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zation, and no conscious personality. We can train him mechanically, but we cannot educate him. This will help to illustrate the difference, spoken of in § 14, between Education and Mechanical training.

We obtain astonishing results, it is true, in our schools for idiots, and yet we cannot fail to perceive that, after all, we have only an external result. We produce a mechanical performance of duties, and yet there seems to be no actual mental growth. It is an exogenous, and not an endogenous, growth, to use the language of Botany. Continual repetition, under the most gentle patience, renders the movements easy, but, after all, they are only automatic, or what the physicians call reflex.

We have the same result produced in a less degree when we attempt to teach an intelligent child something which is beyond his active comprehension. A child may be taught to do or say almost anything by patient training, but, if what he is to say is beyond the power of his mental comprehension, and hence of his active assimilation, we are only training him as we train an animal (§ 14), and not educating him. We call such recitations parrot recitations, and, by our use of the word, express exactly in what position the pupils are placed. An idiot is only a case of permanently arrested development. What in the intelligent child is a passing phase is for the idiot a fixed state. We have idiots of all grades, as we have children

of all ages.

The above observations must not be taken to mean that children should never be taught to perform operations in arithmetic which they do not, in cant phrase, "perfectly understand," or to learn poetry whose whole meaning they cannot fathom. Into this error many teachers have fallen.

There can be no more profitable study for a teacher than to visit one of these numerous idiot schools. He finds the alphabet of his professional work there. As the philologist learns of the formation and growth of language by examining, not

"Perhaps, however slow the growth, there is real progress in liberating the imprisoned soul (?)

the perfectly formed languages, but the dialects of savage tribes, so with the teacher. In like manner more insight into the philosophy of teaching and of the nature of the mind can be acquired by teaching a class of children to read than in any other grade of work.

BOOLE'S LOGICAL METHOD.

BY GEORGE BRUCE HALSTEAD.

Perhaps the possession of absolute originality cannot be better demonstrated than by breaking through the barriers inside which men have hitherto worked, pushing boldly into what was supposed to be outer void and darkness, and, without hint, without help, opening broad roads and showing fertile fields for wholly new, unsuspected sciences. This did George Boole in more than one direction.

The vast Invariantive Algebra, which is now the foundation rock of modern advance in mathematics, was started by him. Says Salmon (3d ed., p. 103): "What I have called Modern Algebra may be said to have taken its origin from a paper in the Cambridge Mathematical Journal for November, 1841, where Dr. Boole established the principles just stated, and made some important applications of them.”

Of the same epoch-making character were his extensive contributions to the Calculus of Operations. Again, in 1862, Russell said before the British Association, in regard to the Calculus of Symbols: "It received a fresh impulse from the very remarkable memoir of Prof. Boole (on a General Method in Analysis.' Phil. Trans., 1844), in which an algebra of non-commutative symbols was invented and applied.” He found many willing and able to follow on these roads, and to settle in the new lands thus laid open; but when, in 1847, he struck the key-note of a generalization of logic, which exhibits it as almost a new science, he seems to have advanced too far beyond his time, and so was left to carry it on alone, which he

did in his great work, "The Laws of Thought," published in 1854. That this, at the present moment, instead of being a thing of the past, is just beginning to attract that attention so well deserved by its extraordinary originality and suggestiveness, carries a plain inference in regard to the character of the mind capable of producing it, unaided, a quarter of a century

ago.

What, then, was his generalization, and what the method he proposed for the solution of the general problem? 1

1

The problem may be very compactly stated, but we cannot guarantee that the reader will be able at once to appreciate its full significance. It is: "Given any assertions, to determine precisely what they affirm, precisely what they deny, and precisely what they leave in doubt, separately and jointly." Or, as Boole himself puts the "statement of the final problem of practical logic. Given a set of premises expressing relations among certain elements, whether things or propositions; required explicitly the whole relation consequent among any of those elements, under any proposed conditions and in any proposed form."

