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In the Light of Logic

In the Light of Logic Author Solomon Feferman
ISBN-10 0195359836
Release 1998-11-19
Pages 352
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In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt G?del, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though G?del has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects.



Mathematics and Mind

Mathematics and Mind Author Alexander George
ISBN-10 9780195079296
Release 1994
Pages 204
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The essays in this volume investigate the conceptual foundations of mathematics illuminating the powers of the mind. Contributors include Alexander George, Michael Dummett, George Boolos, W.W. Tait, Wilfried Sieg, Daniel Isaacson, Charles Parsons, and Michael Hallett.



Three Views of Logic

Three Views of Logic Author Donald W. Loveland
ISBN-10 9781400848751
Release 2014-01-26
Pages 344
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Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic Treats traditional logic in a modern format Presents relevance logic with applications Provides an ideal text for a variety of one-semester upper-level undergraduate courses



The Cultural Logic of Computation

The Cultural Logic of Computation Author David Golumbia
ISBN-10 0674032926
Release 2009-04-30
Pages 257
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In The Cultural Logic of Computation, David Golumbia, who worked as a software designer for more than ten years, argues that computers are cultural "all the way down" - that there is no part of the apparent technological transformation that is not shaped by historical and cultural processes, or that escapes existing cultural politics. The Cultural Logic of Computation provides a needed corrective to the uncritical enthusiasm for computers common today in many parts of our culture.



The Logic of Reliable Inquiry

The Logic of Reliable Inquiry Author Kevin T. Kelly
ISBN-10 9780195091953
Release 1996-01-04
Pages 434
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One scientific method can be said to be more reliable than another if it can arrive at the truth in more possible circumstances than the other can. This extensively ilustrated book shows how to find answers to questions such as whether standard methodological recommendations help or hinder the reliability of inquiry, or how reliable computable methods can be, using techniques and concepts drawn from formal learning theory, topology and the theory of computability.



Computability and Logic

Computability and Logic Author George S. Boolos
ISBN-10 9781107049277
Release 2007-09-17
Pages
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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. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual.



Logical Reasoning with Diagrams

Logical Reasoning with Diagrams Author Gerard Allwein
ISBN-10 9780195104271
Release 1996
Pages 270
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The authors explore the logical properties of diagrams, charts, and maps, and the role these play in problem solving and teaching reasoning skills.



The Logic of Mind

The Logic of Mind Author R.J. Nelson
ISBN-10 9789400925953
Release 2012-12-06
Pages 394
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This book presents a mechanist philosophy of mind. I hold that the human mind is a system of computational or recursive rules that are embodied in the nervous system; that the material presence of these rules accounts for perception, conception, speech, belief, desire, intentional acts, and other forms of intelligence. In this edition I have retained the whole of the fIrst edition except for discussion of issues which no longer are relevant in philosophy of mind and cognitive psychology. Earlier reference to disputes of the 1960's and 70's between hard-line empiricists and neorationalists over the psychological status of grammars and language acquisition, for instance, has simply been dropped. In place of such material I have entered some timely or new topics and a few changes. There are brief references to the question of computer versus distributed processing (connectionist) theories. Many of these questions dissolve if one distinguishes as I now do in Chapter II between free and embodied algorithms. I have also added to my comments on artifIcal in telligence some reflections. on Searle's Chinese Translator. The irreducibility of machine functionalist psychology in my version or any other has been exaggerated. Input, output, and state entities are token identical to physical or biological things of some sort, while a machine system as a collection of recursive rules is type identical to representatives of equivalence classes. This nuld technicality emerges in Chapter XI. It entails that so-called "anomalous monism" is right in one sense and wrong in another.



Computational Philosophy of Science

Computational Philosophy of Science Author Paul Thagard
ISBN-10 0262700484
Release 1993
Pages 240
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By applying research in artificial intelligence to problems in the philosophy of science, Paul Thagard develops an exciting new approach to the study of scientific reasoning. This approach uses computational ideas to shed light on how scientific theories are discovered, evaluated, and used in explanations. Thagard describes a detailed computational model of problem solving and discovery that provides a conceptually rich yet rigorous alternative to accounts of scientific knowledge based on formal logic, and he uses it to illuminate such topics as the nature of concepts, hypothesis formation, analogy, and theory justification.



Logic in Computer Science

Logic in Computer Science Author Michael Huth
ISBN-10 9781139453059
Release 2004-08-26
Pages
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Recent years have seen the development of powerful tools for verifying hardware and software systems, as companies worldwide realise the need for improved means of validating their products. There is increasing demand for training in basic methods in formal reasoning so that students can gain proficiency in logic-based verification methods. The second edition of this successful textbook addresses both those requirements, by continuing to provide a clear introduction to formal reasoning which is both relevant to the needs of modern computer science and rigorous enough for practical application. Improvements to the first edition have been made throughout, with extra and expanded sections on SAT solvers, existential/universal second-order logic, micro-models, programming by contract and total correctness. The coverage of model-checking has been substantially updated. Further exercises have been added. Internet support for the book includes worked solutions for all exercises for teachers, and model solutions to some exercises for students.



