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.[1]
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