## Introduction to the Theory of LogicThis book provides a rigorous introduction to the basic concepts and results of contemporary logic. It also presents, in two unhurried chapters, the mathematical tools (mainly from set theory) that are needed to master the technical aspects of the subject. Methods of definition and proof are also discussed at length, with special emphasis on inductive definitions and proofs and recursive definitions. The book is ideally suited for readers who want to undertake a serious study of logic but lack the mathematical background that other texts at this level presuppose. It can be used as a textbook in graduate and advanced undergraduate courses in logic. Hundreds of exercises are provided. Topics covered include basic set theory, propositional and first-order syntax and semantics, a sequent calculus-style deductive system, the soundness and completeness theorems, cardinality, the expressive limitations of first-order logic, with especial attention to the Loewenheim-Skolem theorems and non-standard models of arithmetic, decidability, complete theories, categoricity and quantifier elimination. |

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### Contents

Chapter | 36 |

Chapter | 76 |

Chapter | 117 |

Chapter | 149 |

FirstOrder Languages without Identity | 162 |

FirstOrder Languages with Identity | 166 |

A WellRounded Set of Sentences Is the Theory of its Canonical Structure | 170 |

Negation Completeness | 174 |

Exponentiation | 201 |

Cardinality | 204 |

Contextual Definition | 206 |

Continuities | 209 |

Denumerable Sets | 215 |

Diagonals | 218 |

ZF | 224 |

11 Choice | 228 |

Henkin Constants and Henkin Axioms | 178 |

Consistent NegationComplete Henkin Sets Are Well Rounded | 183 |

Compactness | 187 |

Chapter | 189 |

Criteria for Size Claims | 190 |

Counting | 191 |

Functional Criteria | 195 |

Big and Small | 196 |

Addition | 197 |

Multiplication | 199 |

12 Zorns Lemma | 236 |

Completeness for Uncountable Languages | 245 |

Chapter | 250 |

Isomorphic Structures | 252 |

Isomorphic Structures and Contextual Definitions | 259 |

Representing Finite Structures | 261 |

The Downward LowenheimSkolem Theorem | 263 |

Chapter | 292 |

Symbols and Notation | 321 |

### Common terms and phrases

admissible PL-assignment Axiom of Choice basic binary relation canonical structure Chapter closed terms composition devices conjunct contains Continuum Hypothesis countable deducibility claim define definition of truth denumerable domain equivalence relation establish Exercise extralogical symbols finite set finite subset first-order language first-order logic first-order propositions free variables function pairing function symbol galaxy guage Hence Henkin set Hint I-formula I-structure I-term indiscernible individual constant inductive clauses infinite cardinal isomorphism L-sequent Let/be logical consequence claims logically equivalent maximal element model the syntactic natural number need to show negation-complete nonstandard Notice notion occur one-place one-to-one correspondence one-to-one function PL-sentence positive integer predicate procedure proof of Lemma prove rational numbers represented result satisfies sequence set of formulas set of I-sentences set of sentences substitutable suffice to show syntactic pattern Theorem tion true truth function truth value tuple two-place universe variable interpretation Vx Px well-rounded set Zorn's Lemma

### Popular passages

Page 73 - ... f ...f •••t •••t ...f ...f ...t ...f ...f •••t •••t •••t •••t •••t •••t •••t •••t ...t •••t •••: ...t...

Page 238 - B be any two sets. The set of all ordered pairs such that the first member of the ordered pair is an element of A and the second member is an element of B is called the cartesian product of A and B and is written as AX B.

Page 8 - The intersection of sets A and B, written A n B, is the set of all elements common to sets A and B.

Page 117 - An argument will be said to be valid if and only if the conjunction of the premises implies the conclusion, ie, if the premises are all true, the conclusion must also be true. It...

Page xii - I have tried to present the material in such a way as to be intelligible and even helpful to graduate students and, perhaps, undergraduates.

Page 175 - Prove that there exists a one-to-one correspondence between the set of positive integers and the set of all positive rational numbers.

Page 8 - The union of sets A and B, written A u B, is the set whose elements are all the elements of A and all the elements of B.

Page 187 - ... then it is not the case that for every set F of sentences of intensional logic, (3) if every finite subset of F is satisfiable, then F is satisfiable. This is obvious in view of the reduction, at which we hinted earlier, of ordinary second-order logic to intensional logic, together with the wellknown failure of the compactness theorem for second-order logic. On the other hand, let us call $ a predicative sentence if...