That this is vastly more general than anything ever attempted by the old logic, needs no pointing out. Its startling breadth makes it seem, at first, absolutely insoluble. To illustrate this, suppose Boole had, as many cursory readers have supposed, made logic depend on the solution of ordinary algebraic equations. With the world of mathematicians to aid him, he could never have solved his problem; for from its very essence it can make no restrictions as to the number or degrees of equations, and mathematicians have never been able to find a general solution for even the equation of the fifth degree, while some of their greatest have given demonstrations of the impossibility of such solution.

1The Revue Philosophique for September 1877, contains an article thirty-three pages long on "La Logique Algèbrique de Boole," by Louis Liard. It is, for the most part, simply a translation of so much of the original, blunders included, into French. Number IV of Mind, October, 1876, contained an article of twelve pages on "Boole's Logical System," by J. Venn. This we enthusiastically recommend to our readers. We only wish it had been three times as long, and that the author had entered somewhat more into detail.

In going on to state how Boole actually did accomplish his purpose, we are met at the outset by a difficulty in the shape of a familiar word, which, as used by him, has been by prominent logicians disastrously misconceived. His critics have always used the term "mathematics" as dealing essentially with quantitative specification, and have drawn their arguments from the supposition that Boole was using the term in that sense. Even his friends have made their fight on this assumed line; which accounts for R. Harley's saying "Logic is never identified or confounded with mathematics," and for Mr. Venn's saying "The prevalent notion about Boole probably is that he regarded logic as a branch of mathematics. This is a very natural mistake."

Boole himself says, p. 11: "Whence is it that the ultimate laws of logic are mathematical in their form;" and, p. 12, says again of logic: "But it is equally certain that its ultimate forms and processes are mathematical." The key to the difficulty is contained in one short sentence, which should have been printed in capitals: "It is not of the essence of mathematics to be conversant with the ideas of number and -quantity."

This simply means that Boole felt strongly the need of some word broad enough to cover the range of sciences expressible by algebras, and thought the facts justified his taking the old word "mathematics" for such a signification.

He says, in regard to it: "The predominant idea has been that of magnitude, or, more strictly, of numerical ratio."

This conclusion is by no means necessary. We might justly assign it as the definitive character of a true calculus; that it is a method resting upon the employment of symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation." In this sense he chooses to use the word "mathematical," and in this sense his symbolic logic is as much a branch of mathematics as the ordinary algebra of number.

His broadened use of the word has been accepted by some -as meeting a real want, among whom we may mention Profes

sor Benjamin Pierce, who adds: "Qualitative relations can be considered by themselves, without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole." By bearing in mind this point we may avoid this pit, which seems to have rendered dangerous all approach to the work under consideration, and into which Stanley Jevons was one of the first to fall.

In any algebra the laws of combination of symbols are allimportant. Upon these depend its particular character and the validity of its processes. So here, in seeking to discover the most natural algebra for logic, though we may convene to represent by letters, x, y, a, b, etc., all ordinary logical classes, we must determine how they combine formally, by careful consideration of the intellectual operations implied in the best use of language as an instrument of reasoning. All thought postulates: I. The law of Identity: x=x. II. The law of Contradiction: It is impossible for any being to possess quality and at the same time not to possess it. III. The law of Excluded Middle: Everything is either x or not x. Reasoning on classes postulates also the axiom: IV. Whatever is predicated of a class may be predicated of the members of that class. Had Boole only referred to these openly, instead of making use of them unconsciously, he would have saved himself a vast amount of trouble and some positive error.

=

Convening, then, to represent any class by a letter-as, men by a and good things by b-we see that, when these are combined in thought or language, one acts as a selective adjective, and that, whichever this be, the result is the same; so that ba, or "good men," gives us the same collection of individuals as ab, or "human good beings." Using the sign as meaning, in the most general way, identity, co-existence, or equality, we say ab=ba. "We are permitted, therefore, to employ the symbols x, y, a, b, etc., in the place of substantives, adjectives, and descriptive phrases, subject to the rule of interpretation that any expression in which several of these symbols are written together shall represent all the objects

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