Simple Type Theory

This unique textbook, in contrast to a standard logic text, provides the reader with a logic that actually 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.

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
• Presents the tools needed to build libraries of mathematical knowledge
• Employs two semantics, one for mathematics and one for logic
• Emphasizes the model-theoretic view of predicate logic

• Includes several important topics, such as definite description and theory morphisms, not usually found in standard logic textbooks Aimed at students of computing and mathematics at the graduate or upper-undergraduate level, this book is also 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.

First book of its kind on Simple Type Theory Provides tools for building mathematics libraries Provides a practical logic

Auteur

William M. Farmer has over 35 years of experience working in industry and academia in computing and mathematics. He received a B.A. in mathematics from the University of Notre Dame in 1978 and an M.A. in mathematics in 1980, an M.S. in computer sciences in 1983, and a Ph.D. in mathematics in 1984 from the University of Wisconsin-Madison. He is currently a Professor in the Department of Computing and Software at McMaster University. Before joining McMaster in 1999, he conducted research in computer science for twelve years at The MITRE Corporation in Bedford, Massachusetts, USA and taught computer programming and networking courses for two years at St. Cloud State University.
Dr. Farmer's research interests are logic, mechanized mathematics, mathematical knowledge management, and formal methods. One of his most significant achievements is the design and implementation of the IMPS proof assistant, which was done at MITRE in partnership with Dr. Joshua Guttman and Dr. Javier Thayer. His work on IMPS has lead to research on developing practice-oriented logics and set theories and on organizing mathematical knowledge as a network of interconnected axiomatic theories. He and Dr. Jacques Carette are currently leading the MathScheme project at McMaster with the aim of developing a framework for integrating axiomatic and algorithmic mathematics. On this project Dr. Farmer has focused on how to reason about the interplay of syntax and semantics, as exhibited in syntax-based mathematical algorithms like symbolic differentiation, within a logic equipped with global quotation and evaluation operators. Dr. Farmer has had a career-long interest in using simple type theory as a practical logic for expressing and reasoning about mathematical ideas. He is the author of "The Seven Virtues of Simple Type Theory", Journal of Applied Logic, 6:267-286, 2008, one of the leading references on simple type theory.

Résumé

"The book's intended audience is advanced undergraduates and graduates in mathematics and computing ... who need a practical logic for expressing and reasoning about mathematical ideas." My reading of this excellent and demanding (in the positive sense) book confirms that modern and applicable type theory has come into its own, one could say with a vengeance. ... I read this book linearly, that is, from first to last page, as a reviewer should." (George Hacken, Computing Reviews, February 16, 2024)

"The book is a good scientific book, and it could even be used as a textbook; that being said, it will work best at colleges and universities where there exists a strong research and education community specializing in simple type theory and Alonzo Church's approach." (Bálint Molnár, Computing Reviews, September 19, 2023)

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