Simple Type Theory

A Practical Logic for Expressing and Reasoning About Mathematical Ideas

Inbunden, Engelska, 2025

AvWilliam M. Farmer

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This unique textbook, in contrast to a standard logic text, provides the reader with a logic that can be used in practice to express and reason about mathematical ideas. The book is an introduction to simple type theory, a classical higher-order version of predicate logic that extends first-order logic. It presents a practice-oriented logic called Alonzo that is based on Alonzo Church's formulation of simple type theory known as Church's type theory. Unlike traditional predicate logics, Alonzo admits undefined expressions. The book illustrates using Alonzo how simple type theory is suited ideally for reasoning about mathematical structures and constructing libraries of mathematical knowledge. For this second edition, more than 400 additions, corrections, and improvements have been made, including a new chapter on inductive sets and types.Topics and features:· Offers the first book-length introduction to simple type theory as a predicate logic· Provides the reader with a logic that is close to mathematical practice· Includes a module system for building libraries of mathematical knowledge· Employs two semantics, one for mathematics and one for logic· Emphasizes the model-theoretic view of predicate logic· Presents several important topics, such as definite description and theory morphisms, not usually found in standard logic textbooksAimed at students of mathematics and computing at the graduate or upper-undergraduate level, this book is well suited for mathematicians, computing professionals, engineers, and scientists who need a practical logic for expressing and reasoning about mathematical ideas.William M. Farmer is a Professor in the Department of Computing and Software at McMaster University in Hamilton, Ontario, Canada.

Produktinformation

Utgivningsdatum2025-04-24
Mått155 x 235 x 26 mm
Vikt646 g
FormatInbunden
SpråkEngelska
SerieComputer Science Foundations and Applied Logic
Antal sidor319
Upplaga2
FörlagBirkhauser Verlag AG
ISBN9783031853517

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William M. Farmer has 40 years of experience working in industry andacademia in computing and mathematics. He received a B.A. inmathematics from the University of Notre Dame in 1978 and an M.A. inmathematics in 1980, an M.S. in computer sciences in 1983, and aPh.D. in mathematics in 1984 from the University of Wisconsin-Madison.He is currently a Professor in the Department of Computing andSoftware at McMaster University. Before joining McMaster in 1999, heconducted research in computer science for twelve years at The MITRECorporation in Bedford, Massachusetts, USA and taught computerprogramming and networking courses for two years at St. Cloud StateUniversity.Dr. Farmer's research interests are logic, mathematical knowledgemanagement, mechanized mathematics, and formal methods. One of hismost significant achievements is the design and implementation of theIMPS proof assistant, which was done at MITRE in partnership withDr. Joshua Guttman and Dr. Javier Thayer. His work on IMPS has led toresearch on developing practical logics based on simple type theoryand NGB set theory and on organizing mathematical knowledge as anetwork of interconnected axiomatic theories. He also hascollaborated with Dr. Jacques Carette for several years at McMaster ondeveloping a framework for integrating axiomatic and algorithmicmathematics. As part of this research, Dr. Farmer has investigatedhow to reason about the interplay of syntax and semantics, asexhibited in syntax-based mathematical algorithms like symbolicdifferentiation, within a logic equipped with global quotation andevaluation operators. Dr. Farmer is currently working on developing acommunication-oriented approach to formal mathematics as analternative to the standard certification-oriented approach employedusing proof assistants.

Chapter 1 Introduction.- Chapter 2 Answers to Readers’ Questions.- Chapter 3 Preliminary Concepts.- Chapter 4 Syntax.- Chapter 5 Semantics.- Chapter 6 Additional Notation.- Chapter 7 Beta-reduction and Substitution.- Chapter 8 Proof Systems.- Chapter 9 Theories.- Chapter 10 Inductive Sets and Types.- Chapter 11 Sequences.- Chapter 12 Developments.- Chapter 13 Real Number Mathematics.- Chapter 14 Morphisms.- Chapter 15 Alonzo Variants.- Chapter 16 Software Support.