Games for Your Mind: The History and Future of Logic Puzzles

by Jason Rosenhouse

Overview

A lively and engaging look at logic puzzles and their role in mathematics, philosophy, and recreation

Logic puzzles were first introduced to the public by Lewis Carroll in the late nineteenth century and have been popular ever since. Games like Sudoku and Mastermind are fun and engrossing recreational activities, but they also share deep foundations in mathematical logic and are worthy of serious intellectual inquiry. Games for Your Mind explores the history and future of logic puzzles while enabling you to test your skill against a variety of puzzles yourself.

In this informative and entertaining book, Jason Rosenhouse begins by introducing readers to logic and logic puzzles and goes on to reveal the rich history of these puzzles. He shows how Carroll's puzzles presented Aristotelian logic as a game for children, yet also informed his scholarly work on logic. He reveals how another pioneer of logic puzzles, Raymond Smullyan, drew on classic puzzles about liars and truthtellers to illustrate Kurt Gödel's theorems and illuminate profound questions in mathematical logic. Rosenhouse then presents a new vision for the future of logic puzzles based on nonclassical logic, which is used today in computer science and automated reasoning to manipulate large and sometimes contradictory sets of data.

Featuring a wealth of sample puzzles ranging from simple to extremely challenging, this lively and engaging book brings together many of the most ingenious puzzles ever devised, including the "Hardest Logic Puzzle Ever," metapuzzles, paradoxes, and the logic puzzles in detective stories.

Product Details

ISBN-13:	9780691174075
Publisher:	Princeton University Press
Publication date:	11/24/2020
Pages:	352
Product dimensions:	6.50(w) x 9.40(h) x 1.20(d)

About the Author

Jason Rosenhouse is professor of mathematics at James Madison University. He is the author of The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser and Among the Creationists: Dispatches from the Anti-Evolutionist Front Line. He is the coauthor (with Laura Taalman) of Taking Sudoku Seriously: The Math behind the World's Most Popular Pencil Puzzle and the coeditor (with Jennifer Beineke) of The Mathematics of Various Entertaining Subjects (Vols. 1–3) (Princeton).

