Proof and Disproof in Formal Logic

An Introduction for Programmers
Author: Richard Bornat
Publisher: Oxford University Press on Demand
ISBN: 0198530277
Category: Mathematics
Page: 243
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Proof and Disproof in Formal Logic is a lively and entertaining introduction to formal logic providing an excellent insight into how a simple logic works. This book concentrates on using logic as a tool: making and using formal proofs and disproofs of particular logical claims. The logic it uses - natural deduction - is very simple and shows how large mathematical universes can be built on small foundations. Aimed at undergraduates and graduates in computerscience, logic, mathematics, and philosophy, the text includes reference to...

Formal Languages in Logic

A Philosophical and Cognitive Analysis
Author: Catarina Dutilh Novaes
Publisher: Cambridge University Press
ISBN: 113978952X
Category: Philosophy
Page: N.A
View: 6992

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

Programming in Martin-Löf's type theory

an introduction
Author: Bengt Nordström,Kent Petersson,Jan M. Smith
Publisher: Oxford University Press, USA
Category: Computers
Page: 221
View: 3330

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In recent years, several formalisms for program construction have appeared. One such formalism is the type theory developed by Per Martin-Lof. Well suited as a theory for program construction, it makes possible the expression of both specifications and programs within the same formalism. Furthermore, the proof rules can be used to derive a correct program from a specification as well as to verify that a given program has a certain property. This book contains a thorough introduction to type theory, with information on polymorphic sets, subsets, monomorphic sets, and a full set of helpful examples.

The Problems of Philosophy

Author: Bertrand Russell
Publisher: Courier Corporation
ISBN: 048612116X
Category: Philosophy
Page: 128
View: 8529

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Accessible, thought-provoking study by Nobel Prize-winner considers distinction between appearance and reality, existence and nature of matter, idealism, inductive logic, intuitive knowledge, many other stimulating subjects.

A First Course in Logic

An Introduction to Model Theory, Proof Theory, Computability, and Complexity
Author: Shawn Hedman
Publisher: Oxford University Press on Demand
ISBN: 9780198529811
Category: Mathematics
Page: 431
View: 6347

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"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, thistext covers the fundamental topics in classical logic in a 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, andmodel theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course."--BOOK JACKET.

Logic Made Easy: How to Know When Language Deceives You

Author: Deborah J. Bennett
Publisher: W. W. Norton & Company
ISBN: 0393347613
Category: Philosophy
Page: 256
View: 6155

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"The best introduction to logic you will find."—Martin Gardner "Professor Bennett entertains as she instructs," writes Publishers Weekly about the penetrating yet practical Logic Made Easy. This brilliantly clear and gratifyingly concise treatment of the ancient Greek discipline identifies the illogical in everything from street signs to tax forms. Complete with puzzles you can try yourself, Logic Made Easy invites readers to identify and ultimately remedy logical slips in everyday life. Designed with dozens of visual examples, the book guides you through those hair-raising times when logic is at odds with our language and common sense. Logic Made Easy is indeed one of those rare books that will actually make you a more logical human being.

A Handbook for Teaching and Learning in Higher Education

Enhancing academic practice
Author: Heather Fry,Steve Ketteridge,Stephanie Marshall
Publisher: Routledge
ISBN: 1317650220
Category: Education
Page: 452
View: 6422

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This entirely new edition of a very successful book focuses on developing professional academic skills for supporting and supervising student learning and effective teaching. It is built on the premise that the roles of those who teach in higher education are complex and multi-faceted. A Handbook for Teaching and Learning in Higher Education is sensitive to the competing demands of teaching, research, scholarship, and academic management. The new edition reflects and responds to the rapidly changing context of higher education and to current understanding of how to best support student learning. Drawing together a large number of expert authors, it continues to feature extensive use of case studies that show how successful teachers have implemented these ideas. It includes key topics such as student engagement and motivation, internationalisation, employability, inclusive strategies for teaching, effective use of technology and issues relating to postgraduate students and student retention. Part 1 explores a number of aspects of the context of UK higher education that affect the education of students, looking at the drivers of institutional behaviours and how to achieve success as a university teacher. Part 2 examines learning, teaching and supervising in higher education and includes chapters on working with diversity, encouraging independent learning and learning gain. Part 3 considers approaches to teaching and learning in different disciplines, covering a full range including arts and humanities, social sciences, experimental sciences through to medicine and dentistry. Written to support the excellence in teaching and learning design required to bring about student learning of the highest quality, this will be essential reading for all new lecturers, particularly anyone taking an accredited course in teaching and learning in higher education, as well as those experienced lecturers who wish to improve their teaching practice. Those working in adult learning and educational development will also find the book to be a particularly useful resource. In addition it will appeal to staff who support learning and teaching in various other roles.

