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Language, Proof and Logic

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Language, Proof and Logic LANGUAGE, PROOF AND LOGIC LANGUAGE, PROOF AND LOGIC JON BARWISE & JOHN ETCHEMENDY In collaboration with Gerard Allwein Dave Barker-Plummer Albert Liu Copyright c 1999, 2000, 2002, 2003, 2007, 2008 CSLI Publications Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States 12 11 10 09 08 14 13 12 11 10 Library of Congress Cataloging-in-Publication Data Barwise, Jon. Language, proof and logic / Jon Barwise and John Etchemendy ; in collaboration with Gerard Allwein, Dave Barker-Plummer, and Albert Liu. p. cm. ISBN-13: 978-1-57586-374-0 (pbk. : alk. paper) ISBN-10: 1-57586-374-X (pbk. : alk. paper) 1. Logic. I. Etchemendy, John, 1952– II. Allwein, Gerard, 1956– III. Barker-Plummer, Dave. IV. Liu, Albert, 1966– V. Title. BC61.B38 1999 160—dc21 99-041113 ∞ 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 three main collaborators on this project: Gerry Allwein, Dave Barker-Plummer, 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 intelli- gence, 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 develop- ment 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 uni- versities. In addition, other colleagues have provided excellent advice that we have tried to incorporate into the final package. We thank Selmer Bringsjord, Renssalaer Polytechnic Institute; Tom Burke, University of South Carolina; Robin Cooper, Gothenburg University; James Derden, Humboldt State Uni- versity; Josh Dever, SUNY Albany; Avrom Faderman, University of Rochester; James Garson, University of Houston; Christopher Gauker, University of Cin- cinnati; Ted Hodgson, Montana State University; John Justice, Randolph- Macon Women’s College; Ralph Kennedy, Wake Forest University; Michael O’Rourke, University of Idaho; Greg Ray, University of Florida; Cindy Stern, California State University, Northridge; Richard Tieszen, San Jose State Uni- versity; 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 Uni- versity), 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 v vi / Acknowledgements publication of LPL. We would also like to thank Stanford’s Center for the Study of Language 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 Contents Acknowledgements v Introduction 1 Thespecialroleoflogicinrationalinquiry.............. 1 Whylearnanartificiallanguage?.................... 2 Consequenceandproof.......................... 4 Instructions about homework exercises (essential!) .......... 5 Totheinstructor............................. 10 Webaddress............................... 15 I Propositional Logic 1 Atomic Sentences 19 1.1Individualconstants........................ 19 1.2Predicatesymbols......................... 20 1.3Atomicsentences.......................... 23 1.4Generalfirst-orderlanguages................... 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 2.1Validandsoundarguments.................... 41 2.2Methodsofproof.......................... 46 2.3Formalproofs............................ 54 2.4ConstructingproofsinFitch.................... 58 2.5Demonstratingnonconsequence.................. 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 3.4Remarksaboutthegame..................... 77 vii viii / Contents 3.5Ambiguityandparentheses.................... 79 3.6Equivalentwaysofsayingthings................. 82 3.7Translation............................. 84 3.8 Alternative notation (optional) .................. 89 4 The Logic of Boolean Connectives 93 4.1Tautologiesandlogicaltruth................... 94 4.2Logicalandtautologicalequivalence...............106 4.3Logicalandtautologicalconsequence...............110 4.4TautologicalconsequenceinFitch.................114 4.5 Pushing negation around (optional) ...............117 4.6 Conjunctive and disjunctive normal forms (optional)......121 5 Methods of Proof for Boolean Logic 127 5.1Validinferencesteps........................128 5.2Proofbycases............................131 5.3Indirectproof:proofbycontradiction...............136 5.4 Arguments with inconsistent premises (optional) ........140 6 Formal Proofs and Boolean Logic 142 6.1Conjunctionrules..........................143 6.2Disjunctionrules..........................148 6.3Negationrules...........................154 6.4 The proper use of subproofs . 163 6.5Strategyandtactics........................167 6.6 Proofs without premises (optional)................173 7 Conditionals 176 7.1 Material conditional symbol: → ..................178 7.2 Biconditional symbol: ↔ ......................181 7.3Conversationalimplicature....................187 7.4 Truth-functional completeness (optional).............190 7.5 Alternative notation (optional) ..................196 8 The Logic of Conditionals 198 8.1Informalmethodsofproof.....................198 8.2 Formal rules of proof for → and ↔ ................206 8.3 Soundness and completeness (optional)..............214 8.4Validarguments:somereviewexercises..............222 Contents Contents /ix II Quantifiers 9 Introduction to Quantification 227 9.1Variablesandatomicwffs.....................228 9.2 The quantifier symbols: ∀, ∃ ....................230 9.3Wffsandsentences.........................231 9.4Semanticsforthequantifiers....................234 9.5ThefourAristotelianforms....................239 9.6Translatingcomplexnounphrases................243 9.7 Quantifiers and function symbols (optional)...........251 9.8 Alternative notation (optional) ..................255 10 The Logic of Quantifiers 257 10.1Tautologiesandquantification...................257 10.2First-ordervalidityandconsequence...............266 10.3 First-order equivalence and DeMorgan’s laws . 275 10.4 Other quantifier equivalences (optional) .............280 10.5 The axiomatic method (optional).................283 11 Multiple Quantifiers 289 11.1Multipleusesofasinglequantifier................289 11.2Mixedquantifiers..........................293 11.3Thestep-by-stepmethodoftranslation..............298 11.4 Paraphrasing English . 300 11.5Ambiguityandcontextsensitivity................304 11.6 Translations using function symbols (optional) .........308 11.7 Prenex form (optional).......................311 11.8Someextratranslationproblems.................315 12 Methods of Proof for Quantifiers 319 12.1Validquantifiersteps........................319 12.2Themethodofexistentialinstantiation..............322 12.3 The method of general conditional proof . 323 12.4Proofsinvolvingmixedquantifiers................329 12.5 Axiomatizing shape (optional) ..................338 13 Formal Proofs and Quantifiers 342 13.1Universalquantifierrules.....................342 13.2Existentialquantifierrules.....................347 13.3Strategyandtactics........................352 13.4 Soundness and completeness (optional)..............361 Contents x/Contents 13.5 Some review exercises (optional) .................361 14 More about Quantification (optional)364 14.1Numericalquantification......................366 14.2Provingnumericalclaims.....................374 14.3 The, both,andneither .......................379 14.4 Adding other determiners to fol .................383 14.5 The logic of generalized quantification . 389 14.6 Other expressive limitations of first-order logic . 397 III Applications and Metatheory 15 First-order Set Theory 405 15.1Naivesettheory..........................406 15.2Singletons,theemptyset,subsets.................412
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