Simple Type Theory: A Practical Logic for Expressing and Reasoning About Mathematical Ideas

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William M. Farmer

Simple Type Theory

A Practical Logic for Expressing and Reasoning About Mathematical Ideas

William M. Farmer

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English

Birkhauser Verlag AG
24 April 2025

Mathematical logic; Set theory; Philosophy of science; Mathematical theory of computation

Series: Computer Science Foundations and Applied Logic

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 textbooks

Aimed 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.

By: William M. Farmer
Imprint: Birkhauser Verlag AG
Country of Publication: Switzerland
Edition: Second Edition 2025
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031853517
ISBN 10: 3031853512
Series: Computer Science Foundations and Applied Logic
Pages: 319
Publication Date: 24 April 2025
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

William M. Farmer has 40 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, mathematical knowledge management, mechanized mathematics, 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 led to research on developing practical logics based on simple type theory and NGB set theory and on organizing mathematical knowledge as a network of interconnected axiomatic theories. He also has collaborated with Dr. Jacques Carette for several years at McMaster on developing a framework for integrating axiomatic and algorithmic mathematics. As part of this research, Dr. Farmer has investigated 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 is currently working on developing a communication-oriented approach to formal mathematics as an alternative to the standard certification-oriented approach employed using proof assistants.