be written down on the unoccupied part of the slate, and the proper series of combinations being chosen, the contradictory combinations can be struck out with the pencil. I have used a slate of this kind, which I call a Logical Slate, for more than ten years, and it has saved me much trouble. It is hardly possible to apply this process to problems of more than six terms, owing to the large number of combinations which would require examination ; thus seven terms would give 128 combinations, eight terms 256, nine terms 512, ten terms 1024, eleven terms 2048, twelve terms 4096, and so on in geometrical progression.

Abstraction of Indifferent Circumstances.

There is a simple but highly important process of inference which enables us to abstract, eliminate or disregard all circumstances indifferently present and absent. Thus if I were to state that a triangle is a figure of three sides, with or without equal angles,' the latter qualification would be superfluous, because by a law of thought I know that angles must be either equal or unequal. To add the qualification gives no new knowledge since the existence of the two alternatives will be understood in the absence of any information to the contrary. Accordingly, when two alternatives differ only as regards a single component term which is positive in one and negative in the other, we may always reduce them to one term by striking out their indifferent part. It is really a process of substitution which enables us to do this; for having any proposition of the form

And as the second member of this is identical with the second member of (1) we may substitute, obtaining

A = AB.

This process of reducing useless alternatives, may be applied again and again; for it is plain that

A = AB (CD + Cd + cD + cd)

communicates no more information than that A is B. This abstraction of indifferent terms is in fact the converse process to that of development described in p. 104; and it is one of the most important operations in the whole sphere of reasoning.

The reader should observe that in the proposition

Illustrations of the Indirect Method.

An infinite variety of arguments and logical problems might be introduced here to show the comprehensive character and powers of the Indirect Method. We can treat either a single premise or a series of premises. Take in the first place a simple definition, such as 'a triangle is a three-sided rectilinear figure.' Let

If we take the series of eight combinations of three letters (see p. 106) and strike out those which are

I

that is, 'a rectilinear figure is either a triangle and threesided, or not a triangle and not three-sided.'

For the class b we have

b = abc abc.

To the second side of this we may apply the process of simplification by abstraction described in the last section; for by the Law of Duality

and as we have two propositions identical in the second side of each we may substitute, getting

b = ab,

or what is not three-sided is not a triangle (whether it be rectilinear or not).

Let us treat by this method the following argument :'Blende is not an elementary substance; elementary

substances are those which are undecomposable; blende, therefore, is decomposable.'

Taking our letters thus

No immediate substitution can be made; but if we take

the contrapositive of (2), namely

we can substitute in (1) obtaining the conclusion

A = Ac.

But the same result may be obtained by taking the eight combinations of A, B, C, of the abecedarium; it will be found that only three combinations, namely

Abc
a BC
abc,

are consistent with the premises, whence it results that A = Abc,

or by the process of Ellipsis before described (p. 69)

A = Ac.

As a somewhat more complex example I take the argument thus stated, one which could not be thrown into the syllogistic form.

All metals except gold and silver are opaque; there

fore what is not opaque is either gold or silver or is not-metal.'

There is more implied in this statement than is distinctly asserted, the full meaning being as follows:

All metals not gold or silver are opaque,
Gold is not opaque but is a metal.
Silver is not opaque but is a metal,

(1)

(2)

(3)

Gold and silver are distinct substances.

(4)

To obtain a complete solution of the question we take

the sixteen combinations of A, B, C, D, and striking out

The expression for not-opaque things consists of the three combinations containing d, thus

In ordinary language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not gold, or else it is not-metal and neither gold nor silver.

A good example for the illustration of the Indirect Method is to be found in De Morgan's Formal Logic (p. 123), the premises being substantially as follows:

From A follows B, and from C follows D; but B and D are inconsistent with each other; therefore A and C are inconsistent.

The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly every C is a D ; but B and D cannot occur together. The premises therefore appear to be of the form

On examining the series of sixteen combinations, but five are found to be consistent with the above conditions,

namely,

ABcd

aBcd

ab CD

abcD

abcd.