## Introduction to Higher-Order Categorical LogicThis work attempts to reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. It contains an introduction to category theory and a set of exercises which accompanies each section. In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic (intuitionistic type theories) is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines. |

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### Contents

I | 3 |

II | 4 |

III | 8 |

IV | 12 |

V | 16 |

VI | 19 |

VII | 27 |

VIII | 35 |

XXXIII | 133 |

XXXIV | 139 |

XXXV | 145 |

XXXVI | 148 |

XXXVII | 153 |

XXXVIII | 160 |

XXXIX | 164 |

XL | 169 |

IX | 41 |

X | 42 |

XI | 47 |

XII | 50 |

XIII | 52 |

XIV | 55 |

XV | 57 |

XVI | 59 |

XVII | 62 |

XVIII | 65 |

XIX | 68 |

XX | 72 |

XXI | 77 |

XXII | 81 |

XXIII | 84 |

XXIV | 88 |

XXV | 93 |

XXVI | 98 |

XXVII | 101 |

XXVIII | 107 |

XXIX | 114 |

XXX | 123 |

XXXI | 124 |

XXXII | 128 |

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### Common terms and phrases

adjoint algebra arrow g asserts associated assume assumption axioms basic bounded C-monoid calculation called canonical subobjects cartesian closed category choice classes closed term completeness computable consider construction containing Corollary correspondence deductive defined Definition discussed easily elements epimorphism equality equations equivalence Example Exercise existence fact filter formula given gives graph hence holds indeterminate induction infer internal language intuitionistic intuitionistic logic isomorphism Lemma limit mapping means models monomorphism Moreover morphism natural numbers object natural transformation Note obtain operations pair partial particular present preserves projective proof Proposition prove pure reader recall recursive functions reflective subcategory relation Remark representable represented result rule satisfies Section Suppose term of type terminal Theorem topos toposes translation type theory unique arrow universal usual variables verify write