Chapter XXVI: Direct Demonstration is superior to Reduction per impossibleEdit
- Negative demonstration is superior to demonstration by reduction to the impossible, for, though both are proved by means of Not-being, in the case of the negative demonstration this Not-being is anterior to the demonstration, in the case of the other it follows. This advantage of priority makes the Negative superior.
Since the affirmative argument is superior to the negative it is clearly superior to the reduction to the impossible. The difference between them should be noticed. Thus, let no B be A, and all B be C. It follows necessarily that no C can be A. When terms are thus placed the negative demonstration shewing that C is not A is direct. The reduction to the impossible on the other hand proceeds as follows. If one have to prove that B is not A one must assume that it is A, and also that C is B; whence it follows that C is A. This is already known and acknowledged to be impossible. Hence the conclusion follows that B cannot be A. If then C be acknowledged to be A, B cannot be A.
The terms then are arranged in a similar way in both methods, but a difference arises according to which of the two negative premises is the better known, whether that shewing that B is not A, or that C is not A. When the conclusion that C is not A is better known we have a demonstration by reduction to the impossible, when the other negative proposition in the syllogism itself (B is not A) is better known, the demonstration is direct. Now the proposition B is A is naturally prior to the proposition C is A, for that from which the conclusion is drawn is prior to the conclusion itself. But the proposition C is not A is the conclusion, the proposition B is not A is a premise from which the conclusion is drawn; and the refutation of any statement does not consist merely in the conclusion but in the premises from which it is drawn. Now that from which a conclusion is drawn is a syllogism so constituted that one premise bears to the other the relation of whole to part or part to whole. The premises CA and BA, however, have not this relation to one another. If then the demonstration from prior and better known premises be superior, and if further both methods of demonstration rest on the assumption that something does not exist, if thirdly one of these methods be derived from a more, another from a less primary source, then negative demonstration is, from this fact alone, superior to reduction to the impossible. Hence, if affirmative be superior to negative demonstration, it is plainly superior to reduction to the impossible.