PREFACE IN preparing these 'Studies' I have tried to carry forward the chief purpose of my Elementary Lessons in Logic, which purpose was the promotion of practical training in Logic. In the preface to those Lessons I said in 1870: 'The relations of propositions and the forms of argument present as precise a subject of instruction and as vigorous an exercise of thought, as the properties of geometrical figures or the rules of Algebra. Yet every schoolboy is made to learn mathematical problems which he will never employ in after life, and is left in total ignorance of those simple principles and forms of reasoning which will enter into the thoughts of every hour. . . In my own classes I have constantly found that the working and solution of logical questions, the examination of arguments and the detection of fallacies, is a not less practicable and useful exercise of mind than is the performance of calculations and the solution of problems in a mathematical class.' The considerable use which has been made of the Elementary Lessons seems to show that they meet an educational want of the present day. The time has now perhaps 118565 arrived when facilities for a more thorough course of logical training may be offered to teachers and students. In For a long time back there have been published books containing abundance of mathematical exercises, and not a few works consist exclusively of such exercises. recent years the teachers of other branches of science, such as Chemistry and the Theory of Heat, have been furnished with similar collections of problems and numerical examples. There can be no doubt about the value of such exercises when they can be had. The great point in education is to throw the mind of the learner into an active, instead of a passive state. It is of no use to listen to a lecture or to read a lesson unless the mind appropriates and digests the ideas and principles put before it. The working of problems and the answering of definite questions is the best, if not almost the only, means of ensuring this active exercise of thought. It is possible that at Cambridge mathematical gymnastics have been pushed to an extreme, the study of the principles and philosophy of Mathematics being almost forgotten in the race to solve the greatest possible number of the most difficult problems in the shortest possible time. But there can be no manner of doubt that from the simple addition sums of the schoolboy up to problems in the Calculus of Variations and the Theory of Probability, the real study of Mathematics must consist in the student cracking his own nuts, and gaining for himself the kernel of understanding. So it must be in Logic. Students of Logic must have logical nuts to crack. Opinions may differ, indeed, as to the value of logical training in any form. That value is twofold, arising both from the general training of the mental powers and from the command of reasoning processes eventually acquired. I maintain that in both ways Logic, when properly taught, need not fear comparison with the Mathematics, and in the second point of view Logic is decidedly superior to the sciences of quantity. Many students acquire a wonderful facility in integrating differential equations, and cracking other hard mathematical nuts, who will never need to solve an equation again, after they settle down in the conveyancer's chambers or the vicar's parsonage. With the ordinary forms of logical inference and of logical combination they will ceaselessly deal for the rest of their lives; yet for the knowledge of the forms and principles of reasoning they generally trust to the light of nature. I do not deny that a mind of first-rate ability has considerable command of natural logic, which is often greatly improved by a severe course of mathematical study. But I have had abundant opportunities, both as a teacher and an examiner, of estimating the logical facility of minds of various training and capacity, and I have often been astonished at the way in which even well-trained students break down before a simple logical problem. A man who is very ready at integration begins to hesitate and flounder. when he is asked such a simple question as the following: 'If all triangles are plane figures, what information, if any, does this proposition give us concerning things which are not triangles?' As to untrained thinkers, they seldom discriminate between the most widely distinct assertions. De Morgan has remarked in more than one place1 that a beginner, when asked what follows from 'Every A is B,' answers 'Every B is A of course.' The fact that such a converse is often true in geometry, although it cannot be inferred by pure logic, tends to mystify the student. Although all mathematical reasoning must necessarily be logical if it be correct, yet the conditions of quantitative reasoning are often such as actually to mislead the reasoner who confuses them with the conditions of argumentation in ordinary life. A mathematical education requires, in short, to be corrected and completed, if indeed it should not be preceded, by a logical education. There was never a greater teacher of mathematics than De Morgan; but from his earliest essay on the Study of Mathematics to his very latest writings, he always insisted upon the need of logical as well as purely mathematical training. This was the purpose of his tract of 1839, entitled, First Notions of Logic preparatory to the Study of Geometry, subsequently reprinted as the first chapter of the Formal Logic. A like idea inspired his valuable essays On the Method of Teaching Geometry, quoted above. 1 The Schoolmaster: Essays on Practical Education, 1836, vol. ii. p. 120, note. This excellent essay On the Method of Teaching Geometry' was originally printed in the Quarterly Journal of Education, No. XI. 1833, vol. vi. pp. 237-251. Similar views are put forth in De Morgan's earlier work, On the Study and Difficulties of Mathematics, published in 1831 by the Society for the Diffusion of Useful Knowledge. See chapter xiv. See also De Morgan's Fourth Memoir on the Syllogism, p. 4, in the Cambridge Philosophical Transactions for 1860. |