does it destroy, but it transmutes and throws the same matter into a new form.

The difficult question still remains, Where does novelty of form begin? Is it a case of inference when we pass from Sincerity is the parent of truth' to The parent of truth is sincerity?' The old logicians would have called this change conversion, one case of immediate inference. But as all identity is necessarily reciprocal, and the very meaning of such a proposition is that the two terms are identical in their signification, I fail to see any difference between the statements whatever. As well might we say that ab and ba are different equations.

Another point of difficulty is to decide when a change is merely grammatical and when it involves a real logical · transformation. Between a table of wood and a wooden table there is no logical difference (p. 37), the adjective being merely a convenient substitute for the prepositional phrase. But it is uncertain to my mind whether the change from All men are mortal' to 'No men are not mortal' is purely grammatical. Logical change may perhaps be best described as consisting in the determination of a relation between certain classes of objects from a relation between certain other classes. Thus I consider it a truly logical inference when we pass from 'All men are mortal' to All immortals are not-men,' because the classes immortals and not-men are different from mortals and men, and yet the propositions contain at the bottom the very same truth, as shown in the combinations of the Abecedarium.

From logical inference we must discriminate the passage from the qualitative to the quantitative form of a proposition. We state the same truth when we say that 'mortality belongs to all men,' as when we assert that 'all men are mortals.' Here we do not pass from class to class, but from one kind of term, the abstract, to another

kind, the concrete. But inference probably enters when we pass from either of the above propositions to the assertion that the class of immortal men is zero, or contains no objects.

It is really a question of words to what processes we shall or shall not apply the name 'inference,' and I have no wish to continue the trifling discussions which have already taken place upon the subject. We shall not commit any serious error, provided that we always bear in mind that two propositions may be connected together in four different ways. They may be

1. Tautologous or identical, involving the same relation between the same terms and classes, and only differing in the order of statement; thus Victoria is the Queen of England' is tautologous with 'The Queen of England is Victoria.'

2. Grammatically equivalent, in which the classes or objects are the same and similarly related, and the only difference is in the words; thus Victoria is the Queen of England' is grammatically equivalent to 'Victoria is England's Queen.'

3. Equivalent in qualitative and quantitative form, the classes being the same, but viewed in a different

manner.

4. Logically equivalent, when the classes and relations are different, but involve the same knowledge of the possible combinations.

CHAPTER VII.

INDUCTION.

WE enter in this chapter upon the second great department of logical method, that of Induction or the Inference of general from particular truths. It cannot be said that the Inductive process is of greater importance than the Deductive process already considered, because the latter process is absolutely essential to the existence of the former. Each is the complement and counterpart of the other. The principles of thought and existence which underlie them are at the bottom the same, just as subtraction of numbers necessarily rests upon the same principles as addition. Induction is, in fact, the inverse operation to deduction, and cannot be conceived to exist without the corresponding operation, so that the question of relative importance cannot arise. Who thinks of asking whether addition or subtraction is the more important process in arithmetic? But at the same time much difference in difficulty may exist between a direct and inverse operation; the integral calculus, for instance, is almost infinitely more difficult than the differential calculus of which it is the inverse. It must be allowed that in logic inductive investigations are of a far higher degree of difficulty, variety, and complexity than any questions of deduction; and it is this fact no doubt which has led some logicians to erroneous opinions concerning the exclusive importance of induction.

Hitherto we have been engaged in considering how

from certain conditions, laws, or identities governing the combinations of qualities, we may deduce the nature of the combinations agreeing with those conditions. Our work has been to unfold the results of what is contained in any statements, and the process has been one of Synthesis. The terms or combinations of which the character has been determined have usually, though by no means always, involved more qualities, and therefore, by the relation of extension and intension, fewer objects than the terms in which they were described. The truths inferred were thus usually less general than the truths from which they were inferred.

In induction all is inverted. The truths to be ascertained are more general than the data from which they are drawn. The process by which they are reached is analytical, and consists in separating the complex combinations in which natural phenomena are presented to us, and determining the relations of separate qualities. Given events obeying certain unknown laws, we have to discover the laws obeyed. Instead of the comparatively easy task of finding what effects will follow from a given law, the effects are now given and the law is required. We have to interpret the will by which the conditions of creation were laid down.

Induction an Inverse Operation.

I have already asserted that induction is the inverse operation of deduction, but the difference is one of such great importance that I must dwell upon it. There are many cases where we can easily and infallibly do a certain thing but may have much trouble in undoing it. A person may walk into the most complicated labyrinth or the most extensive catacombs, and turn hither and thither at his will; it is when he wishes to return that doubt and

difficulty commence. In entering, any path served him in leaving, he must select certain definite paths, and in this selection he must either trust to memory of the way he entered or else make an exhaustive trial of all possible ways. The explorer entering a new country makes sure his line of return by barking the trees.

The same difficulty arises in many scientific processes. Given any two numbers, we may by a simple and infallible process obtain their product, but it is quite another matter when a large number is given to determine its factors. Can the reader say what two numbers multiplied together will produce the number 8,616,460,799 ? I think it unlikely that any one but myself will ever know; for they are two large prime numbers, and can only be rediscovered by trying in succession a long series of prime divisors until the right one be fallen upon. The work would probably occupy a good computer for many weeks, but it did not occupy me many minutes to multiply the two factors together. Similarly there is no direct process for discovering whether any number is a prime or not; it is only by exhaustingly trying all inferior numbers which could be divisors, that we can show there is none, and the labour of the process would be intolerable were it not performed systematically once for all in the process known as the Sieve of Eratosthenes, the results being registered in tables of prime numbers.

The immense difficulties which are encountered in the solution of algebraic equations are another illustration. Given any algebraic factors, we can easily and infallibly arrive at the product, but given a product it is a matter of infinite difficulty to resolve it into factors. Given any series of quantities however numerous, there is very little trouble in making an equation which shall have those quantities as roots. Let a, b, c, d, &c., be the quantities; then (x —α) (x —b) (x−c) (x-d).....

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