Logic, Inductive and Deductive

by William Minto

but denied that Some S is not in P. The denial of a proposition implies the admission of its Contradictory. If it is not true that Some S is not in P, it must be true that All S is in P. Take this along with the admission that All P is in M, and you have a syllogism in BArbArA,

All P is in M

All S is in P,

yielding the conclusion All S is in M. If then the original conclusion is denied, it follows that All S is in M. But this contradicts the Minor Premiss, which has been admitted to be true. It is thus shown that an opponent cannot admit the premisses and deny the conclusion without contradicting himself.

The same process may be applied to Bokardo.

Some M is not in P.

All M is in S.

Some S is not in P.

Deny the conclusion, and you must admit that All S is in P. Syllogised in Barbara with All M is in S, this yields the conclusion that All M is in P, the contradictory of the Major Premiss.

The beginner may be reminded that the argument ad absurdum is not necessarily confined to Baroko and Bokardo. It is applied to them simply because they are not reducible by the ordinary processes to the First Figure. It might be applied with equal effect to other Moods, DImArIs, e.g., of the Third.