## 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 Detailed historical references are provided throughout, and each section concludes with a set of exercises.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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adjoint functors asserts assume C-monoid calculation called canonical subobjects cartesian closed category category theory characteristic morphism closed term completes the proof computable construction Corollary deductive system defined Definition diagram element epimorphism equations Example Exercise finite free topos free variables Freyd functional completeness functor categories given graph hence Heyting algebra holds indeterminate arrow induction initial object internal language intuitionistic type theory isomorphism kernel Lambek Lawvere left adjoint Lemma logicians mapping monoid monomorphism Moreover morphism natural numbers object natural transformation numerical functions obtain polynomial preordered set preserves prime filter primitive recursive functions proof of Proposition Proposition 6.1 provable prove pullback pure type theory reader representable rule of choice rules of inference Section sheaf strict logical functor subset Suppose surjective term forming operations term of type terminal object Top0 topos toposes triple typed A-calculus unique arrow universal property variable of type weak natural numbers write