Part 8

Similarly, according as a particular case (M) is or is not Z, we have

Every K is Z Every K is Z M is a Z M is not a Z ————————————— ———————————— No conclusion M is not a K

That is to say: The assertion of an hypothesis is the assertion of its necessary consequence, and the denial of the necessary consequence is the denial of the hypothesis; but the assertion of the necessary consequence gives no right to assert the hypothesis, nor does the denial of the hypothesis give any right to deny the truth of that which would (were the hypothesis true) be its necessary consequence.

Demonstration is of two kinds: which arises from this, that every proposition has a contradictory; and of these two, one must be true and the other must be false. We may then either prove a proposition to be true, or its contradictory to be false. ‘It is true that Every A is B,’ and, ‘it is false that there are some As which are not Bs,’ are the same proposition; and the proof of either is called the indirect proof of the other.

But how is any proposition to be proved false, except by proving a contradiction to be true? By proving a necessary consequence of the proposition to be false. But this is not a complete answer, since it involves the necessity of doing the same thing; or, so far as this answer goes, one proposition cannot be proved false unless by proving another to be false. But it may happen, that a necessary consequence can be obtained which is obviously and self-evidently false, in which case no further proof of the falsehood of the hypothesis is necessary. Thus the proof which Euclid gives that all equiangular triangles are equilateral is of the following structure, logically considered.

(1.) If there be an equiangular triangle not equilateral, it follows that a whole can be found which is not greater than its part.

Footnote 1:

This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.

(2.) It is false that there can be any whole which is not greater than its part (self evident).

(3.) Therefore it is false that there is any equiangular triangle which is not equilateral; or all equiangular triangles are equilateral.

When a proposition is established by proving the truth of the matters it contains, the demonstration is called direct; when by proving the falsehood of every contradictory proposition, it is called indirect. The latter species of demonstration is as logical as the former, but not of so simple a kind; whence it is desirable to use the former whenever it can be obtained.

The use of indirect demonstration in the Elements of Euclid is almost entirely confined to those propositions in which the converses of simple propositions are proved. It frequently happens that an established assertion of the form

Every A is B (1)

may be easily made the means of deducing,

Every (thing not A) is not B (2)

which last gives

Every B is A (3)

The conversion of the second proposition into the third is usually made by an indirect demonstration, in the following manner. If possible, let there be one B which is not A, (2) being true. Then there is one thing which is not A and is B; but every thing not A is not B; therefore there is one thing which is B and is not B: which is absurd. It is then absurd that there should be one single B which is not A; or, Every B is A.

The following proposition contains a method which is of frequent use.

HYPOTHESIS.—Let there be any number of propositions or assertions,—three for instance, A, B, and C,—of which it is the property that one or the other must be true, and one only. Let there be three other propositions, P, Q, and R, of which it is also the property that one, and one only, must be true. Let it also be a connexion of those assertions, that

When A is true, P is true When B is true, Q is true When C is true, R is true

CONSEQUENCE: then it follows that

When P is true, A is true When Q is true, B is true When R is true, C is true

For, when P is true, then Q and R must be false; consequently, neither B nor C can be true, for then Q or R would be true. But either A, B, or C must be true, therefore A must be true; or, when P is true, A is true. In a similar way the remaining assertions may be proved.

Case 1. If │When P is Q, A is B │When P is not Q, A is not B It follows that│When A is B, P is Q │When A is not B, P is not Q

Case 2. If │When A is greater than B, P is greater than Q „ │When A is equal to B, P is equal to Q „ │When A is less than B, P is less than Q

It follows that│When P is greater than Q, A is greater than B „ │When P is equal to Q, A is equal to B „ │When P is less than Q, A is less than B

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