Commentary Aristotle Posterior Analytics
Lecture 39
Affirmative demonstration is stronger than negative
a32. That affirmative demonstration excels negative may be shown as follows.
(1) We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates or hypotheses — in short from fewer premises; for, given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum. The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form as follows. Assuming that in both cases alike the middle terms are known, and that middles which are prior are better known than such as are posterior, we may suppose two demonstrations of the inherence of A in E, the one proving it through the middles B, C and D, the other through F and G. Then A-D is known to the same degree as A-E (in the second proof), but A-D is better known than and prior to A-E (in the first proof); since A-E is proved through A-D, and the ground is more certain than the conclusion. Hence demonstration by fewer premises is ceteris paribus superior. Now, both affirmative and negative demonstration operate through three terms and two premises, but whereas the former assumes only that something is, the latter assumes both that something is and that something else is not, and thus operating through more kinds of premise is inferior.
b10. (2) It has been proved that no conclusion follows if both premises are negative, but that one must be negative, the other affirmative.
b12. So we are compelled to lay down the following additional rule: as the demonstration expands, the affirmative premises must increase in number, but there cannot be more than one negative premise in each complete proof. Thus, suppose no B is A, and all C is B. Then if both the premises are to be again expanded, a middle must be interposed. Let us interpose D between A and B, and E between B and C. Then clearly E is affirmatively related to B and C, while D is affirmatively related to B but negatively to A; for all B is D, but there must be no D which is A. Thus there proves to be a single negative premise, A-D. In the further prosyllogisms too it is the same, because in the terms of an affirmative syllogism the middle is always related affirmatively to both extremes; in a negative syllogism it must be negatively related only to one of them, and so this negation comes to be a single negative premise, the other premises being affirmative. If, then, that through which a truth is proved is a better known and more certain truth, and if the negative proposition is proved through the affirmative and not vice versa, affirmative demonstration, being prior and better known and more certain, will be superior.
b30. (3) The basic truth of demonstrative syllogism is the universal immediate premise, and the universal premise asserts in affirmative demonstration and in negative denies: and the affirmative proposition is prior to and better known than the negative (since affirmation explains denial and is prior to denial, just as being is prior to not-being). It follows that the basic premise of affirmative demonstration is superior to that of negative demonstration, and the demonstration which uses superior basic premises is superior.
b39. (4) Affirmative demonstration is more of the nature of a basic form of proof, because it is a sine qua non of negative demonstration.
That affirmative demonstration. After showing that universal demonstration is more powerful than particular, the Philosopher here shows that affirmative demonstration is more powerful than negative.
In support of this he presents five reasons.
In the first reason (86a32) he presupposes that, all else being equal, the better demonstration is the one which proceeds from fewer postulates or suppositions or propositions.
How these differ is clear from what has been stated above. For propositions might even be taken to mean those per se known statements which are neither suppositions nor postulates, as stated above. But a supposition is not the same as a postulate: for a supposition is a proposition which is not per se known, but is taken by the learner as opined; a postulate, on the other hand, is a proposition which is not per se known and has not been opined by the learner, nothing being said about whether he has opinions to the contrary or not.
Accordingly, he shows in two ways that, other things being equal, the demonstration which employs fewer of these is better.
First, because if it happens that both propositions from which one proceeds are equally known, it follows that one knows more quickly through fewer propositions than through more, because a discourse through fewer propositions comes to an end sooner than one through more. For it is more desirable or more advantageous that a man learn more quickly. Hence, it remains that a demonstration which proceeds from fewer propositions, provided they are equally known, is better.
Second, he proves the same proposition universally without the above supposition, namely, that all the propositions used are equally known. To prove this he assumes this supposition, namely, that the middles which are of one order are equally known, but the middles which are prior are better known. For this must be universally true. Therefore, supposing this, let there be one demonstration in which it is demonstrated that A is in E in virtue of three middles, BCD, so that a conclusion is reached from four propositions, namely, “Every B is A,” “Every C is B,” “Every D is C” and “Every E is D.” Then let the other demonstration be one which concludes the same conclusion, namely, that A is in E, through two middles which are Z and H. Under these conditions, from the fact that the order of knowing is proportionate to the order of middles, since the prior known are the better known, as has been said, it is obvious that the proposition, “Every D is A” in the first demonstration must be as well known as “Every E is A” in the second, because two middles are used for obtaining each. However, it is also obvious that in the first demonstration this proposition, “Every D is A,” is prior and better known than the proposition, “Every E is A,” because this latter is demonstrated from something prior in the first demonstration—and from what has been established above, it is apparent that that through which something is demonstrated is more credible and more known than that which is demonstrated through it. What remains, therefore, is that this proposition, “Every E is A,” so far as it is concluded by the second demonstration, is better known than the same proposition so far as it is concluded by the first demonstration which used more middles.
