This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it is not a self-evident result, independent of the steps of deduction by which we have reached it. For where two classes are coincident like A and B, whatever is true of the one is true of the other what is excluded from :; the one must be excluded from the other similarly. Now as a bears to A exactly the same relation that b bears to B, the identity of either pair follows from the identity of the other pair. In every identity, equality, or similarity, we may argue from the negative of the one side to the negative of the other. Thus at ordinary temperatures

it follows that whatever star is not the brightest is not Sirius, and vice versa. Every correct definition is of the form A = B, and may often require to be applied in the equivalent negative form.

Let us take as an illustration of the mode of using this result the argument following:

Vowels are letters which can be sounded alone, (1)
The letter w cannot be sounded alone ;
Therefore the letter w is not a vowel.

(2)

(3)

Here we have a definition (1), and a comparison of a thing with that definition (2), leading to exclusion of the thing from the class defined.

Taking the terms

A =
vowel,

B = letter which can be sounded alone,

C = letter w,

Now by the Indirect method we obtain from (1) the Contrapositive

b = α,

and inserting in (2) the equivalent for b we have

C = aC,

or the letter w is not a vowel.'

(3)

Miscellaneous Examples of the Method.

We can apply the Indirect Method of Inference however many may be the terms involved or the premises containing those terms. As the working of the method is best learnt from examples, I will take a case of two premises forming the syllogism Barbara thus

Iron is a metal

Metal is element.

(1)

(2)

If we want to ascertain what inference is possible concerning the term Iron, we develop the term by the Law of Duality. Iron must be either metal or not-metal; iron which is metal must be either element or not-element; and similarly iron which is not-metal must be either element or not-element. There are then altogether four alternatives among which the description of iron must be contained; thus

Our first premise informs us that iron is a metal, and if we substitute this description in (7) and (3) we shall have self-contradictory combinations. Our second premise

likewise informs us that metal is element, and applying this description to (8) we again have self-contradiction, so that there remains only (a) as a description of ironour inference is

To represent this process of reasoning in general symbols, let

A = iron

B = metal

C = element.

Wherever the letters A or B appear in the second side of (5) substitute their equivalents given in (1) and (2) and the results at full length are

The last three alternatives break the Law of Contradiction, so that

A = ABC+0+0+0
A = ABC.

This conclusion is, indeed, no more than we could obtain by the direct process of substitution; it is the characteristic of the Indirect process that it gives all possible logical conclusions, both those which we have previously obtained, and an almost infinite number of others of which the ancient logic took little or no account. From the same premises, for instance, we can obtain a description of the

class not-element or c. By the Law of Duality we can develop c into four alternatives, thus

c = ABC + Abc + aBc + abc.

Now if we substitute for A and B as before, we get
C = ABCc+ABbc + aBCc + abc,

and striking out the terms which break the Law of Contradiction there remains

C = abc,

or what is not element is also not iron and not metal. This Indirect Method of Inference thus furnishes a complete solution of the following problem-Given any number of logical premises or conditions, required the description of any class of objects, or any term, as governed by those conditions.

The steps of the process of inference may thus be concisely stated:

1. By the Law of Duality develop the utmost number of alternatives which may exist in the description of the required class or term as regards the terms involved in the premises.

2. For each term in these alternatives substitute its description as given in the premises.

3. Strike out every alternative which is then found to break the Law of Contradiction.

4. The remaining terms may be equated to the term in question as the desired description or inference.

Abbreviation of the Process.

Before proceeding to illustrations of the use of this method, I must point out how much its practical employment can be simplified, and how much more easy it is than would appear from the description. When we want to effect at all a complete solution of a logical

problem it is best to form, in the first place, a complete series of all the combinations of terms involved in it. If there be two terms A and B, the utmost variety of combinations in which they can appear are

АВ

Ab

aB

ab.

The term A appears in the first and second; B in the first and third; a in the third and fourth; and b in the second and fourth. Now if we have any premise, say A = B,

we must ascertain which of these combinations would be rendered self-contradictory by substitution; the second and third would have to be struck out, and there would remain

AB ab.

Hence we draw the following inferences

= ab.

A = AB, B = AB, a = ab, b Exactly the same method must be followed where a question involves a greater number of terms. Thus by the Law of Duality the three terms A, B, C, give rise to eight conceivable combinations, namely

The development of the term A is formed by the first four of these; for B we must select (a), (B), (e), (); C consists of (a), (y), (e) (n); b of (y), (d), (n), (0), and so on.