of computer science and particularly of complexity theory. It is both a textbook for introductory courses in the above-mentioned disciplines as well as a monograph in which further results of new research are systematically presented and where an attempt is made to make explicit the connections and analogies between a variety of concepts and constructions. (shrink)
The classic results obtained by Gödel, Tarski, Kleene, and Church in the early thirties are the finest flowers of symbolic logic. They are of fundamental importance to those investigations of the foundations of mathematics via the concept of a formal system that were inaugurated by Frege, and of obvious significance to the mathematical disciplines, such as computability theory, that developed from them. Derived from courses taught by the author over several years, this new exposition presents all of the (...) results with their original proofs and central concepts in a manner that is unified by a systematic grounding of the notion of effectiveness in the semantics of the existential quantifier. Logicians and non-mathematicians, repelled by detail which is not obviously relevant in the standard textbooks, will be able to reach the heart of the matter with a minimum of fuss. (shrink)
This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy - as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas. The volume will be of interest to logicians, computer (...) scientists, philosophers, and linguists. The contributions concern classical first-order logic, higher-order logic, non-classical theories of implication, set theories with universal sets, the logical and semantical paradoxes, the lambda-calculus, especially as it is used in computation, philosophical issues about meaning and ontology in the abstract sciences and in natural language, and much else. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields. (shrink)
Gödel's relation to the work of Plato, Leibniz, Kant, and Husserl is examined, and a new type of platonic rationalism that requires rational intuition, called ...
The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and (...) model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course. (shrink)
Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text. Reprint of 1982 ed.
This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. 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 will provide the reader with a solid foundation in set theory and provides a context for the presentation of advanced topics such as absoluteness, relative consistency results, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing. (shrink)
Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the excercises and the end of the volume. This is an ideal introduction to mathematics and (...) logic for the advanced undergraduate student. (shrink)
This volume is the product of the Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science and contains the text of most of ...
This_classic undergraduate text_elegantly acquaints students with the_fundamental concepts and methods of mathematics. In addition to introducing_many noteworthy historical figures_from the 18th through the mid-20th centuries, it examines_the axiomatic method, set theory, infinite sets, the linear continuum and the real number system, groups, intuitionism,_formal systems, mathematical logic, and other topics.
Logic With Trees is a new and original introduction to modern formal logic. It contains discussions on philosophical issues such as truth, conditionals and modal logic, presenting the formal material with clarity, and preferring informal explanations and arguments to intimidatingly rigorous development. Worked examples and exercises guide beginners through the book, with answers to selected exercises enabling readers to check their progress. Logic With Trees equips students with: a complete and clear account of the truth-tree system for first order logic; (...) the importance of logic and its relevance to many different disciplines; the skills to grasp sophisticated formal reasoning techniques necessary to explore complex metalogic; the ability to contest claims that "ordinary" reasoning is well represented by formal first order logic. (shrink)
CHAPTER 1. INTENSIONAL LOGIC §1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, ...
INTENSIONAL LOGIC §1. Natural Language and Intensional Logic When we speak of a theory of meaning for a natural language such as English, we have in mind an ...
SYMBOLIC LOGIC. CHAPTER I. ON THE FORMS OF LOGICAL PROPOSITION. IT has been mentioned in the Introduction that the System of Logic which this work is ...