What is a Logical System?

Author: Dov M. Gabbay
Publisher: Oxford University Press
ISBN: 0198538596
Category: History
Page: 454
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Collection of papers addressing this fundamental question - what is a logical system? The world famous contributors present a spectrum of views on the answer.

Mathematical Logic

Author: Ian Chiswell,Wilfrid Hodges
Publisher: Oxford University Press on Demand
ISBN: 0198571003
Category: Mathematics
Page: 250
View: 7857

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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 becalculated; 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, assumingMatiyasevich'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. Optional 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 andcomputer 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.

Human Reasoning and Cognitive Science

Author: Keith Stenning,Michiel van Lambalgen
Publisher: MIT Press
ISBN: 0262293536
Category: Medical
Page: 422
View: 4426

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In Human Reasoning and Cognitive Science, Keith Stenning and Michiel van Lambalgen--a cognitive scientist and a logician--argue for the indispensability of modern mathematical logic to the study of human reasoning. Logic and cognition were once closely connected, they write, but were "divorced" in the past century; the psychology of deduction went from being central to the cognitive revolution to being the subject of widespread skepticism about whether human reasoning really happens outside the academy. Stenning and van Lambalgen argue that logic and reasoning have been separated because of a series of unwarranted assumptions about logic. Stenning and van Lambalgen contend that psychology cannot ignore processes of interpretation in which people, wittingly or unwittingly, frame problems for subsequent reasoning. The authors employ a neurally implementable defeasible logic for modeling part of this framing process, and show how it can be used to guide the design of experiments and interpret results.

Computer-Aided Reasoning

An Approach
Author: Matt Kaufmann,Panagiotis Manolios,J Strother Moore
Publisher: Springer
ISBN: 9780792377443
Category: Computers
Page: 270
View: 9600

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Computer-Aided Reasoning: An Approach is a textbook introduction to computer-aided reasoning. It can be used in graduate and upper-division undergraduate courses on software engineering or formal methods. It is also suitable in conjunction with other books in courses on hardware design, discrete mathematics, or theory, especially courses stressing formalism, rigor, or mechanized support. It is also appropriate for courses on artificial intelligence or automated reasoning and as a reference for business and industry. Current hardware and software systems are often very complex and the trend is towards increased complexity. Many of these systems are of critical importance; therefore making sure that they behave as expected is also of critical importance. By modeling computing systems mathematically, we obtain models that we can prove behave correctly. The complexity of computing systems makes such proofs very long, complicated, and error-prone. To further increase confidence in our reasoning, we can use a computer program to check our proofs and even to automate some of their construction. In this book we present: A practical functional programming language closely related to Common Lisp which is used to define functions (which can model computing systems) and to make assertions about defined functions; A formal logic in which defined functions correspond to axioms; the logic is first-order, includes induction, and allows us to prove theorems about the functions; The computer-aided reasoning system ACL2, which includes the programming language, the logic, and mechanical support for the proof process. The ACL2 system has been successfully applied to projects of commercial interest, including microprocessor, modeling, hardware verification, microcode verification, and software verification. This book gives a methodology for modeling computing systems formally and for reasoning about those models with mechanized assistance. The practicality of computer-aided reasoning is further demonstrated in the companion book, Computer-Aided Reasoning: ACL2 Case Studies. Approximately 140 exercises are distributed throughout the book. Additional material is freely available from the ACL2 home page on the Web, including solutions to the exercises, additional exercises, case studies from the companion book, research papers, and the ACL2 system with detailed documentation.