Table of Contents

Preface xi

I The Pain and Pleasure of Logic 1

1 Is Logic Boring and Pointless? 3

1.1 Logic in Practice, Logic in Theory 3

1.2 Enter the Philosophers 8

1.3 Notes and Further Reading 14

2 Logic Just for Fun 15

2.1 Sudoku and Mastermind 15

2.2 Some Classic Logic Puzzles 18

2.3 Puzzles in Propositional Logic 21

2.4 Notes and Further Reading 22

2.5 Solutions 23

II Lewis Carroll and Aristotelian Logic 29

3 Aristotle's Syllogistic 31

3.1 The Beginning of Formal Logic 31

3.2 Proposition Jargon 34

3.3 Operations on Propositions 37

3.4 Figures and Moods 41

3.5 Aristotle's Proof Methods 45

3.6 Notes and Further Reading 48

4 The Empuzzlement of Aristotelian Logic 50

4.1 Diagrams for Propositions 50

4.2 Playing the Game 53

4.3 A Closer Look at Placing Counters 56

4.4 One More Example 59

4.5 Are We Having Fun Yet? 61

4.6 Puzzles for Solving 64

4.7 Solutions 65

5 Sorites Puzzles 68

5.1 A Quadriliteral Diagram? 69

5.2 Notation and Formulas 72

5.3 The Formalization in Action 76

5.4 The Method of Underscoring 78

5.5 The Method of Trees 81

5.6 Puzzles for Solving 88

5.7 Notes and Further Reading 90

5.8 Solutions 90

6 Carroll's Contributions to Mind 93

6.1 The Barbershop Puzzle 93

6.2 Achilles and the Tortoise 97

6.3 Scholarly Responses to Carroll's Regress 99

6.4 Does the Tortoise Have a Point? 107

6.5 Notes and Further Reading 110

III Raymond Smullyan and Mathematical Logic 113

7 Liars and Truthtellers 115

7.1 Propositional Logic 115

7.2 A Knight/Knave Primer 120

7.3 A Selection of Knight/Knave Puzzles 122

7.4 Sane or Mad? 123

7.5 The Lady or the Tiger? 125

7.6 Some Unusual Knights and Knaves 127

7.7 Two Elaborate Puzzles 128

7.8 Notes and Further Reading 129

7.9 Solutions 131

8 From Aristotle to Russell 137

8.1 Aristotle's Organon 138

8.2 Medieval Logic 140

8.3 Mill's A System of Logic 144

8.4 Boole and Venn 146

8.5 Russell's The Principles of Mathematics 151

8.6 Notes and Further Reading 153

9 Formal Systems in Life and Math 154

9.1 What Is a Formal System? 154

9.2 What Can Your Formal Language Say? 159

9.3 Formalizations of Arithmetic 161

9.4 Notes and Further Reading 163

10 The Empuzzlement of Gödel's Theorems 164

10.1 Established Knights and Knaves 164

10.2 A Sentence That Is True but Unprovable 166

10.3 Establishment, Revisited 169

10.4 A Gödelian Machine 172

10.5 Gödel's Second Incompleteness Theorem 174

10.6 Puzzles for Solving 177

10.7 Notes and Further Reading 180

10.8 Solutions 181

11 Question Puzzles 184

11.1 Three Warm-Ups 184

11.2 The Power of Indexical Questions 185

11.3 The Heaven/Hell Puzzle 186

11.4 The Nelson Goodman Principle 189

11.5 Generalized Nelson Goodman Principles 191

11.6 Coercive Logic 194

11.7 Smullyan as a Writer 195

11.8 Solutions 196

IV Puzzles Based on Nonclassical Logics 199

12 Should "Logics" Be a Word? 201

12.1 Logical Pluralism? 202

12.2 Is Classical Logic Correct? 206

12.3 Applications of Nonclassical Logic 208

12.4 Notes and Further Reading 210

13 Many-Valued Knights and Knaves 212

13.1 The Transitional Phase 212

13.2 The Three-Valued Island 214

13.3 The Fuzzy Island 220

13.4 Modus Ponens and Sorites 225

13.5 Puzzles for Solving 228

13.6 Solutions 231

V Miscellaneous Topics 237

14 The Saga of the Hardest Logic Puzzle Ever 239

14.1 Boolos Introduces the Puzzle 239

14.2 Is There a Simpler Solution? 245

14.3 Trivializing the Hardest Puzzle Ever 250

14.4 Are Three Questions Necessary? 253

14.5 Two Questions When Random Is Really Random 255

14.6 What If Random Can Remain Silent? 259

14.7 Notes and Further Reading 265

15 Metapuzzles 266

15.1 A Warm-Up Puzzle 266

15.2 The Playful Children and Caliban's Will 267

15.3 Knight/Knave Metapuzzles 269

15.4 Solutions 270

16 Paradoxes 274

16.1 What Is a Paradox? 275

16.2 Paradoxes of Predication 276

16.3 The Paradox of the Preface 279

16.4 The Liar 282

16.5 Miscellaneous Paradoxes 289

16.6 Notes and Further Reading 290

17 A Guide to Some Literary Logic Puzzles 292

17.1 The Nine Mile Walk 293

17.2 The Early Days of "Logic Fiction" 295

17.3 A Gallery of Eccentric Detectives 300

17.4 The Anti-Logicians 303

17.5 Carr and Queen 305

17.6 The Thinking Machine 307

Glossary 311

References 319

Index 327

What People are Saying About This

From the Publisher

"Logic is one of the most important subjects in mathematics and Jason Rosenhouse's welcoming style makes it easy to understand. A fantastic achievement!"—James Grime, presenter of Numberphile

"Games for Your Mind combines the whimsy of Lewis Carroll with the cleverness of Raymond Smullyan along with very accessible mathematics, delivered with clarity and humor by award-winning author Jason Rosenhouse. Highly recommended for anyone interested in learning about logic in an engaging way."—Arthur Benjamin, Harvey Mudd College

"Rosenhouse deftly traces the entire history of logic from Aristotle to Raymond Smullyan. For readers who quail at the thought of logic or math yet love Sudoku, puzzles, paradoxes, and the deductive world of Sherlock Holmes, this book delivers an accessible introduction while exploring logic's perils and joys in greater depth."—Mark Burstein, president emeritus, Lewis Carroll Society of North America

"What starts out as a garden tour of classical logic puzzles takes a turn down the rabbit hole into nonclassical logic, paradoxes, and metapuzzles, with Rosenhouse as an expert and enjoyable guide."—Laura Taalman, coauthor of Taking Sudoku Seriously: The Math behind the World's Most Popular Pencil Puzzle

"A uniquely personal whirlwind tour of logic ancient and modern, spiced with puzzles from two of the greats, Lewis Carroll and Raymond Smullyan. Almost every reader will find material he or she would never otherwise have encountered."—Peter Winkler, author of Mathematical Puzzles: A Connoisseur's Collection and Mathematical Mind-Benders

"This book is a pleasure. All aspects of logic are explained in a clear and engaging manner, with many humorous touches, and Rosenhouse handles the philosophical aspects of logic just as skillfully as the mathematical ones. Games for Your Mind is a great introduction to logic for anyone who likes puzzles."—John Stillwell, author of Reverse Mathematics: Proofs from the Inside Out

"Rosenhouse presents a rich variety of logic puzzles while introducing readers to the history of logic. Games for Your Mind is an engaging and accessible book that will appeal to people attracted to logic and can open a gate to more serious study as well."—Michał Walicki, author of Introduction to Mathematical Logic

Editorial Reviews

"Fascinating. . . . Part philosophy, part maths, part activity book; Games for Your Mind is an ingenious thing."—-Amy Barrett, BBC Science Focus

"Excellent."—-Elizabeth Palmer, Christian Century

"It’s a serious and at times technical book, specifically about logic puzzles, though beneath its concern with matters such as obversion and epistemic obligations it has an unexpected jauntiness."—-Henry Hitchings, Times Literary Supplement

"Jason Rosenhouse’s Games for Your Mind is an engaging popular mathematics book written to enlighten the reader on the mathematics and logic behind popular puzzles. . . .overall, the reviewer would recommend this book to all people who want a puzzling challenge. Although the puzzles towards the end of the book feel impossible, the thrill of that ‘ah!’ moment when you work through Rosenhouse’s solution is surely a high for any mathematician out there."—-Holly A. J. Middleton-Spencer, London Mathematical Society

From the Publisher