The Handbook of the Logic of Argument and Inference is an authoritative reference work in a single volume, designed for the attention of senior undergraduates, graduate students and researchers in all the leading research areas concerned with the logic of practical argument and inference. After an introductory chapter, the role of standard logics is surveyed in two chapters. These chapters can serve as a mini-course for interested readers, in deductive and inductive logic, or as a refresher. Then follow two chapters (...) of criticism; one the internal critique and the other the empirical critique. The first deals with objections to standard logics (as theories of argument and inference) arising from the research programme in philosophical logic. The second canvasses criticisms arising from work in cognitive and experimental psychology. The next five chapters deal with developments in dialogue logic, interrogative logic, informal logic, probability logic and artificial intelligence. The last chapter surveys formal approaches to practical reasoning and anticipates possible future developments. Taken as a whole the Handbook is a single-volume indication of the present state of the logic of argument and inference at its conceptual and theoretical best. Future editions will periodically incorporate significant new developments. (shrink)
Discussion of Wittgenstein's Tractatus is currently dominated by two opposing interpretations of the work: a metaphysical or realist reading and the 'resolute' reading of Diamond and Conant. Marie McGinn's principal aim in this book is to develop an alternative interpretative line, which rejects the idea, central to the metaphysical reading, that Wittgenstein sets out to ground the logic of our language in features of an independently constituted reality, but which allows that he aims to provide positive philosophical insights into how (...) language functions. McGinn takes as a guiding principle the idea that we should see Wittgenstein's early work as an attempt to eschew philosophical theory and to allow language itself to reveal how it functions. By this account, the aim of the work is to elucidate what language itself makes clear, namely, what is essential to its capacity to express thoughts that are true or false. However, the early Wittgenstein undertakes this descriptive project in the grip of a set of preconceptions concerning the essence of language that determine both how he conceives the problem and the approach he takes to the task of clarification. Nevertheless, the Tractatus contains philosophical insights, achieved despite his early preconceptions, that form the foundation of his later philosophy. -/- The anti-metaphysical interpretation that is presented includes a novel reading of the problematic opening sections of the Tractatus, in which the apparently metaphysical status of Wittgenstein's remarks is shown to be an illusion. The book includes a discussion of the philosophical background to the Tractatus, a comprehensive interpretation of Wittgenstein's early views of logic and language, and an interpretation of the remarks on solipsism. The final chapter is a discussion of the relation between the early and the later philosophy that articulates the fundamental shift in Wittgenstein's approach to the task of understanding how language functions and reveal the still more fundamental continuity in his conception of his philosophical task. (shrink)
Highlights of this volume from the 2004 Annual European Meeting of the Association for Symbolic Logic (ASL) include a tutorial survey of the recent highpoints of universal algebra, written by a leading expert; explorations of foundational questions; a quartet of model theory papers giving an excellent reflection of current work in model theory, from the most abstract aspect "abstract elementary classes" to issues around p-adic integration.
Logic and Reality is a collection of essays by philosophers, logicians, mathematicians, and computer scientists, celebrating the work of the late distinguished philosopher Arthur Prior on the eightieth anniversary of his birth. Topics range from philosophical discussions of the nature of time and of the nature of logic itself, to descriptions of computer systems that can reason and take account of the fact that they exist in a temporal world.
Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.
Pt. I. Writings on the theory of logic: I. Pure logic or the logic of quality apart from quantity. II. The substitution of similars. III. On the mechanical performance of logical inference. IV. On a general system of numerically definite reasoning.--Pt. II. John Stuart Mill's philosophy tested: I. On geometrical reasoning. II. On resemblance. III. The experimental methods. IV. Utilitarianism. V. On the method of difference.
Lucid, non-intimidating presentation of propositional logic, propositional calculus and predicate logic by Russian scholar. Topics of concern in a variety of fields, including computer science, systems analysis, linguistics, etc. Accessible to high school students; valuable review of fundamentals for professionals. Exercises (no solutions). Preface. Three appendices. Indices. Bibliogaphy. 14 figures.
Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen (...) has brought together an outstanding collection of essays in a volume which will be essential for philosophers and historians of mathematics who are interested in the nature of reasoning and justification. A companion volume, Proof, Knowledge and Formalization is also available from Routledge. (shrink)
This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these (...) theories provide a new outlook on classical topics, such as inductive definitions and predicative mathematics; (iii) they are particularly promising with regard to applications. Research arising from paradoxes has moved progressively closer to the mainstream of mathematical logic and has become much more prominent in the last twenty years. A number of significant developments, techniques and results have been discovered. Academics, students and researchers will find that the book contains a thorough overview of all relevant research in this field. (shrink)
ELEMENTARY LOGIC GR. C. MOISIL Institute of Mathematics, Rumanian Academy, Bucharest, Rumania 1. We shall consider a typified logic of propositions. ...
Introduction Alexius Meinong and his circle of students and collaborators at the Phi- losophisches Institut der Universitat Graz formulated the basic ...
INTRODUCTION In ita original form the present paper was presented to the American Mathematical Society, April 2k,, as a companion piece to the writer's ...
" There are 31 chapters in 5 parts and approximately 320 exercises marked by difficulty and whether or not they are necessary for further work in the book.
Janusz Czelakowski Elements of Formal Action Theory 1. Elementary Action Systems 1.1 Introductory Remarks. In contemporary literature one may distinguish ...