Mathematical Logic

Mathematical Logic Author George Tourlakis
ISBN-10 9781118030691
Release 2011-03-01
Pages 294
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A comprehensive and user-friendly guide to the use of logic in mathematical reasoning Mathematical Logic presents a comprehensive introduction to formal methods of logic and their use as a reliable tool for deductive reasoning. With its user-friendly approach, this book successfully equips readers with the key concepts and methods for formulating valid mathematical arguments that can be used to uncover truths across diverse areas of study such as mathematics, computer science, and philosophy. The book develops the logical tools for writing proofs by guiding readers through both the established "Hilbert" style of proof writing, as well as the "equational" style that is emerging in computer science and engineering applications. Chapters have been organized into the two topical areas of Boolean logic and predicate logic. Techniques situated outside formal logic are applied to illustrate and demonstrate significant facts regarding the power and limitations of logic, such as: Logic can certify truths and only truths. Logic can certify all absolute truths (completeness theorems of Post and Gödel). Logic cannot certify all "conditional" truths, such as those that are specific to the Peano arithmetic. Therefore, logic has some serious limitations, as shown through Gödel's incompleteness theorem. Numerous examples and problem sets are provided throughout the text, further facilitating readers' understanding of the capabilities of logic to discover mathematical truths. In addition, an extensive appendix introduces Tarski semantics and proceeds with detailed proofs of completeness and first incompleteness theorems, while also providing a self-contained introduction to the theory of computability. With its thorough scope of coverage and accessible style, Mathematical Logic is an ideal book for courses in mathematics, computer science, and philosophy at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners who wish to learn how to use logic in their everyday work.



Computational Logic and Human Thinking

Computational Logic and Human Thinking Author Robert Kowalski
ISBN-10 9781139498159
Release 2011-07-21
Pages
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The practical benefits of computational logic need not be limited to mathematics and computing. As this book shows, ordinary people in their everyday lives can profit from the recent advances that have been developed for artificial intelligence. The book draws upon related developments in various fields from philosophy to psychology and law. It pays special attention to the integration of logic with decision theory, and the use of logic to improve the clarity and coherence of communication in natural languages such as English. This book is essential reading for teachers and researchers who may be out of touch with the latest developments in computational logic. It will also be useful in any undergraduate course that teaches practical thinking, problem solving or communication skills. Its informal presentation makes the book accessible to readers from any background, but optional, more formal, chapters are also included for those who are more technically oriented.



Logical Foundations of Mathematics and Computational Complexity

Logical Foundations of Mathematics and Computational Complexity Author Pavel Pudlák
ISBN-10 9783319001197
Release 2013-04-22
Pages 695
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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.



Computing and Philosophy in Asia

Computing and Philosophy in Asia Author Soraj Hongladarom
ISBN-10 9781443807432
Release 2009-03-26
Pages 295
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This volume is a collection of selected papers presented at the Second Asia-Pacific Computing and Philsosophy Conference, which was held in Bangkok, Thailand in January 2005. The conference was organized by the Center for Ethics of Science and Technology, Chulalongkorn University on behalf of the International Association of Computing and Philosophy (www.ia-cap.org). Computing have had a long relationship with philosophy, starting from the problem of how symbols being manipulated in computing bear a relation to the outside world, to those of artificial intelligence, robotics, computer simulation, and so on. Moreover, as computer technologies have become thoroughly pervasive in today's environment, there are also issues concerning social and ethical impacts brought about by them. The papers in the volume represent a wide variety of concerns and various dimensions within which computing and philosophy are related. Furthermore, it also represents some of the first attempts to highlight cultural dimensions of computing and philosophy, which became prominent when the conference was held for the first time within the milieu of an Asian culture. (The First Asia-Pacific Computing and Philosophy was held in Canberra, Australia.) Hence, many of the papers in the volume address this added dimension. Apart form the usual problems of how computers and human lives are interconnected, the papers here also discuss how computers are related to human lives as lived in a specific culture. Thus the book breaks a new ground and should be of interest to a wide range of scholars and students who are interested, not only on computing and philosophy generally construed, but also on this exciting new dimension of how the cultures of Asia, the West, and others bear upon the traditional issues in computing and philosophy, and on how this dimension raises some new concerns and agenda. Among the topics discussed in this volume are: political online forums in Saudi Arabia, e-democracy and structural transformation of public sphere, the Buddhist informational person, a glance into the lives of computerized generation in Thailand, technology and journalism in the market, local approaches and global potential (?) of information ethics, computer-enhanced good life, computer teaching ethics, and many others.



The Great Formal Machinery Works

The Great Formal Machinery Works Author Jan von Plato
ISBN-10 9781400885039
Release 2017-08-02
Pages 400
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The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution. Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schröder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Gödel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later. Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.



Conditionals

Conditionals Author Gabriella Crocco
ISBN-10 9780198538615
Release 1995
Pages 368
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Internationally recognized logicians present current thinking on the understanding of the role of deduction in human reasoning.



Proofs and Algorithms

Proofs and Algorithms Author Gilles Dowek
ISBN-10 0857291211
Release 2011-01-11
Pages 156
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Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel’s incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.