Elements of Intuitionism

Author: Michael A. E. Dummett

Publisher: Oxford University Press

ISBN: 9780198505242

Category: Mathematics

Page: 331

View: 4820

This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.

Philosophy of Language: The Key Thinkers

Author: Barry Lee

Publisher: Bloomsbury Publishing

ISBN: 1441131396

Category: Language Arts & Disciplines

Page: 320

View: 2492

Philosophers have raised and struggled with questions relating to human language for more than 2000 years. Philosophy of Language: The Key Thinkers offers a comprehensive historical overview of this fascinating field. Thirteen specially commissioned essays introduce and explore the contributions of those philosophers who have shaped the subject and the central issues and arguments therein. Philosophical questions relating to language have been subjected to particularly intense scrutiny since the work of Gottlob Frege in the nineteenth and early twentieth centuries. This book concentrates on the development of philosophical views on language over the last 130 years, offering coverage of all the leading thinkers in the field including Frege, Russell, Wittgenstein, Austin, Quine, Chomsky, Grice, Davidson, Dummett and Kripke. Crucially the book demonstrates how the ideas and arguments of these key thinkers have contributed to our understanding of the theoretical account of language use and its central concepts. Ideal for undergraduate students, the book lays the necessary foundations for a complete and thorough understanding of this fascinating subject.

Pluralism in Mathematics: A New Position in Philosophy of Mathematics

Author: Michèle Friend

Publisher: Springer Science & Business Media

ISBN: 9400770588

Category: Science

Page: 291

View: 3955

This book is about philosophy, mathematics and logic, giving a philosophical account of Pluralism which is a family of positions in the philosophy of mathematics. There are four parts to this book, beginning with a look at motivations for Pluralism by way of Realism, Maddy’s Naturalism, Shapiro’s Structuralism and Formalism. In the second part of this book the author covers: the philosophical presentation of Pluralism; using a formal theory of logic metaphorically; rigour and proof for the Pluralist; and mathematical fixtures. In the third part the author goes on to focus on the transcendental presentation of Pluralism, and in part four looks at applications of Pluralism, such as a Pluralist approach to proof in mathematics and how Pluralism works in regard to together-inconsistent philosophies of mathematics. The book finishes with suggestions for further Pluralist enquiry. In this work the author takes a deeply radical approach in developing a new position that will either convert readers, or act as a strong warning to treat the word ‘pluralism’ with care.

Epistemology versus Ontology

Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf

Author: P. Dybjer,Sten Lindström,Erik Palmgren,B.G. Sundholm

Publisher: Springer Science & Business Media

ISBN: 9400744358

Category: Philosophy

Page: 388

View: 966

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice (ZFC). This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics?

Interpolation and Definability

Modal and Intuitionistic Logics

Author: Dov M. Gabbay,Larisa Maksimova

Publisher: Oxford University Press on Demand

ISBN: 0198511744

Category: Computers

Page: 508

View: 4497

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language.Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (second edition), J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and NonmonotonicReasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control.


An Introduction to Formal Methods

Author: Antoni Diller

Publisher: N.A


Category: Computers

Page: 309

View: 7252

Offers a thorough and comprehensive tutorial introduction to Z, taking as its starting point the practical use of this formal specification language. After briefly explaining what Z is and why it is a good method for specifying software components, covers Z's basic mathematical toolkit, the fundamental ideas of the schema calculus, and the methods of reasoning. Specific case studies illustrate all aspects of Z and the mathematical notations that must be covered. Also discusses how Z specifications can be animated using a high-level modern functional programming language and Prolog. Standard notation is used throughout, and many proofs and fully explained exercises are included. With its helpful appendices and an annotated bibliography that contains many pointers to the ever-increasing literature on Z, this work will serve as a ready reference manual of the most frequently used Z notations

A New Application for Explanation-based Generalisation Within Automated Deduction

Author: Siani L. Baker

Publisher: N.A


Category: Automatic theorem proving

Page: 18

View: 5712

Abstract: "Generalisation is currently a major theorem-proving problem. This paper proposes a new method of generalisation, involving the use of explanation-based generalization within a new domain, which may succeed when other methods fail. The method has been implemented for simple arithmetical examples."

Reductive Logic and Proof-search

Proof Theory, Semantics, and Control

Author: David J. Pym,Eike Ritter

Publisher: Oxford University Press on Demand

ISBN: 0198526334

Category: Mathematics

Page: 208

View: 5106

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes 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. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects , J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning , and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2 .

A proof environment for arithmetic with the omega rule

Author: Siani L. Baker,Alan Smaill,University of Edinburgh. Department of Artificial Intelligence

Publisher: N.A


Category: Computers

Page: 17

View: 1083

Abstract: "An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the [omega]-rule. This paper exploits this notion within the domain of automated theorem-proving and discusses the implementation of such a proof environment, namely the CORE system which implements a version of the primitive recursive [omega]-rule. This involves providing an appropriate representation for infinite proofs, and a means of verifying properties of such objects. By means of the CORE system, from a finite number of instances a conjecture for a proof of the universally quantified formula is automatically derived by an inductive inference algorithm, and checked for correctness. In addition, candidates for cut formulae are generated by an explanation-based learning algorithm."

LATEX line by line

tips and techniques for document processing

Author: Antoni Diller

Publisher: Wiley

ISBN: 9780471979180

Category: Computers

Page: 311

View: 8698

This is an easy-to-follow tutorial on the most popular text processing system used in the academic community. It explains formatting fundamentals and the more complex techniques for typesetting mathematical formulas. It is useful as a resource for those with access to the previous version (LATEX 2.09) who want to update themselves on the latest version - LATEX 2. The book is aimed at anyone interested in text processing and in particular those wanting to use LATEX to produce high quality documents. LATEX 2e is suitable for people with no previous LATEX experience. Written from the users point of view, this edition features many entirely new commands, replacing obsolete material as well as an appendix describing the main differences between old version LATEX 2.09 and the new version. There is also a glossary of all basic LATEX 2 commands. Many of the typesetting examples from the book are coded as templates and are available on the accompanying Website.


Author: Paul Tomassi

Publisher: Routledge

ISBN: 1134705913

Category: Philosophy

Page: 432

View: 9344

Bringing elementary logic out of the academic darkness into the light of day, Paul Tomassi makes logic fully accessible for anyone attempting to come to grips with the complexities of this challenging subject. Including student-friendly exercises, illustrations, summaries and a glossary of terms, Logic introduces and explains: * The Theory of Validity * The Language of Propositional Logic * Proof-Theory for Propositional Logic * Formal Semantics for Propositional Logic including the Truth-Tree Method * The Language of Quantificational Logic including the Theory of Descriptions. Logic is an ideal textbook for any logic student: perfect for revision, staying on top of coursework or for anyone wanting to learn about the subject. Related downloadable software for Macs and PCs is available for this title at www.logic.routledge.com.