Syllogistic and Its Extensions by Otto Bird, A Modern Formal Logic by Milton Fisk (review)

530 BOOK REVIEWS Syllogistic and Its Extensions. By OTTO BIRD. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1964. Pp. xii and 116. A Modern Formal Logic. By MILTON FrsK. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1964. Pp. xi and 116. These two monographs are the first to appear in the Prentice-Hall Fundamentals of Logic Series. The five authors wrote or are writing these monographs while members of the faculty of the University of Notre Dame. The other titles and authors in this series are: Language and Logic by Ernan McMullin, Method in the Sciences: An Introduction by Harry Nielsen, and History of Logic by Ivo Thomas, 0. P. The series, under the general editorship of Ernan McMullin, is intended to present to the teacher and the student the general principles, scope, and development of logic. Though there are many and varied texts in the field of logic, this series has merit in that many authors are able to treat subjects best known to them. These first two monographs deal with logic as a formal system. Otto Bird in Syllogistic and Its Extensions treats syllogistic in the first two chapters and he reserves the last three chapters for the extensions of basic syllogistic. Since his concern is syllogistic, he does not proceed in the same manner as the authors of traditional texts on logic in which the treatment of logic is divided into the three acts of the mind. Rather, Bird begins his study of syllogistic with an introduction of all the necessary elements of syllogistic argumentation. The notion of syllogism itself is introduced by way of description and analysis and its definition remains for a later section. Syllogistic in the first part of the book is restricted to general, affirmative and referential names. Since Bird stresses the formal character of logic, he does not discuss concrete statements but functors and variables. For functors he uses the traditional symbols A, I, E, and 0. The subject and predicate are replaced by name-variables b and a. In explaining the laws of the square of opposition Bird uses Euler circles and a truth table. Having defined syllogism in Aristotle's terms, Bird discusses validity. Validity does not depend on the truth or falsity of the premises. A valid syllogism occurs if the premises are true, then the conclusion will be true. If from true premises one infers a false conclusion, then that argument is invalid. Using these as criteria Bird analyzes validity in the various figures and moods of syllogistic arguments. In explaining the valid moods he uses Venn diagrams and the traditional verse containing all the valid moods. Realizing that syllogistic laws are numerous and difficult to handle, Bird in the second chapter discusses the systematizing of syllogistics. There is a short treatment of Aristotle's reduction to the first figure. The notion of indirect reduction is used as a bridge to introduce antilogism which applies the law of compound transposition. The remainder of the chapter is concerned with logic as a formal axiomatic system. The expressions, rules, axioms, and definitions of system CS, categorical syllogism, are enumerated BOOK REVIEWS 531 and explained. Employing this system Bird derives all the traditional laws of basic syllogistic. The final part of this chapter deals with the properties of a formal deductive syStem: the independence of its axioms, its consistency , and its completeness. System CS can not handle negative terms. And in his explanation of negative terms Bird mentions the difficulties that arise in using transcendental terms and terms that mutually exclude everything else in the universe. However, with the introduction of negative terms more operations are possible: contraposition, obversion, and inversion. Employing these operations and the octagon of opposition, Bird shows that negative terms can be reduced to affirmative terms and with some restrictions on system CS, system CS (n) is introduced. Syllogistic with nonreferential terms is the subject for the fourth chapter. Empty terms in system CS (n) make some of its operations invalid. In explaining the admission of empty terms Bird uses Venn diagrams and Boolean algebra. In this chapter there is a detailed discussion of the notion of existential import. The final chapter contains short discussions on four...