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Logic: The Theory of Formal Inference

Alice Ambrose and Morris Lazerowitz
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a concise introduction to symbolic logic, aimed more at philosophy students than math students. The present volume is a Dover 2015 unaltered reprint of the 1961 Holt, Rinehart and Winston edition.

The first chapter covers the propositional calculus. It makes some use of truth tables for proofs, but primarily teaches rules of inference. The second chapter covers first-order logic. It makes heavy use of Venn diagrams, but also does some work with rules of inference. The third chapter deals with classes. Class is a synonym for set, but is treated here as a notation. Classes are not studied as objects in themselves, but are used as a shorthand for working with properties, and what is presented is a calculus of classes analogous to the calculus of propositions and quantifiers that was covered earlier.

There are numerous examples in the text, and a modest number of exercises at the end of each chapter. The exercises usually ask the student to check the correctness of a logical argument. Exercises cover both everyday verbal reasoning and symbolic forms.

A serious drawback of this book, especially for the beginning students it is aimed at, is that the notation is that of Russell & Whitehead’s 1910–1913 Principia Mathematica. This notation has been superseded over the years and is now mostly obsolete. Here’s a sample (quantifier distribution rule F5 from p. 48): \( (x)(fx \supset gx) \,. \supset \, : (\exists x)fx \, . \supset . (\exists x) gx \). In modern mathematical notation this would be \( (\forall x(fx \implies gx) ) \implies ((\exists x \, fx ) \implies (\exists x \, gx)) \). Another drawback is that there is no index. Even such a short book as this one needs an index, especially a book that introduces many new terms.

Bottom line: Probably has outlived its usefulness.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

I. Truth-Functions

II. Quantification

III. Classes