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Introduction to Higher-Order Categorical Logic

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Paperback | 26 September 1988

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304 Pages

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22.86 x 15.24 x 1.73

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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 Detailed historical references are provided throughout, and each section concludes with a set of exercises.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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'A readable and timely account of important results, most of which were not previously available in book form.' Bulletin of the London Mathematical Society

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You Can Find This Book In

Non-Fiction Mathematics Groups & Group Theory Mathematical Foundation Mathematical Logic Philosophy Philosophy & Logic

Preface
Introduction to category theory
Introduction to Part 0	p. 3
Categories and functors	p. 4
Natural transformations	p. 8
Adjoint functors	p. 12
Equivalence of categories	p. 16
Limits in categories	p. 19
Triples	p. 27
Examples of cartesian closed categories	p. 35
Cartesian closed categories and [lambda]-calculus
Introduction to Part I	p. 41
Historical perspective on Part I	p. 42
Propositional calculus as a deductive system	p. 47
The deduction theorem	p. 50
Cartesian closed categories equationally presented	p. 52
Free cartesian closed categories generated by graphs	p. 55
Polynomial categories	p. 57
Functional completeness of cartesian closed categories	p. 59
Polynomials and Kleisli categories	p. 62
Cartesian closed categories with coproducts	p. 65
Natural numbers objects in cartesian closed categories	p. 68
Typed [lambda]-calculi	p. 72
The cartesian closed category generated by a typed [lambda]-calculus	p. 77
The decision problem for equality	p. 81
The Church-Rosser theorem for bounded terms	p. 84
All terms are bounded	p. 88
C-monoids	p. 93
C-monoids and cartesian closed categories	p. 98
C-monoids and untyped [lambda]-calculus	p. 101
A construction	p. 107
Historical comments on Part I	p. 114
Type theory and toposes
Introduction to Part II	p. 123
Historical perspective on Part II	p. 124
Intuitionistic type theory	p. 128
Type theory based on equality	p. 133
The internal language of a topos	p. 139
Peano's rules in a topos	p. 145
The internal language at work	p. 148
The internal language at work II	p. 153
Choice and the Boolean axiom	p. 160
Topos semantics	p. 164
Topos semantics in functor categories	p. 169
Sheaf categories and their semantics	p. 177
Three categories associated with a type theory	p. 186
The topos generated by a type theory	p. 189
The topos generated by the internal language	p. 193
The internal language of the topos generated	p. 196
Toposes with canonical subobjects	p. 200
Applications of the adjoint functors between toposes and type theories	p. 205
Completeness of higher order logic with choice rule	p. 212
Sheaf representation of toposes	p. 217
Completeness without assuming the rule of choice	p. 223
Some basic intuitionistic principles	p. 226
Further intuitionistic principles	p. 231
The Freyd cover of a topos	p. 237
Historical comments on Part II	p. 244
Supplement to Section 17	p. 250
Representing numerical functions in various categories
Introduction to Part III	p. 253
Recursive functions	p. 253
Representing numerical functions in cartesian closed categories	p. 257
Representing numerical functions in toposes	p. 264
Representing numerical functions in C-monoids	p. 271
Historical comments on Part III	p. 277
Bibliography	p. 279
Author index	p. 289
Subject index	p. 291
Table of Contents provided by Syndetics. All Rights Reserved.

Introduction to Higher-Order Categorical Logic

At a Glance

Paperback

Industry Reviews

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How to return your order

Defective items

You Can Find This Book In

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