Metatheory - University of Cambridge

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In 2005, P.D. Magnus released the first version of his open-source logic textbook, forallx. In 2012, Tim Button adapted this for a first-year course at the University of Cambridge. Metatheory takes up exactly where the Cambridge implementation of forallx leaves off. Tim would, once again, like to thank Magnus for making forallx freely available. With this in mind, Metatheory is released under the same open-source license as forallx. Tim would also like to thank Brian King for comments and corrections. © 2013 by Tim Button. Some rights reserved. This book is offered under a Creative Commons license (Attribution-ShareAlike 3.0). 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/ Typesetting was carried out entirely in L A TEX2ε. The style for typesetting proofs employs forallx.sty by P.D. Magnus, University at Albany, State University of New York; Magnus’s file is based upon fitch.sty (v0.4) by Peter Selinger, University of Ottawa.
Contents 1 Induction 3 1.1 Stating and justifying the Principle of Induction . . . . . . . . 3 1.2 Arithmetical proofs involving induction . . . . . . . . . . . . . 4 1.3 Strong Induction . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Induction on length . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Unique readability . . . . . . . . . . . . . . . . . . . . . . . . . 9 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Substitution 12 2.1 Two crucial lemmas about substitution . . . . . . . . . . . . . . 12 2.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Normal forms 20 3.1 Disjunctive Normal Form . . . . . . . . . . . . . . . . . . . . . 20 3.2 Proof of DNF Theorem via truth tables . . . . . . . . . . . . . 21 3.3 Proof of DNF Theorem via substitution . . . . . . . . . . . . . 22 3.4 Cleaning up DNF sentences . . . . . . . . . . . . . . . . . . . . 27 3.5 Conjunctive Normal Form . . . . . . . . . . . . . . . . . . . . . 27 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Expressive adequacy 30 4.1 The expressive adequacy of TFL . . . . . . . . . . . . . . . . . 30 4.2 Individually expressively adequate connectives . . . . . . . . . . 32 4.3 Failures of expressive adequacy . . . . . . . . . . . . . . . . . . 33 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Soundness 37 5.1 Soundness defined, and setting up the proof . . . . . . . . . . . 37 5.2 Checking each rule . . . . . . . . . . . . . . . . . . . . . . . . . 38 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Completeness 43 6.1 Completeness defined, and setting up the proof . . . . . . . . . 43 6.2 An algorithmic approach . . . . . . . . . . . . . . . . . . . . . . 44 6.3 A more abstract approach: polarising sentences . . . . . . . . . 54 6.4 The superior generality of the abstract proof . . . . . . . . . . 58 Practice exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 iii
Page 1: Metatheory for truth-functional log
Page 5 and 6: Contents 2 The corresponding notion
Page 7 and 8: 1. Induction 4 knocking the first o
Page 9 and 10: 1. Induction 6 So the induction pro
Page 11 and 12: 1. Induction 8 sentences. But thing
Page 13 and 14: 1. Induction 10 Case 1: A is atomic
Page 15 and 16: Substitution 2 In this chapter, we
Page 17 and 18: 2. Substitution 14 2.2 Interpolatio
Page 19 and 20: 2. Substitution 16 Step 3: Repeat s
Page 21 and 22: 2. Substitution 18 We shall use the
Page 23 and 24: Normal forms 3 In this chapter, I p
Page 25 and 26: 3. Normal forms 22 Case 2: S is tru
Page 27 and 28: 3. Normal forms 24 until no (bi)con
Page 29 and 30: 3. Normal forms 26 4a: B ∗ contai
Page 31 and 32: 3. Normal forms 28 (cnf2) Every occ
Page 33 and 34: Expressive adequacy 4 In chapter 3,
Page 35 and 36: 4. Expressive adequacy 32 4.2 Indiv
Page 37 and 38: 4. Expressive adequacy 34 In fact,
Page 39 and 40: 4. Expressive adequacy 36 negations
Page 41 and 42: 5. Soundness 38 Again, it is easy t
Page 43 and 44: 5. Soundness 40 m i j k l A ∨ B A
Page 45 and 46: 5. Soundness 42 assumptions. So, ap
Page 47 and 48: 6. Completeness 44 6.2 An algorithm
Page 49 and 50: 6. Completeness 46 been expanded. N
Page 51 and 52: 6. Completeness 48 We have managed
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6. Completeness 50 1 (A ∨ (¬B
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6. Completeness 52 Step 1. Start a
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6. Completeness 54 Now let D be any
Page 59 and 60:
6. Completeness 56 Now, when we are
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6. Completeness 58 Finally: by Lemm
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6. Completeness 60 Practice exercis
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Metatheory - University of Cambridge

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