BOOK REVIEWS Prior, A.N., Formal Logic. Oxford: The Clarendon Press, 1955. Pp. ix—329. $5.60. In the preface, the author indicates that this book is designed primarily as a textbook, though he hopes that it will prove to be of interest to others who are not Logic students. The content, which includes; Part I — The Classical Prepositional and Functional Calculi; Part II — The Traditional Logic of Terms; and Part HI — Modal, Three-valued and Extensional Systems ; is of such a nature that it should attract a variety of critico-analytical, perceptive readers other than those solely interested in Logic. Part I of the triparted arrangement commences with 'Truth-Functions and Tautologies' and is a systematic study of tautologous implications derived from fixed rules of transformation or inference and their relation to systems of tautologous alterations and conjunctions. In the axiomatizations of the Propositional Calculus to obtain a mechanical accuracy, a rigorously fixed symbolism is employed to express the laws laid down and established, as well as to represent the processes of proof. Using axioms and rules the author shows that these axioms are 'complete' and sufficient, not only to prove all tautologous formulae in the propositional calculus, but also to disprove all non-tautologous ones. Besides the fundamental axioms found in the Principia Mathematica of Whitehead and Russell, the newer and more economical ones derived by Lukasiewicz, and Hubert and Ackerman are employed. In answer to the question 'Why should the logician, whose primary interest is in inference, study other tautologous formulae beside those which are implications ?', the chapter entitled 'System with Propositional Constants and Functional Variables', is an attempt to show that all other tautologous formulae are either implications or definitional abbreviations for them. In defense of this thesis there are sections about The Implicational Calculus and Wajsberg's System in C and 0; Quine's Completeness Proof for the Full Calculus; Wajsberg's Completeness Proof for the Restricted Calculus; and Logic, Metalogic, and the Functorial Variables. Especially well expressed and explained in connection with the Theory of Quantification is the portion of the text on propositional operators, which form propositions, not out of other propositions, but out of names. The author illucidates the manner in which these new types of propositional formulae are introduced, and substituted for propositional variables in theses of the propositional calculus. With keen and astute logical acumen, he draws attention to the fact that there is a close resemblance (1) between the laws of equipollence and the principles now known as de Morgan Laws ; and (2) between universal quantification and conjunction, and existential quantification and alteration. 407 4o8Book-Reviews A. N. Prior's critical ability is especially pronounced in connection with 'Quantification and Equivalential Calculi: Definition Re-examined.' The work on extension, by the use of quantifiers and functional variables, of versions of the propositional calculus in which the only undefined truth-factor is E, 'If and only if,' includes the interesting note that Leániewski's special interest in a protothenic with an 'equivalential' basis arose from the personal view that he held about the nature of definition in formal systems. The author's account of definitions as assertions of equivalence include the contributions of Leániewski, Tarski and Sobocinski, who espoused the theory in its entirety, and of Lukasiewicz and Meredith, who partially adopted it. Despite the opinions of this distinguished company, the author clearly states that the conclusion of the account is wrong-headed and he shows that it makes it difficult to distinguish the definitions of a system from additional axioms. Indeed worthy of critical study, is Part II which is a truly remarkable synthesis of the Traditional Logic of Terms, that commences with the Aristotelian Syllogistic; is followed by a comprehensive treatment of Categorical Forms with Negative, Complex, and Quantified Terms; and concludes with Singular and Existential Propositions. When Professor Prior discusses Singulars and Generals in Traditional and Modern Logic, he explains with keen perception that there is a large element of anachronism in attributing to Aristotle any satisfactory way of interpreting his forms in the modern functional calculus and at the same time 'saving' all or most of his laws. The elaborate body of thought of...