In this volume, Lambek and Scott 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. Part II demonstrates 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 Detailed historical references are provided throughout, and each section concludeds with a set of exercises.
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"...important monograph...very clearly written..." SciTech Book News
"...a readable and timely account of important results, most of which were not previously available in book form." London Mathematical Society
"[The authors] present their material as persuasively and as lucidly as possible. In addition, they have included many useful exercises and illuminating historical and philosophical remarks, which should make the book attractive to an audience not confined to the already expert. This is an excellent and timely work on a subject that is assuming an increasingly important role in the foundations of mathematics." J.L. Bell, Journal of Symbolic Logic
"...a readable and timely account of important results, most of which were not previously available in book form." London Mathematical Society
"[The authors] present their material as persuasively and as lucidly as possible. In addition, they have included many useful exercises and illuminating historical and philosophical remarks, which should make the book attractive to an audience not confined to the already expert. This is an excellent and timely work on a subject that is assuming an increasingly important role in the foundations of mathematics." J.L. Bell, Journal of Symbolic Logic
Book Description
Part I indicates that typed-calculi are a formulation of higher-order logic, and cartesian closed categories are essentially the same. Part II demonstrates that another formulation of higher-order logic is closely related to topos theory.
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5.0 out of 5 stars
This is definitely Higher!,
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This review is from: Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics) (Paperback)
I was looking for a book for my girlfriend this Christmas and stumbled upon this one. At first I thought it would be too light but was I ever mistaken!! This book is so high that it would make Jack Kerouac dizzy. It begins with a treatment of basic category theory and ccc's and then goes on to present toposes and intuitionistic type theory. The authors take care to annotate their turnstile with the set of free variables (Hah! I bet you thought I had no idea what this book was about!) so that they can deal with empty types in a reasonable way. The treatment of presheaf models is very lucid and the discussion of internal languages and lambda-calculi is excellent. In fact many papers of Koymans are just exercises from this book worked out. The book is slightly out of date, no treatment of linear logic or symmetric monoidal-closed categories. Overall this book is highly recommended for the beginner and expert alike.
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Inside This Book
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First Sentence:
In Part 0 we recall the basic background in category theory which may be required in later portions of this book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
canonical subobjects, weak natural numbers, unique existence property, pure type theory, inner reduction, term forming operations, inductional assumption, indeterminate arrow, free topos, conjunction calculus, logical functor, intuitionistic principles, characteristic morphisms, classical type theory, first order intuitionistic logic, sheaf spaces, intuitionistic type theory, preordered set, free cartesian, subobject classifier, functional completeness, external equality, reflective subcategory, canonical monomorphism, terminal object Key Phrases - Capitalized Phrases (CAPs): (learn more)
Dana Scott, Yoneda's Lemma, Marie-France Thibault, Peter Freyd, Underlying Proposition New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In Part 0 we recall the basic background in category theory which may be required in later portions of this book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
canonical subobjects, weak natural numbers, unique existence property, pure type theory, inner reduction, term forming operations, inductional assumption, indeterminate arrow, free topos, conjunction calculus, logical functor, intuitionistic principles, characteristic morphisms, classical type theory, first order intuitionistic logic, sheaf spaces, intuitionistic type theory, preordered set, free cartesian, subobject classifier, functional completeness, external equality, reflective subcategory, canonical monomorphism, terminal object Key Phrases - Capitalized Phrases (CAPs): (learn more)
Dana Scott, Yoneda's Lemma, Marie-France Thibault, Peter Freyd, Underlying Proposition New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 1
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24
books
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- Summer School Pb by Dick Rogers in Back Matter
- Algol-like Languages (Progress in Theoretical Computer Science) by Peter O'Hearn on page 63, and page 161
- A Primer of Infinitesimal Analysis by John L. Bell in Back Matter
- Computer Science Logic: 8th Workshop, CSL '94, Kazimierz, Poland, September 25 - 30, 1994. Selected Papers (Lecture Notes in Computer Science) by Leszek Pacholski on page 426
- Linear Logic in Computer Science (London Mathematical Society Lecture Note Series) by Thomas Ehrhard on page 63
- Typed Lambda Calculi and Applications: 6th International Conference, TLCA 2003, Valencia, Spain, June 10-12, 2003, Proceedings (Lecture Notes in Computer Science) by Martin Hofmann on page 194
See all 24 books citing this book
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