Studies and Exercises in Formal Logic

by John Neville Keynes

This is an example of direct or ostensive reduction.

257. Indirect Reduction.—A proposition is established indirectly when its contradictory is proved false; and this is effected if it can be shewn that a consequence of the truth of its contradictory would be self-contradiction.

The method of indirect proof is in several cases adopted by Euclid; and it may be employed in the reduction of syllogisms from one mood to another. Thus, AOO in figure 2 is usually reduced in this manner. The argument may be stated as follows:—
From the premisses,—

	All P is M,
	Some S is not M,
it follows that	Some S is not P ;

for if this conclusion is not true, then, by the law of excluded 319 middle, its contradictory (namely, All S is P) must be so; and, the premisses being given true, the three following propositions must all be true, namely,

All P is M,

Some S is not M,

All S is P.

But combining the first and the third of these we have a syllogism in figure 1, namely,

	All P is M,
	All S is P,
yielding the conclusion	All S is M.

Some S is not M and All S is M are, therefore, true together; but, by the law of contradiction, this is absurd, since they are contradictories.
Hence it has been shewn that the consequence of supposing Some S is not P false is a self-contradiction; and we may accordingly infer that it is true.

It will be observed that the only syllogism made use of in the above argument is in figure 1; and the process may, therefore, be regarded as a reduction of the reasoning to figure 1.