forallx in Lurch An Introduction to Formal Logic in Lurch P.D. Magnus University at Albany, State University of New York Nathan Carter Bentley University http://web.bentley.edu/empl/c/ncarter/faxil, version 1.1 [171002] This book is offered under a Creative Commons license. (Attribution-ShareAlike 3.0) In his original text, P.D. Magnus thanked the people who made his project possible. Notable among these are Cristyn Magnus, who read many early drafts; Aaron Schiller, who was an early adopter and provided considerable, helpful feedback; and Bin Kang, Craig Erb, Nathan Carter, Wes McMichael, Selva Samuel, Dave Krueger, Brandon Lee, and the students of Introduction to Logic, who detected various errors in previous versions of the book. Nathan Carter thanks P.D. for creating the text in the first place and Ken Monks for being the other half of the Lurch project. Thanks also go to Bentley University for supporting me for a summer of working on this textbook, and to all my logic students, who help shape my teaching and the development of Lurch. Original copyright notice: c 2005{2017 by P.D. Magnus. Some rights reserved. You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (a) Attribution. You must give the original author credit. (b) Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under a license identical to this one. | For any reuse or distribution, you must make clear to others the license terms of this work. Any of these conditions can be waived if you get permission from the copyright holder. Your fair use and other rights are in no way affected by the above. | This is a human-readable summary of the full license, which is available on-line at http://creativecommons.org/licenses/by-sa/3.0/ This version: c 2013{2017 by Nathan Carter. Some rights reserved; the original Creative Common license given above is preserved. 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. This copy of forallx in Lurch is current as of October 2, 2017. The most recent version is available on-line at http://web.bentley.edu/empl/c/ncarter/faxil Contents Preface . .6 1 Getting started with Lurch 7 1.1 What is Lurch?............................................7 1.2 Installing the software . .7 1.3 A math word processor . .9 1.4 Beyond word processing . 11 2 What is logic? 12 2.1 Arguments . 12 2.2 Sentences . 13 2.3 Two ways that arguments can go wrong . 14 2.4 Deductive validity . 15 2.5 Other logical notions . 16 2.6 Formal languages . 18 2.7 Typing arguments in Lurch ..................................... 20 Practice Exercises . 20 3 Sentential logic 23 3.1 Sentence letters . 23 3.2 Connectives . 24 3.3 Other symbolization . 31 3.4 Sentences of SL . 32 3.5 How Lurch can help . 35 Practice Exercises . 37 4 Truth tables 40 4.1 Truth-functional connectives . 40 4.2 Complete truth tables . 40 4.3 Using truth tables . 43 4.4 Partial truth tables . 44 4.5 Lurch and truth tables . 46 Practice Exercises . 48 5 Proofs in SL 51 5.1 Basic rules for SL . 52 5.2 Derived rules . 59 5.3 Derived rules of logical equivalence . 61 Practice Exercises . 61 6 Proofs in Lurch 63 6.1 Having Lurch check your work . 63 6.2 Creating your own proofs . 70 6.3 Proof-theoretic concepts . 71 3 4 CONTENTS 6.4 Proofs and truth tables . 71 Practice Exercises . 73 7 Quantified logic 75 7.1 From sentences to predicates . 75 7.2 Building blocks of QL . 76 7.3 Quantifiers . 79 7.4 Translating to QL . 82 7.5 Sentences of QL . 90 7.6 Identity . 93 Practice Exercises . 97 8 Formal semantics 103 8.1 Semantics for SL . 104 8.2 Interpretations and models in QL . 107 8.3 Semantics for identity . 110 8.4 Working with models . 111 8.5 Truth in QL . 114 Practice Exercises . 118 9 Proofs in QL 122 9.1 Rules for quantifiers . 122 9.2 Rules for identity . 126 9.3 QL Proofs in Lurch .......................................... 128 9.4 Proof-theoretic concepts . 130 9.5 Soundness and completeness . 131 Practice Exercises . 132 10 Real numbers 136 10.1 An important transition . 136 10.2 Shortcuts about statements . 137 10.3 Axioms . 139 10.4 Theorems . 139 10.5 Shortcuts in Lurch .......................................... 144 Practice Exercises . 148 11 Mathematical Induction 150 11.1 Why introduce induction? . 150 11.2 One new rule . 150 11.3 A first induction theorem . 152 11.4 Completing the example with new privileges . 153 Practice Exercises . 155 12 Calculus 157 12.1 Absolute Value . 157 12.2 Working with Limits . 158 12.3 The foundations of calculus . 162 Practice Exercises . 163 13 Abstract Mathematics 166 13.1 Sets . 166 13.2 Proofs about Sets . 170 13.3 Functions . 171 Practice Exercises . ..