Language, Proof and Logic

Language, Proof and Logic Language, Proof and Logic Second Edition Dave Barker-Plummer, Jon Barwise and John Etchemendy in collaboration with Albert Liu, Michael Murray and Emma Pease Copyright © 1999, 2000, 2002, 2003, 2007, 2008, 2011 CSLI Publications Center for the Study of Language and Information Leland Stanford Junior University First Edition 1999 Second Edition 2011 Printed in the United States 15 14 13 12 11 1 2 3 4 5 Library of Congress Cataloging-in-Publication Data Barker-Plummer, Dave. Language, proof, and logic. { 2nd ed. / Dave Barker-Plummer, Jon Barwise, and John Etchemendy in collaboration with Albert Liu, Michael Murray, and Emma Pease. p. cm. { Rev. ed. of: Language, proof, and logic / Jon Barwise & John Etchemendy. Includes index. ISBN 978-1-57586-632-1 (pbk. : alk. paper) 1. Logic. I. Barwise, Jon. II. Etchemendy, John, 1952- III. Barwise, Jon. Language, proof, and logic. IV. Title. BC71.B25 2011 160{dc23 2011019703 CIP 1 The acid-free paper used in this book meets the minimum requirements of the American National Standard for Information Sciences|Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984. Acknowledgements Our primary debt of gratitude goes to our main collaborators on this project: Gerry Allwein and Albert Liu. They have worked with us in designing the entire package, developing and implementing the software, and teaching from and refining the text. Without their intelligence, dedication, and hard work, LPL would neither exist nor have most of its other good properties. In addition to the five of us, many people have contributed directly and in- directly to the creation of the package. First, over two dozen programmers have worked on predecessors of the software included with the package, both earlier versions of Tarski's World and the program Hyperproof, some of whose code has been incorporated into Fitch. We want especially to mention Christopher Fuselier, Mark Greaves, Mike Lenz, Eric Ly, and Rick Wong, whose outstand- ing contributions to the earlier programs provided the foundation of the new software. Second, we thank several people who have helped with the development of the new software in essential ways: Rick Sanders, Rachel Farber, Jon Russell Barwise, Alex Lau, Brad Dolin, Thomas Robertson, Larry Lemmon, and Daniel Chai. Their contributions have improved the package in a host of ways. Prerelease versions of LPL have been tested at several colleges and univer- sities. In addition, other colleagues have provided excellent advice that we have tried to incorporate into the final package. We thank Selmer Bringsjord, Rens- selaer Polytechnic Institute; Tom Burke, University of South Carolina; Robin Cooper, Gothenburg University; James Derden, Humboldt State University; Josh Dever, SUNY Albany; Avrom Faderman, University of Rochester; James Garson, University of Houston; Christopher Gauker, University of Cincinnati; Ted Hodgson, Montana State University; John Justice, Randolph-Macon Wom- en's College; Ralph Kennedy, Wake Forest University; Michael O'Rourke, University of Idaho; Greg Ray, University of Florida; Cindy Stern, Califor- nia State University, Northridge; Richard Tieszen, San Jose State University; Saul Traiger, Occidental College; and Lyle Zynda, Indiana University at South Bend. We are particularly grateful to John Justice, Ralph Kennedy, and their students (as well as the students at Stanford and Indiana University), for their patience with early versions of the software and for their extensive comments and suggestions. We would also like to thank the many instructors and students who have offered useful feedback since the initial publication of LPL. We would also like to thank Stanford's Center for the Study of Language v vi / Acknowledgements and Information and Indiana University's College of Arts and Sciences for their financial support of the project. Finally, we are grateful to our publisher, Dikran Karagueuzian and his team at CSLI Publications, for their skill and enthusiasm about LPL, and to Lauri Kanerva for his dedication and skill in the preparation of the final manuscript. Acknowledgements for the Second Edition One part of developing courseware packages for publication is the continual challenge of maintaining software in the face of developments in the commer- cial computer market. Since the initial publication of LPL, many operating system versions have come and gone, each requiring modifications, small and large, to the applications that are part of the package. With the publication of the second edition of LPL we are releasing the 3.x series of the Fitch, Boole and Submit applications, and the 7.x series of the Tarski's World applica- tion. While retaining the same functionality, these are essentially complete rewrites of the applications that appeared