An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof

by Peter B. Andrews

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.

Genres: Mathematics, Logic

304 pages, Hardcover

First published January 1, 1986

About the author

Peter Bruce Andrews was an American mathematician and Professor of Mathematics, Emeritus at Carnegie Mellon University in Pittsburgh, Pennsylvania, and the creator of the mathematical logic Q0. He received his Ph.D. from Princeton University in 1964 under the tutelage of Alonzo Church. He received the Herbrand Award in 2003. His research group designed the TPS automated theorem prover. A subsystem ETPS (Educational Theorem Proving System) of TPS is used to help students learn logic by interactively constructing natural deduction proofs.

Reviews

[Ivo Spigel · Rating 5 out of 5]

A great read but not for the faint hearted :-)