Principia Mathematica

Author: Alfred North Whitehead,Bertrand Russell
Publisher: N.A
Category: Logic, Symbolic and mathematical
Page: N.A
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The Argument of Mathematics

Author: Andrew Aberdein,Ian J Dove
Publisher: Springer Science & Business Media
ISBN: 9400765347
Category: Philosophy
Page: 393
View: 8012

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Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. The book begins by first challenging the assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​

A Philosophical Guide to Conditionals

Author: Jonathan Francis Bennett,Jonathan Bennett
Publisher: Oxford University Press on Demand
ISBN: 0199258872
Category: Philosophy
Page: 387
View: 7450

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The author, one of the world's leading authorities on the subject of conditional sentences, distils many years' work and teaching into 'A Philosophical Guide to Conditionals', an authoritative treatment of the subject.

The Research Imagination

An Introduction to Qualitative and Quantitative Methods
Author: Paul S. Gray,John B. Williamson,David A. Karp,John R. Dalphin
Publisher: Cambridge University Press
ISBN: 1139467026
Category: Social Science
Page: N.A
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The idea that science is a blueprint for research, and imagination gives research its life and purpose inspired this comprehensive explanation of research methodology. The authors' decades of experience have revealed that research is a craft requiring judgment and creativity, not simply memorization and application of the rules of science. Whether one is conducting an intimate one-on-one interview or a large-scale examination of an entire society, human imagination and scientific principles of inquiry go hand in hand. To that end, this book emphasizes scientific method, but also acknowledges its critics. It covers a wide variety of data-collection techniques, but presents them as reinforcing rather than competing with one another, thus striking a balance between qualitative and quantitative methods. It is designed for students and instructors who want a comprehensive treatment of a variety of research techniques with special emphasis on qualitative approaches.

Logical Reasoning

Author: Bradley Harris Dowden
Publisher: Bradley Dowden
ISBN: 9780534176884
Category: Philosophy
Page: 404
View: 4215

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This book is designed to engage students' interest and promote their writing abilities while teaching them to think critically and creatively. Dowden takes an activist stance on critical thinking, asking students to create and revise arguments rather than simply recognizing and criticizing them. His book emphasizes inductive reasoning and the analysis of individual claims in the beginning, leaving deductive arguments for consideration later in the course.

Essays on Paradoxes

Author: Terence Horgan
Publisher: Oxford University Press
ISBN: 019985842X
Page: 328
View: 8766

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This volume brings together many of Terence Horgan's essays on paradoxes: Newcomb's problem, the Monty Hall problem, the two-envelope paradox, the sorites paradox, and the Sleeping Beauty problem. Newcomb's problem arises because the ordinary concept of practical rationality constitutively includes normative standards that can sometimes come into direct conflict with one another. The Monty Hall problem reveals that sometimes the higher-order fact of one's having reliably received pertinent new first-order information constitutes stronger pertinent new information than does the new first-order information itself. The two-envelope paradox reveals that epistemic-probability contexts are weakly hyper-intensional; that therefore, non-zero epistemic probabilities sometimes accrue to epistemic possibilities that are not metaphysical possibilities; that therefore, the available acts in a given decision problem sometimes can simultaneously possess several different kinds of non-standard expected utility that rank the acts incompatibly. The sorites paradox reveals that a certain kind of logical incoherence is inherent to vagueness, and that therefore, ontological vagueness is impossible. The Sleeping Beauty problem reveals that some questions of probability are properly answered using a generalized variant of standard conditionalization that is applicable to essentially indexical self-locational possibilities, and deploys "preliminary" probabilities of such possibilities that are not prior probabilities. The volume also includes three new essays: one on Newcomb's problem, one on the Sleeping Beauty problem, and an essay on epistemic probability that articulates and motivates a number of novel claims about epistemic probability that Horgan has come to espouse in the course of his writings on paradoxes. A common theme unifying these essays is that philosophically interesting paradoxes typically resist either easy solutions or solutions that are formally/mathematically highly technical. Another unifying theme is that such paradoxes often have deep-sometimes disturbing-philosophical morals.

An Imaginary Tale

The Story of √-1
Author: Paul J. Nahin
Publisher: Princeton University Press
ISBN: 9781400833894
Category: Mathematics
Page: 296
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Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics. Some images inside the book are unavailable due to digital copyright restrictions.