Therefore, a demonstration which proceeds from fewer is better than one which proceeds from more.
Therefore, having proved the major proposition, Aristotle assumes that an affirmative proposition proceeds from fewer things than a negative: not indeed from fewer terms or fewer propositions materially, because every demonstration, whether affirmative or negative, demonstrates through three terms and two propositions; but a negative demonstration is said to proceed from more according to the quality of the propositions. For an affirmative demonstration takes only “being,” that is, proceeds only from affirmative propositions, but a negative demonstration takes being and non-being, that is, uses an affirmative and a negative. Therefore, the affirmative is stronger than the negative.
Then, at: it has been proved that no conclusion (86b10) he presents the second reason.
Now this second reason is introduced to support the first, which might seem deficient on the ground that it was not subsumed under the major proposition in the way in which it was proved. Therefore, to forestall any subterfuge he adds this second reason to confirm the first.
For it has been shown in Prior Analytics I that a syllogism cannot be formed out of two negative propositions, but one proposition at least must be affirmative and the other may be negative. This clearly shows that affirmative propositions have greater efficacy for syllogizing than do negatives. Hence it follows that an affirmative demonstration which proceeds from affirmatives only, is more powerful than the negative demonstration which proceeds from a negative and an affirmative.
Then, at: so we are compelled (86b12) he presents the third reason.
He says that as a consequence of the foregoing reasons, we can assert that when a demonstration is augmented, that is, by resolving propositions into their principles, it is necessary that there be several affirmative propositions, but it is impossible for there to be more than one negative. Thus, let us take the following negative demonstration: “No B is A; Every C is B: therefore, No C is A.” Therefore, let this demonstration be augmented as to both propositions, if each is mediate, by taking a middle for each, letting the middle of the major proposition, “No B is A,” be D, and the middle of the minor proposition, “Every C is B,” be E. Now, since this proposition, “Every C is B,” is affirmative and was concluded without benefit of a negative, it is necessary that its middle, namely, E, be affirmative to each extreme. Consequently, there are two affirmative propositions, namely, “Every E is B,” and “Every C is E,” from which “Every C is B” is concluded. But the major proposition is negative, namely, “No B is A.” Now, a negative is not concluded from two negatives; but in the first figure, in which demonstration is best made, according to what has been said, that major must be negative and the minor affirmative. Consequently, this middle, D, must be affirmative to B and negative to A, as in the following demonstration: “No D is A; Every B is D; therefore No B is A.” Hence by augmenting a negative demonstration by resolving the propositions into their principles, there will be four propositions: only one of these is negative, namely, “No D is A,” and the other three will be affirmative, namely, “Every B is D,” “Every E is B,” and “Every C is E.” And the same thing will happen in all other syllogisms, because it is always required that the middle, through which the affirmative propositions are proved, be affirmative to both extremes. But the middle through which the negative proposition is proved must be negative to one of the extremes only. Thus it follows that only one proposition is negative and the others all affirmative. From this it is clear that it is mainly through affirmatives that a negative proposition is demonstrated. Therefore, if that through which something is demonstrated is more known and more credible than that which is demonstrated through it (since a negative proposition is mainly proved by affirmatives and not vice versa), it follows that the affirmative proposition is prior and more known and more credible than the negative. Accordingly, the affirmative demonstration will be more valuable.
Then, at: the basic truth of demonstrative syllogism (86b30) he presents the fourth reason.
He says that the principle of a demonstrative syllogism is an immediate universal proposition, in the sense that the proper principle of an affirmative syllogism is an affirmative proposition and the proper principle of a negative syllogism is a universal negative proposition. But the effect of a nobler principle is itself more noble. Therefore, as an affirmative proposition is to a negative, so is an affirmative demonstration to a negative one. But an affirmative proposition is more powerful than a negative one. (He proves this in two ways: first, because the affirmative is prior and better known, since the negative is proved by the affirmative and not vice versa. Second, because affirmation naturally precedes negation, as being is prior to non-being—for although in one and the same thing which passes from non-being to being, the non-being is prior in the order of time, yet in the order of nature, being is prior and, absolutely speaking, is prior even in time, because non-beings are not brought into existence except by something that is). Therefore, it is clear that affirmative demonstration is more powerful than negative.
Then, at: affirmative demonstration is more (86b39) he presents the fifth reason and it is this:
That upon which something depends is more principal. But negative demonstration depends on affirmative, because there cannot be a negative demonstration without an affirmative proposition, which is not proved except by an affirmative demonstration. Therefore, affirmative demonstration is more principal than negative.