Commentary Aristotle Posterior Analytics
Lecture 40
Negative ostensive demonstration is stronger than demonstration leading to the impossible a1. Since affirmative demonstration is superior to negative, it is clearly superior also to reductio ad impossibile.
a2. We must first make certain what is the difference between negative demonstration and reductio ad impossibile. Let us suppose that no B is A, and that all C is B: the conclusion necessarily follows that no C is A. If these premises are assumed, therefore, the negative demonstration that no C is A is direct.
a7. Reductio ad impossibile, on the other hand, proceeds as follows. Supposing we are to prove that A does not inhere in B, we have to assume that it does inhere, and further that B inheres in C, with the resulting inference that A inheres in C. This we have to suppose a known and admitted impossibility; and we then infer that A cannot inhere in B. Thus if the inherence of B in C is not questioned, A’s inherence in B is impossible.
a13. The order of the terms is the same in both proofs: they differ according to which of the negative propositions is the better known, the one denying A of B or the one denying A of C. When the falsity of the conclusion is the better known, we use reductio ad impossibile; when the major premise of the syllogism is the more obvious, we use direct demonstration.
All the same, the proposition denying A of B is, in the order of being, prior to that denying A of C; for premises are prior to the conclusion which follows from them, and ‘no C is A’ is the conclusion, ‘no B is A’ one of its premises.
a19. For the destructive result of reductio ad impossibile is not a proper conclusion, nor are its antecedents proper premises. On the contrary: the constituents of syllogism are premises related to one another as whole to part or part to whole, whereas the premises A-C and A-B are not thus related to one another. Now, the superior demonstration is that which proceeds from better known and prior premises, and while both these forms depend for credence on the not-being of something, yet the source of the one is prior to that of the other. Therefore, negative demonstration will have an unqualified superiority to reductio ad impossibile, and affirmative demonstration, being superior to negative, will consequently be superior also to reductio ad impossibile.
Since affirmative demonstration is superior. After showing that universal demonstration ranks higher than particular, and affirmative higher than negative, the Philosopher here shows, third, that ostensive demonstration is more powerful than that which leads to the impossible.
Concerning this he does three things.
First, he proposes what he intends.
Second, he prefaces certain matters needed for showing the proposition, at: he must first make certain (87a2; [350]).
Third, he proves his proposition, at: all the same, the proposition (87a17; [353]).
He says therefore first (87a1) that since we have shown that affirmative demonstration is more powerful than negative, from this it further follows that an affirmative ostensive demonstration is more powerful than one which leads to the impossible.
Then, at: we must first make certain (87a2) he lays down certain things that are necessary for showing his proposition.
In regard to this he does three things.
First, he shows what a negative demonstration is.
Second, he shows what a demonstration to the impossible is, at: reductio ad impossibile, on the other hand (87a7; [351]).
Third, he concludes the comparison of the one with the other, at: the order of the terms is the same (87a13; [352]).
He says therefore first (87a2) that in order to manifest the proposition it is necessary to point out the difference between them, that is, between the negative demonstration and the one leading to the impossible. If, therefore, one assumes that A is in no B, and B is in every C, and concludes that A is in no C, it will be a negative demonstration.
Then, at: reductio ad impossibile, on the other hand (87a7) he elucidates what a demonstration leading to the impossible is.
And he says that a demonstration leading to the impossible is as follows: Suppose we are to prove that A is not in B. Let us assume the opposite of what we wish to prove, namely, assume that every B is A; then assume that B is in C, using the proposition, “Every C is B”; from these the conclusion follows that “Every C is A,” which is such that everyone knows and admits that it is impossible. From this we conclude that the first proposition, namely, “Every B is A,” is false. Consequently, it will be necessary either that no B is A or at least that some B is not A.
But it must be understood that “A is not in B” follows, when it is clear that B is in C, because if it is obvious that “Every C is A” is false, but not obvious that “Every C is B” is true, it would not, just in virtue of the inference, be plain that “Every B is A” is false; for the falsity of the conclusion might have proceeded from either premise, as was stated above.
Then, at: the order of the terms (87a13) he concludes the comparison of each of the aforesaid demonstrations.
First, he shows wherein they agree, namely, in having a like ordering of their terms. For just as in the negative demonstration, B is taken as the middle between A and C, so too in that which leads to the impossible. Second, he shows wherein they differ, because it makes a difference which negative proposition in each demonstration is better known, namely, whether it be the proposition, “No B is A,” or the proposition, “No C is A”: for in a demonstration leading to the impossible, the proposition “C is not A” is taken to be better known, since from the fact that A is not in C one shows that A is not in B. Consequently, the proposition, “C is not A,” is taken as better known. But when that which is set down as a premise of the syllogism is taken as better known, there is a demonstrative, that is, an ostensive, negative demonstration.
Then, at: all the same, the proposition (87a17) he shows his proposition in the following way: The proposition, “B is not A,” is naturally prior to the proposition, “C is not A.” And this is proved by the fact that the premises, from which the conclusion is inferred, are naturally prior to the conclusion. But in the order of the syllogism, “C is not A” is set down as the conclusion, whereas “B is not A” is set down as that from which the conclusion is inferred. Therefore, “B is not A” is naturally prior.
Then, at: for the destructive result (87a19) he removes an objection.
For someone could say that even the negative “C is not A” is the one from which one concludes “B is not A” in a demonstration to the impossible.
But he excludes this, saying that by the fact that the conclusion is destroyed and from its destruction something in the premises is destroyed, it does not follow that what was first the conclusion is now a principle and vice versa, absolutely and according to nature, but only in a qualified sense. For the relation of conclusion to principle is such that the principle is destroyed by the conclusion’s being destroyed. But that functions as the principle from which a syllogism proceeds, which is related to the conclusion as a whole to a part; while the conclusion is to the principle, as part to whole. For the subject of a negative conclusion is subsumed under the subject of the first proposition. But the propositions AC and AB are not so related to each other that AC is to AB as whole to part. For BA is not subsumed under CA; rather it is just the opposite. Hence it remains that although from the destruction of CA, one concludes to the destruction of BA, nevertheless CA is naturally the conclusion and BA the principle. Consequently, “B is not A” is naturally better known than “C is not A.”
From this, one argues in the following way: That demonstration is the worthier which proceeds from better known and prior principles. But a negative demonstration proceeds from something better known and prior than does a demonstration leading to the impossible. For each causes one to know something in virtue of a negative proposition: but the negative demonstration proceeds to cause belief from the negative proposition, “B is not A,” which is naturally prior. Demonstration to the impossible, on the other hand, proceeds to cause belief from the negative proposition, “C is not A,” which is naturally posterior. What remains, therefore, is that the negative demonstration is more powerful than one which leads to the impossible. Furthermore, as was shown above, the affirmative is stronger than the negative. Therefore, an affirmative ostensive demonstration is much stronger than one which leads to the impossible.