Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics, Series Number 7)

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 The book is slightly out of date, no treatment of linear logic or symmetric monoidal-closed categories.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. This 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 Detailed historical references are provided throughout, and each section concludes with a set of exercises.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

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.

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and categony theony. 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. 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. In Part 0 we recall the basic background in category theory which may be required in later portions of this book.