## Advanced Logic for ApplicationsThis book is intended to be a survey of the most important results in mathematical logic for philosophers. It is a survey of results which have philosophical significance and it is intended to be accessible to philosophers. I have assumed the mathematical sophistication acquired· in an introductory logic course or in reading a basic logic text. In addition to proving the most philosophically significant results in mathematical logic, I have attempted to illustrate various methods of proof. For example, the completeness of quantification theory is proved both constructively and non-constructively and relative ad vantages of each type of proof are discussed. Similarly, constructive and non-constructive versions of Godel's first incompleteness theorem are given. I hope that the reader· will develop facility with the methods of proof and also be caused by reflect on their differences. I assume familiarity with quantification theory both in under standing the notations and in finding object language proofs. Strictly speaking the presentation is self-contained, but it would be very difficult for someone without background in the subject to follow the material from the beginning. This is necessary if the notes are to be accessible to readers who have had diverse backgrounds at a more elementary level. However, to make them accessible to readers with no background would require writing yet another introductory logic text. Numerous exercises have been included and many of these are integral parts of the proofs. |

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### Contents

Henkin Sets and the Fundamental Theorem | 1 |

Derivation Rules and Completeness | 13 |

Gentzen Systems and Constructive Completeness Proofs | 21 |

Quantification Theory with Identity and Functional Constants | 37 |

First Order Theories with Equality | 43 |

Godels Second Incompleteness Theorem | 49 |

Undecidability and Incompleteness | 55 |

Godels Second Incompleteness Theorem | 72 |

Intuitionistic Logic | 105 |

Second Order Logic | 116 |

Algebraic Logic | 131 |

Anadic Logic | 151 |

161 | |

163 | |

164 | |

168 | |

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### Common terms and phrases

A B(x A D B argument assigned atomic formula axiom schema Bew(b branching quantifier Chapter computable functions constants construction COROLLARY cylindrification deduction theorem Den(x derivable domain element equation equivalent EXERCISE existential extension of Q Ez)Bew(z finite number finite sequences finite subset follows free variables function symbols Godel number Godel's Henkin set identity inconsistent individual variable induction hypothesis infinite list interpretation intuitionistic LEMMA metalanguage modus ponens n-formula n-tuples natural numbers negation number theory numeralwise represents object language omega-consistent operations order theories predicate letters prenex prenex normal form primitive recursive function proof provable iff prove quantification theory realize recursively enumerable relation result of substituting rules rules of inference Sat(x second order logic sentence set of axioms set of formulas set of n-tuples set of sequences Skolem subformula Suppose true truth functional vocabulary weakly n.r.