Logical Labyrinths

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Editorial, Sales, and Customer Service Office A K Peters, Ltd. 888 Worcester Street, Suite 230 Wellesley, MA 02482 www.akpeters.com Copyright c○ 2009 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Smullyan, Raymond M. <strong>Logical</strong> labyrinths / Raymond M. Smullyan. p. cm. Includes bibliographical references and index. ISBN 978-1-56881-443-8 (alk. paper) 1. Logic, Symbolic and mathematical. 2. Mathematical recreations. 3. Puzzles. I. Title. QA9.S575 2009 511.3–dc22 2008039863 Printed in Canada 13 12 11 10 09 10 9 8 7 6 5 4 3 2 1
Page 2 and 3: Logical Labyrinths
Page 6 and 7: Contents Preface ix I Be Wise, Gene
Page 8 and 9: Preface This book serves as a bridg
Page 11: - Part I - Be Wise, Generalize!
Page 14 and 15: 4 I. Be Wise, Generalize! Problem 1
Page 16 and 17: 6 I. Be Wise, Generalize! Problem 1
Page 18 and 19: 8 I. Be Wise, Generalize! Three Met
Page 20 and 21: 10 I. Be Wise, Generalize! be a kni
Page 22 and 23: 12 I. Be Wise, Generalize! Remarks.
Page 24 and 25: 14 I. Be Wise, Generalize! 1.19. Th
Page 27 and 28: - Chapter 2 - Male or Female? The n
Page 29 and 30: 2. Male or Female? 19 At this point
Page 31 and 32: - Chapter 3 - Silent Knights and Kn
Page 33 and 34: 3. Silent Knights and Knaves 23 3.4
Page 35 and 36: - Chapter 4 - Mad or Sane? When Abe
Page 37 and 38: 4. Mad or Sane? 27 “I don’t qui
Page 39 and 40: 4. Mad or Sane? 29 Reason. Suppose
Page 41 and 42: - Chapter 5 - The Difficulties Doub
Page 43 and 44: 5. The Difficulties Double! 33 reme
Page 45: 5. The Difficulties Double! 35 assu
Page 48 and 49: 38 I. Be Wise, Generalize! signify
Page 50 and 51: 40 I. Be Wise, Generalize! Case 2:
Page 53 and 54: - Chapter 7 - Beginning Proposition
Page 55 and 56:
7. Beginning Propositional Logic 45
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Page 63 and 64:
Page 65 and 66:
- Chapter 8 - Liars, Truth-Tellers,
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8. Liars, Truth-Tellers, and Propos
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- Chapter 9 - Variable Liars Note.
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9. Variable Liars 69 The next islan
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9. Variable Liars 71 We will call a
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74 II. Be Wise, Symbolize! Problem
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76 II. Be Wise, Symbolize! Some Oth
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78 II. Be Wise, Symbolize! p, q is
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80 II. Be Wise, Symbolize! can get
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82 II. Be Wise, Symbolize! It is im
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84 II. Be Wise, Symbolize! The Meth
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86 II. Be Wise, Symbolize! lead to
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88 II. Be Wise, Symbolize! (1) F[p
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90 II. Be Wise, Symbolize! (d) ((p
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92 II. Be Wise, Symbolize! Remark.
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94 II. Be Wise, Symbolize! (1) T(X
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96 II. Be Wise, Symbolize! Syntheti
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- Chapter 12 - All and Some Proposi
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12. All and Some 101 Problem 12.7.
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12. All and Some 103 Problem 12.18.
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12. All and Some 105 doesn’t smok
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12. All and Some 107 Charles. Thus
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12. All and Some 109 he would fail
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- Chapter 13 - Beginning First-Orde
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13. Beginning First-Order Logic 113
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- Part III - Infinity
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136 III. Infinity one-to-one corres
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138 III. Infinity Now, with each of
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140 III. Infinity sets, and “coun
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142 III. Infinity “Well, let’s
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144 III. Infinity Boolean Operation
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146 III. Infinity Better still, it
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148 III. Infinity Theorem 14.5. For
Page 160 and 161:
150 III. Infinity braces versus par
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152 III. Infinity inhabitants of Pr
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154 III. Infinity 14.17. Suppose A
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156 III. Infinity Or still again, y
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158 III. Infinity on any day withou
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160 III. Infinity Now, in geometry,
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162 III. Infinity (1) For any n, if
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164 III. Infinity Theorem 15.8 has
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166 III. Infinity Solutions 15.1. I
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168 III. Infinity Actually, this is
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170 III. Infinity its proper subset
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- Chapter 16 - Generalized Inductio
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16. Generalized Induction, König
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- Part IV - Fundamental Results in
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192 IV. Fundamental Results in Firs
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Page 227 and 228:
- Chapter 19 - The Regularity Theor
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19. The Regularity Theorem 219 Trut
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19. The Regularity Theorem 221 (b)
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19. The Regularity Theorem 223 (1)
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19. The Regularity Theorem 225 Simp
Page 237 and 238:
- Part V - Axiom Systems
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230 V. Axiom Systems called the axi
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232 V. Axiom Systems Some History T
Page 243 and 244:
234 V. Axiom Systems that are unifo
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236 V. Axiom Systems (b) Show that
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238 V. Axiom Systems Example (conti
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240 V. Axiom Systems A 2 . (X⇒Y)
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242 V. Axiom Systems Now suppose th
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244 V. Axiom Systems Re (2): By B 4
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246 V. Axiom Systems “Is there so
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248 V. Axiom Systems F 7 . X⇒(X
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250 V. Axiom Systems P 7 . ∼Y⇒
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252 V. Axiom Systems M 5 . X⇒(X
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254 V. Axiom Systems (1) X⇒(X⇒X
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256 V. Axiom Systems exercise 21.5.
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258 V. Axiom Systems “As to the s
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260 V. Axiom Systems ∼∼p is not
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262 V. Axiom Systems Solutions 21.1
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264 V. Axiom Systems P 4 : (X⇒Y)
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266 V. Axiom Systems so is X⇒Z, b
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268 V. Axiom Systems Inference Rule
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270 V. Axiom Systems (a) If X⇒φ(
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272 V. Axiom Systems 22.4. The axio
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- Part VI - More on First-Order Log
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278 VI. More on First-Order Logic T
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280 VI. More on First-Order Logic t
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282 VI. More on First-Order Logic X
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- Chapter 24 - Robinson’s Theorem
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24. Robinson’s Theorem 287 This t
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24. Robinson’s Theorem 289 satisf
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- Chapter 25 - Beth’s Definabilit
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25. Beth’s Definability Theorem 2
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25. Beth’s Definability Theorem 2
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298 VI. More on First-Order Logic I
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300 VI. More on First-Order Logic (
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308 VI. More on First-Order Logic x
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310 VI. More on First-Order Logic P
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314 VI. More on First-Order Logic s
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318 VI. More on First-Order Logic F
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References [1] E. W. Beth. “On Pa
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