## Mathematical Logic: Part 1: Propositional Calculus, Boolean Algebras, Predicate Calculus, Completeness TheoremsLogic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. In particular, it is a major element in theoretical computer science and has undergone a huge revival with the explosion of interest in computers and computer science. This book provides students with a clear and accessible introduction to this important subject. The concept of model underlies the whole book, giving the text a theoretical coherence whilst still covering a wide area of logic. |

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### Contents

314 Formulas of the language | 122 |

315 Free variables bound variables and closed formulas | 125 |

316 Substitutions in formulas | 127 |

32 Structures | 130 |

321 Models of a language | 131 |

322 Substructures and restrictions | 133 |

323 Homomorphisms and isomorphisms | 135 |

33 Satisfaction of formulas in structures | 137 |

122 Tautologies and logically equivalent formulas | 27 |

123 Some tautologies | 31 |

13 Normal forms and complete sets of connectives | 34 |

132 Normal forms | 38 |

133 Complete sets of connectives | 40 |

14 The interpolation lemma | 42 |

142 The definability theorem | 44 |

15 The compactness theorem | 45 |

152 The compactness theorem for propositional calculus | 48 |

EXERCISES FOR CHAPTER 1 | 53 |

Boolean algebras | 63 |

21 Algebra and topology review | 64 |

212 Topology | 67 |

213 An application to propositional calculus | 71 |

221 Properties of Boolean algebras order relations | 72 |

222 Boolean algebras as ordered sets | 75 |

23 Atoms in a Boolean algebra | 79 |

24 Homomorphisms isomorphisms subalgebras | 81 |

242 Boolean subalgebras | 86 |

25 Ideals and filters | 88 |

252 Maximal ideals | 91 |

253 Filters | 93 |

254 Ultrafilters | 94 |

255 Filterbases | 96 |

26 Stones theorem | 97 |

261 The Stone space of a Boolean algebra | 98 |

262 Stones theorem | 102 |

EXERCISES FOR CHAPTER 2 | 106 |

Predicate calculus | 112 |

31 Syntax | 113 |

312 Terms of the language | 115 |

313 Substitutions in terms | 121 |

332 Satisfaction of the formulas in a structure | 140 |

34 Universal equivalence and semantic consequence | 147 |

35 Prenex forms and Skolem forms | 157 |

352 Skolem forms | 160 |

36 First steps in model theory | 165 |

362 Elementary equivalence | 170 |

363 The language associated with a structure and formulas with parameters | 174 |

364 Functions and relations definable in a structure | 176 |

37 Models that may not respect equality | 179 |

EXERCISES FOR CHAPTER 3 | 182 |

The completeness theorems | 193 |

41 Formal proofs or derivations | 194 |

412 Formal proofs | 196 |

413 The finiteness theorem and the deduction theorem | 200 |

42 Henkin models | 202 |

421 Henkin witnesses | 203 |

422 The completeness theorem | 205 |

43 Herbrands method | 209 |

432 The avatars of a formula | 212 |

44 Proofs using cuts | 217 |

442 Completeness of the method | 221 |

45 The method of resolution | 224 |

452 Proofs by resolution | 230 |

EXERCISES FOR CHAPTER 4 | 241 |

Solutions | 245 |

Solutions to the exercises for Chapter 2 | 270 |

Solutions to the exercises for Chapter 3 | 293 |

Solutions to the exercises for Chapter 4 | 320 |

Bibliography | 330 |

332 | |

### Other editions - View all

Mathematical Logic: A Course with Exercises, Part 1 René Cori,Daniel Lascar No preview available - 2000 |

Mathematical Logic: A Course with Exercises, Part 1 René Cori,Daniel Lascar No preview available - 2000 |

Mathematical Logic: Propositional calculus, Boolean algebras, predicate calculus René Cori,Daniel Lascar No preview available - 2000 |

### Common terms and phrases

A A B a„_i arbitrary assignment of truth atomic formulas axiom base set belong bijection binary connective binary relation Boolean algebra Chapter cl(F clause clopen clopen sets closed formula compactness theorem conclude consequence consider constant symbol contradictory defined definition derivable disjunction distribution of truth element a e elementarily equivalent empty equal equivalence classes example Exercise exists F and G filter finite subset formula F free variables hence homomorphism homomorphism of Boolean ideal induction hypothesis infinite integer interpretation L-structure Lemma logically equivalent modus ponens natural number necessary and sufficient non-zero notation obtain open sets open subsets operations pairwise distinct principal unifier Proof propositional calculus propositional variables prove quantifiers sequence set of formulas structure sub-formula substitution substructure Suppose tautology theory topological space truth table truth values ultrafilter unary function symbol underlying set unique universally valid verify w„_i words