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Logic for Mathematicians

J. Barkley Rosser
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
Mark Bollman
, on

I like a book that is what it says it is, and Logic for Mathematicians fits that description. This is an immersion into pure logic for non-specialists, and it succeeds in its declared goal.

Those who are fascinated by dense mathematical notation will be cheered by this book. Rosser has made a conscious decision to use symbolic logic, “because we do not know otherwise to attain the desired precision” (p. vii). This reasonable decision leads to such compact expressions as

 (A,γ)(α,β):α || γ on α.β || γ.A on β. ⊃ .α = β

for the parallel postulate (p. 177). While there is certainly a case to be made for the clear exposition of language, it is at times fascinating to see the economy of notation in the symbolic logic version of common mathematical statements.

As mathematicians, we occasionally find ourselves defending our subject for its logical clarity, yet many of us have not studied logic formally. Rosser’s book is a comprehensive introduction to logic and very suitable for self-study. If you are interested in sharpening your logic skills and not afraid of some dense mathematical sentences, this is a fine book for that purpose.

Mark Bollman ( is associate professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

List of Symbols
1. What Is Symbolic Logic?
2. The Statement Calculus
3. The use of Names
4. Axiomatic Treatment of the Statement Calculus
5. Clarification
6. The Restricted Predicate Calculus
7. Equality
8. Descriptions
9. Class Membership
10. Relations and Functions
11. Cardinal Numbers
12. Ordinal Numbers
13. Counting
14. The Axiom of Choice
15. We Rest Our Case
A Proof of the Axiom of Infinity
The Axiom of Counting
The Axiom of Choice
Nonstandard Analysis