✍ Scribed by Leon C. Robbins Jr.
No coin nor oath required. For personal study only.
As its title suggests, this book, by one of the most distinguished logicians of our day, is a text which develops in great detail a particular system of symbolic logic with the interests and needs of mathematicians always providing the motivation. It is a condensed and modernized version of the three volume work of Whitehead and Russell, Principia Mathematica. The fundamental relationship of class membership is not dealt with by means of a strict type theory, as in Principia Mathematica, but by a modification of type theory due to W. V. Quine, "New Foundations for Mathematical Logic," American Mathematical Monthly, Vol. 44, pp. 70-80 (1937).
In the first chapter, the need for and the benefits of a formal logical system are convincingly and engagingly set forth. Chapters II and IV treat the basic sentential connectives, "not," "and," "or," "if-then," and "if and only if," which are fundamental for mathematics and the other sciences. Here the basic units are sentences, and their combinations into compound sentences is presented axiomatically and by truth tables. Quantification theory, which is roughly the logic of "for all" and "for some," is the subject of Chapter VI. This logic allows the internal structure of sentences to be analyzed into predicates, variables, (or constants), sentential connectives, and quantifiers.
Professor Rosser next turns to the treatment of equality, descriptions (which are technical devices for introducing into a formal system names for variously needed entities), and class membership, in Chapters XI, XII, and XII[. The axiom of choice and related topics are discussed in Chapter XIV. Seven other statements, some of which go by the name of "Zorn's Lemma," are given and are proven to be pairwise equivalent and equivalent to the axiom of choice. Thus this part of the book is particularly timely. Professor Rosser rests his case in Chapter XV. He indicates a course of study leading from his book straight into the heart of classical analysis, and he incidentally clears up an hiatus that occurs in a well known function theory text.
On the negative side the reviewer would like to caution the uninitiated reader against an inaccuracy in Chapter III, page 51, sixth paragraph. There are other ambiguities of the use vs. mention variety, for example, page 83, third paragraph. These may be due to the author's attempt to follow mathematical usage as closely as possible instead of current syntactical conventions.
Limitations of space prevent the reviewer from noting more than a few features which he found of special interest. A theorem about proof procedures is established which justifies the common practice in mathematical reasoning of dropping existential quantlfiers and using some arbitrarily selected individual as the basis for further reasoning. The Russell paradox is very clearly explained, and several methods of avoiding it are indicated. There is an excellent discussion of function vs. function value, a distinction which is blurred by many mathematicians. This is particularly true of the authors of books on the differential and integral calculus. It is the reviewer's opinion that many of the difficulties encountered by college mathematics students could be traced to these fundamental confusions, and he heartily recommends Professor Rosser's analysis to students and teachers. Several principles of induction are presented, and the often neglected question of the justification of the procedure of definition by induction is given thorough consideration. The chapters on cardinals, ordinals, and the axiom of choice offer extremely valuable information for any one interested in foundational mathematics. Throughout Professor Rosser profusely illustrates the formal principles and theorems by examples from standard mathematical texts. A wealth of interesting exercises is given. The typography is excellent.
📜 SIMILAR VOLUMES
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