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 HUGH MACCOLL, Symbolic Logic and Its Applications. 255 Symbolic Logic and Its Applications. By HUGH MACCOLL, B.A. Lend. London, 1906. Pp. xi, 141. EEADEBS of Mr. MacColFs papers in MIND and elsewhere will be already familiar with most of the contents .of this volume, but they will be glad to have his system in a more connected and accessible form than hitherto. For reaching those who do not follow closely the development of symbolic logic, publication in book form is almost essential ; and it is much to be hoped that this book will be widely read. From this point of view, it has the great merit of being by no means difficult, and of demanding absolutely no previous knowledge of the subject. It appears to be intended to be suitable for beginners, and in this it certainly succeeds. The distinctive characteristic of Mr. MacColl's writing is that he deals always with whole statements or propositions, not, like most writers, with classes. " The complete statement or proposition," he says, "is the real unit of all reasoning" (p. 2). Hence he is primarily concerned with implication, not with inclusion ; his formulae state that one statement implies another, not (directly) that one class is contained in another. The relation of inclusion between classes is for him derivative, being in fact the relation of implication between the statement that a thing belongs to the one class, and the statement that it belongs to the other. 1 He was, I believe, the first to found symbolic logic on propositions and implication, and in this respect he seems to me to have made an important advance upon his predecessors. In the present work, he gives the elementary formulae of his method, re-states some parts of traditional formal logic, deals very briefly with some contro- verted points, and explains his interesting "calculus of limits". 2 His discussion of the syllogism and the would-be canons (about not having two negative premises and so on) should serve to show philosophers how far formal logic has travelled from the pedantic trivialities of Barbara Celarent. There is an excellent analysis of induction (pp. 84-35), and some illuminating definitions of received terms are given. The habitual confusion (in more or less subtilised forms) between a hypothetical and an inference 3 is shown to be a cause of wide-spread and important errors. Mr. MacColl rightly pro- tests against the practice of stating syllogisms as if they contained inferences instead of mere hypotheticals. This might be thought to be a purely verbal question, but it is in fact, as he points out, a 1 In MIND, N.S., No. 55, p. 400, I spoke of this as Peano's interpreta- tion. It was, however, given by Mr. MacColl as early as 1878, and should therefore be ascribed to him. 2 A good account of this will be found in vol. ii., part ii. (posthu- mously published) of Schroder's Alyebra der Logik (Leipzig, 1905), pp. 515-563. (Schroder points out and rectifies a mathematical error in Mr. MacColl's method of changing the order in a multiple integral.) 3 In a hypothetical, it is merely asserted that if the hypothesis is true, the conclusion is true ; in an inference, the hypothesis being known to be true, the conclusion is also asserted.