with the initial publication of LPL. The Grade Grinder too has undergone many updates and changes | it is now running in its third major incarnation, having been ported from the Solaris operating system to Mac OS X along the way. Three generations of the LPL web site have also come and gone. All of this maintenance and development requires the talent of skilled programmers, and we have been fortunate in the people that have contributed to the project. Albert Liu contributed to many aspects of the software over a long period. Michael Murray's work is evident in all aspects of the desktop applications, and is primarily responsible for the web site as it currently ap- pears. Emma Pease has served as the project's system administrator for the past two years. We have modernized the appearance of the LPL applications with this new release, and the graphical design work of Aaron Kalb is evident in every aspect of the software including the web site design, which was realized by Deonne Castaneda. Nik Swoboda of the Universidad Politécnica de Madrid created and maintains the Linux ports of the LPL software. Leslie Rogers served as the lead QA engineer for the new applications. The LPL package would not be what it is without the dedication and hard work of all of these people, and many others, and we thank them all. Dave Barker-Plummer frequently teaches using LPL in Stanford's Philos- ophy 150 class. He uses this as an excuse to try out new teaching material and sometimes beta versions of the software. Dave would like to thank all of the students who have enrolled in those classes over the years for their patience and good humor. Acknowledgements Acknowledgements for the Second Edition / vii We have benefitted greatly from the feedback of the many instructors who have adopted the LPL package in their teaching. We would particularly like to thank Richard Zach, University of Calgary; S. Marc Cohen, University of Washington and Bram van Heuveln, Rensselaer Polytechnic Institute for much appreciated comments on the package. Bram suggested to us the addition of the \Add Support Steps" feature of the new Fitch program. Richard Johns of the University of British Columbia suggested the new \goggles" features which are also included in that program. The Openproof project continues to benefit from generous funding from Stanford University and from its home in the intellectually stimulating en- vironment of Stanford's Center for the Study of Language and Information (CSLI). As always, we are grateful to our publisher, Dikran Karagueuzian, and his team at CSLI Publications for their continued enthusiasm for LPL. Acknowledgements for the Second Edition viii / Acknowledgements Acknowledgements Contents Acknowledgements v Introduction 1 The special role of logic in rational inquiry .............. 1 Why learn an artificial language? .................... 2 Consequence and proof .......................... 4 Instructions about homework exercises (essential!) .......... 5 To the instructor ............................. 11 Web address ............................... 16 I Propositional Logic 1 Atomic Sentences 19 1.1 Individual constants ........................ 19 1.2 Predicate symbols ......................... 20 1.3 Atomic sentences .......................... 23 1.4 General first-order languages ................... 28 1.5 Function symbols (optional) ................... 31 1.6 The first-order language of set theory (optional) ........ 37 1.7 The first-order language of arithmetic (optional) ........ 38 1.8 Alternative notation (optional) .................. 40 2 The Logic of Atomic Sentences 41 2.1 Valid and sound arguments .................... 41 2.2 Methods of proof .......................... 46 2.3 Formal proofs ............................ 54 2.4 Constructing proofs in Fitch ................... 58 2.5 Demonstrating nonconsequence .................. 63 2.6 Alternative notation (optional) .................. 66 3 The Boolean Connectives 67 3.1 Negation symbol: : ........................ 68 3.2 Conjunction symbol: ^ ...................... 71 3.3 Disjunction symbol: _ ....................... 74 3.4 Remarks about the game ..................... 77 ix x / Contents 3.5 Ambiguity and parentheses .................... 79 3.6 Equivalent ways of saying things ................. 82 3.7 Translation ............................. 84 3.8 Alternative notation (optional) .................. 90 4 The Logic of Boolean Connectives 93 4.1 Tautologies and logical truth ................... 94 4.2 Logical and tautological equivalence ............... 106 4.3 Logical and tautological consequence ............... 110 4.4 Tautological consequence in Fitch ................ 114 4.5 Pushing negation around (optional) ............... 118 4.6 Conjunctive and

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