You are here

Foundations of Mathematical Logic

Haskell B. Curry
Dover Publications
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a textbook, aimed at graduate students, about what lies beneath mathematical logic. It is not a text in mathematical logic itself, and assumes the reader already has a modest knowledge of that subject. The book is a 1977 Dover corrected reprint of the 1963 McGraw-Hill work. The type is tiny but still clear, and the book appears to have been reproduced from a larger page size; I estimate the type is about 8.5 points on a 10.5 point line spacing.

The preface describes the book as covering “the constructive theory of the first-order predicate calculus”, and it uses only constructive methods for its proofs. The book draws from many schools of thought, with the greatest influence being the work of Gerhard Gentzen (1909–1945). Roughly the first half of the book is devoted to background material in formal systems in general, with the rest dealing with nature of the key features of the predicate calculus, namely implication, negation, and quantification.

As logic books go, this one has mostly narrative and only a moderate amount of formula. It’s more philosophical than mathematical and deals with the nature of these things more than their mechanics. The introductory sections describing the problem to be solved and the approaches that have been taken are the most interesting parts. The body of the work is harder to follow, and uses a lot of non-standard terminology such as “epitheory”.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

Preface; Explanation of Conventions

Chapter 1. Introduction
  1. The nature of mathematical logic
  2. The logical antinomies
  3. The nature of mathematics
  4. Mathematics and logic
  5. Supplementary topics
Chapter 2. Formal Systems
  1. Preliminaries
  2. Theories
  3. Systems
  4. Special forms of systems
  5. Algorithms
  6. Supplementary topics
Chapter 3. Epitheory
  1. The nature of epitheory
  2. Replacement and monotone relations
  3. The theory of definition
  4. Variables
  5. Supplementary topics
Chapter 4. Relational logical algebra
  1. Logical algebras in general
  2. Lattices
  3. Skolem lattices
  4. Classical Skolem lattices
  5. Supplementary topics
Chapter 5. The Theory of Implication
  1. General principles of assertional logical algebra
  2. Propositional algebras
  3. The systems LA and LC
  4. Equivalence of the systems
  5. L deducibility
  6. Supplementary topics
Chapter 6. Negation
  1. The nature of negation
  2. L systems for negation
  3. Other formulations of negation
  4. Technique of classical negation
  5. Supplementary topics
Chapter 7. Quantification
  1. Formulation
  2. Theory of the L systems
  3. Other forms of quantification theory
  4. Classical epitheory
  5. Supplementary topics
Chapter 8. Modality
  1. Formulation of necessity
  2. The L theory of necessity
  3. The T and H formulations of necessity
  4. Supplementary topics
Bibliography; Index