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An Open Introduction to Logic

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An Open Introduction to Logic For All X The Lorain County Remix P.D. Magnus University at Albany, State University of New York and Cathal Woods Virginia Wesleyan University and J. Robert Loftis Lorain County Community College Licensed under a Creative Commons license. (Attribution-NonComercial-ShareAlike 4.0 International ) https://creativecommons.org/licenses/by-nc-sa/4.0/ This is version 0.1 of An Open Introduction to Logic. It is current as of December 22, 2017. © 2005{2017 by Cathal Woods, P.D. Magnus, and J. Robert Loftis. Some rights reserved. Licensed under a Creative Commons license. (Attribution-NonComercial-ShareAlike 4.0 International ) https://creativecommons.org/licenses/by-nc-sa/4.0/ This book incorporates material from An Introduction to Reasoning by Cathal Woods, available at sites.google.com/site/anintroductiontoreasoning/ and For All X by P.D. Magnus (version 1.27 [090604]), available at www.fecundity.com/logic. Introduction to Reasoning © 2007{2014 by Cathal Woods. Some rights reserved. Licensed under a Creative Commons license: Attribution-NonCommercial-ShareAlike 3.0 Unported. http://creativecommons.org/licenses/by-nc-sa/3.0/ For All X © 2005{2010 by P.D. Magnus. Some rights reserved. Licensed under a Creative Commons license: Attribution ShareAlike http://creativecommons.org/licenses/by-sa/3.0/ J. Robert Loftis compiled this edition and wrote original material for it. He takes full responsibility for any mistakes remaining in this version of the text. Typesetting was carried out entirely in LATEX2". The style for typesetting proofs is based on fitch.sty (v0.4) by Peter Selinger, University of Ottawa. \When you come to any passage you don't understand, read it again: if you still don't understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired. In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy." { Charles Dodgson (Lewis Carroll) Symbolic Logic (1896) iii Contents About this Book ix Acknowledgments xi Part I Basic Concepts 1 What Is Logic? 3 1.1 Introduction......................................... 3 1.2 Statement, Argument, Premise, Conclusion ....................... 6 Practice Exercises ..................................... 13 1.3 Arguments and Nonarguments .............................. 15 Practice Exercises ..................................... 18 1.4 Arguments and Explanations ............................... 19 Practice Exercises ..................................... 24 2 The Basics of Evaluating Argument 29 2.1 Two Ways an Argument Can Go Wrong......................... 29 2.2 Valid, Sound ........................................ 30 Practice Exercises ..................................... 35 2.3 Strong, Cogent, Deductive, Inductive........................... 37 Practice Exercises ..................................... 44 3 What is Formal Logic? 49 3.1 Formal as in Concerned with the Form of Things.................... 49 3.2 Formal as in Strictly Following Rules........................... 51 3.3 More Logical Notions for Formal Logic.......................... 53 Practice Exercises ..................................... 57 Part II Categorical Logic 4 Categorical Statements 65 4.1 Quantified Categorical Statements ............................ 65 Practice Exercises ..................................... 66 4.2 Quantity, Quality, Distribution, and Venn Diagrams .................. 68 iv Practice Exercises ..................................... 73 4.3 Transforming English into Logically Structured English ................ 75 Practice Exercises ..................................... 80 4.4 Conversion, Obversion, and Contraposition ....................... 83 Practice Exercises ..................................... 93 4.5 The Traditional Square of Opposition .......................... 96 Practice Exercises .....................................101 4.6 Existential Import and the Modern Square of Opposition . 105 Practice Exercises .....................................108 5 Categorical Syllogisms 113 5.1 Standard Form, Mood, and Figure ............................113 Practice Exercises .....................................116 5.2 Testing Validity.......................................119 Practice Exercises .....................................129 5.3 Existential Import and Conditionally Valid Forms ...................132 Practice Exercises .....................................137 5.4 Rules and Fallacies.....................................139 Practice Exercises .....................................146 Part III Sentential Logic 6 Sentential Logic 151 6.1 Sentence Letters ......................................151 6.2 Sentential Connectives...................................153 6.3 More Complicated Translations..............................161 6.4 Recursive Syntax for SL..................................167 Practice Exercises .....................................171 7 Truth Tables 177 7.1 Basic Concepts.......................................177 7.2 Complete Truth Tables...................................178 Practice Exercises .....................................182 7.3 Using Truth Tables.....................................184 Practice Exercises .....................................187 7.4 Partial Truth Tables....................................190 Practice Exercises .....................................193 7.5 Expressive Completeness..................................196 Practice Exercises .....................................197 8 Proofs in Sentential Logic 199 8.1 Substitution Instances and Proofs.............................199 Practice Exercises .....................................203 8.2 Basic Rules for Sentential Logic..............................207 Practice Exercises .....................................213 8.3 Conditional Proof......................................220 v Practice Exercises .....................................225 8.4 Indirect Proof........................................227 Practice Exercises .....................................229 8.5 Tautologies and Equivalences ...............................232 Practice Exercises .....................................234 8.6 Derived Rules........................................235 Practice Exercises .....................................238 8.7 Rules of Replacement ...................................239 Practice Exercises .....................................241 8.8 Proof Strategy .......................................241 Practice Exercises .....................................242 8.9 Soundness and completeness................................242 Practice Exercises .....................................246 Part IV Quantificational Logic 9 Quantified Logic 251 9.1 From Sentences to Predicates...............................251 9.2 Building Blocks of Quantified Logic............................253 Practice Exercises .....................................256 9.3 Quantifiers .........................................257 9.4 Translating to Quantified Logic..............................261 Practice Exercises .....................................265 9.5 Recursive Syntax for QL..................................267 Practice Exercises .....................................270 9.6 Tricky Translations.....................................270 Practice Exercises .....................................275 9.7 Identity ...........................................278 Practice Exercises .....................................282 10 Semantics for Quantified Logic 285 10.1 Creating models in Quantified Logic ...........................285 Practice Exercises .....................................289 10.2 Working with Models ...................................291 Practice Exercises .....................................295 11 Proofs in Quantified Logic 297 11.1 Rules for Quantifiers....................................297 Practice Exercises .....................................301 11.2 Rules for Identity......................................305 Practice Exercises .....................................306 A Other Symbolic Notation 309 B Bibliography 313 vi C Glossary 317 D Quick Reference 327 vii viii About this Book This book was created by combining two previous books on logic and critical thinking, both made available under a Creative Commons license, and then adding some material so that the coverage matched that of commonly used logic textbooks. P.D. Magnus' For All X(2008) formed the basis of PartB: Formal Logic. I began using For All X in my own logic classes in 2009, but I quickly realized I needed to make changes to make it appropriate for the community college students I was teaching. In 2010 I began developing For All X : The Lorain County Remix and using it in my classes. The main change I made was to separate the discussions of sentential and quantificational logic and to add exercises. It is this remixed version that became the basis for PartB: Formal Logic complete version of this text. Similarly, Part ??: Critical Thinking grew out of Cathal Woods' Introduction to Reasoning. In the Spring of 2011, I began to use an early version of this text (Woods and Roche 2011) in my critical thinking courses. I kept up with the updates and changes to the text until the release of Woods 2014, all the while gradually merging the material with the work in For All X. After that point, my version forks from Woods's. On May 20, 2016, I posted the combined textbook to Github and all subsequent changes have been tracked there: https://github.com/rob-helpy-chalk/openintroduction
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