12. REDUCTION OF THE DISJUNCTIVE ARGUMENT TO THE HYPOTHETICAL AND THEN TO THE CATEGORICAL.
It would seem that the laws of the disjunctive contradict those of the categorical syllogism; for we apparently derive from two affirmatives a negative conclusion, and we also derive an affirmative conclusion when one premise is negative. This objection is seen to be nugatory when the disjunctive is reduced to the categorical form. The reduction involves the two steps of first changing the disjunctive to the hypothetical form and then to the categorical form. The following illustrations will suffice to make the matter clear:
(1) Disjunctive.
A is either B or C
A is B
∴ A is not C
Hypothetical.
If A is B, then A is not C
A is B
∴ A is not C
Categorical.
The case of A being B is the case of A not being C
In this case A is B
∴ A is not C
(2) Disjunctive.
The defendant is either guilty or innocent;
He is not innocent,
∴ He is guilty.
Hypothetical.
If the defendant is guilty, then he is not innocent;
But he is guilty,
∴ He is not innocent.
Categorical.
The case of the defendant being guilty is the case of the defendant not being innocent,
In this case the defendant is guilty,
∴ In this case the defendant is not innocent.