Simple Type Theory
Description
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.
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Person
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.
Content
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.