Commentary Aristotle Posterior Analytics
Lecture 26
Demonstrative syllogisms best made in the first figure; on mediate and immediate negative propositions a17. Of all the figures the most scientific is the first. Thus, it is the vehicle of the demonstrations of all the mathematical sciences, such as arithmetic, geometry, and optics, and practically all of all sciences that investigate causes: for the syllogism of the reasoned fact is either exclusively or generally speaking and in most cases in this figure—a second proof that this figure is the most scientific; for grasp of a reasoned conclusion is the primary condition of knowledge.
a23. Third, the first is the only figure which enables us to pursue knowledge of the essence of a thing. In the second figure no affirmative conclusion is possible, and knowledge of a thing’s essence must be affirmative; while in the third figure the conclusion can be affirmative, but cannot be universal, and essence must have a universal character: e.g., man is not two-footed animal in any qualified sense, but universally.
a30. Finally, the first figure has no need of the others, while it is by means of the first that the other two figures are developed, and have their intervals close-packed until immediate premises are reached. Clearly, therefore, the first figure is the primary condition of knowledge.
a33. Just as an attribute A may (as we saw) be atomically connected with a subject B, so its disconnexion may be atomic. I call ‘atomic’ connexions or disconnexions which involve no intermediate term; since in that case the connexion or disconnexion will not be mediated by something other than the terms themselves.
a36. It follows that if either A or B, or both A and B, have a genus, their disconnexion cannot be primary. Thus: let C be the genus of A. Then, if C is not the genus of B — for A may well have a genus which is not the genus of B — there will be a syllogism proving A’s disconnexion from B thus: all A is C; no B is C; therefore no B is A. Or if it is B which has a genus D, we have: all B is D; no D is A; therefore no B is A, by syllogism; and the proof will be similar if both A and B have a genus.
b5. That the genus of A need not be the genus of B and vice versa, is shown by the existence of mutually exclusive coordinate series of predication. If no term in the series ACD... is predicable of any term in the series BEF..., and if G — a term in the former series — is the genus of A, clearly G will not be the genus of B; since, if it were, the series would not be mutually exclusive. So also if B has a genus, it will not be the genus of A.
b12. If, on the other hand, neither A nor B has a genus and A does not inhere in B, this disconnexion must be atomic. If there be a middle term, one or other of them is bound to have a genus, for the syllogism will be either in the first or the second figure. If it is in the first, B will have a genus — for the premise containing it must be affirmative: if in the second, either A or B indifferently, since syllogism is possible if either is contained in a negative premise, but not if both premises are negative.
b21. Hence it is clear that one thing may be atomically disconnected from another, and we have stated when and how this is possible.
Of all the figures the most scientific. After determining about the material of the syllogism, the Philosopher here determines about its form, showing in which figure chiefly the demonstrative syllogism is formed.
His treatment is divided into two parts.
In the first he shows that the demonstrative syllogism is formed first and foremost in the first figure.
In the second, because it is possible in the first figure to proceed from negatives and because a demonstration must proceed from things immediate, he shows how a negative proposition happens to be immediate, at: just as an attribute A (79a33; [217]).
He shows the first with three reasons, the first of which (79a17) is this.
In whatever figure the 'propter quid' syllogism is best made, that figure is the best for causing scientific knowledge and for that reason is most suitable for demonstrations, since demonstration is a syllogism which causes scientific knowledge. But a syllogism 'propter quid' is best made in the first figure, and this is evidenced by the fact that the mathematical sciences of arithmetic and geometry and all other sciences which demonstrate 'propter quid' employ the first figure in most cases. Therefore, the first figure is the one which first and foremost causes scientific knowledge and is most suitable for demonstrations.
But the reason why demonstration 'propter quid' is best made in the first figure is this: in the first figure the middle term is both subjected to the major extreme, which is the predicate of the conclusion, and predicated of the minor extreme, which is the subject of the conclusion. Now, in demonstrations 'propter quid' the middle must be the cause of the proper attribute which is predicated of the subject in the conclusion. Furthermore, one of the modes of “saying per se” is when the subject is the cause of the predicate, as “being butchered, he died,” as has been explained above; and this is verified in the first figure in which the middle is subject to the major extreme, as has been said.
The second reason, at: third, the first is the only figure (79a23) is this: the quod quid est [that is, the definition as signifying the essence] plays a most important role in demonstrative sciences because, as has been said, a definition is either the principle of a demonstration or the conclusion or a demonstration differing in the position of its terms. Now, when it is a question of formulating a definition the first figure alone is suitable. For that is the only figure in which a universal affirmative is concluded, which alone produces science of the quod quid est. (For the quod quid est is known through an affirmation. Furthermore, the definition is predicated affirmatively and universally of the defined: for it is not some man that is a two-legged animal, but every man.) Therefore, the first figure is foremost in causing scientific knowledge and is best accommodated to demonstrations.
Then, at: finally, the first figure (79a30) he gives the third reason which is this: For purposes of demonstration the other figures need the first, but the first does not need them. Therefore, the first figure is better equipped for causing scientific knowledge than the others. That the others need the first is evidenced by the fact that if complete scientific knowledge is to be had, the mediate propositions present in demonstrations must be reduced to immediate ones. But this reduction is made in two ways, namely, by condensing middles [that is, by inserting fresh middles, the movement proceeding from the extremes toward the middle first employed] and by expanding outwards [that is, by going from the middle toward the remote extreme]. It is done by condensing when the middle actually employed is joined mediately to each of the extremes or only to the minor extreme. Hence when other middles are introduced between the first middle and the extremes, the middles, as it were, close in. Thus, if someone first says, “Every E is C,” [minor], “Every C is A,” [major], and then a middle, say D, is introduced between C and E, and a middle B between C and A. It is done by expanding when the middle is immediate to the minor extreme and mediate to the major: for then it is necessary to introduce several other middles more general than the middle first taken. Thus, if one says, “Every E is D,” “Every D is A,” and then other middles more general than D are introduced [between D and A]. Now, these processes of condensing and expanding can be performed only in the first figure, both because the first figure is the only one which yields a universal affirmative conclusion, and because it is only in the first figure that the middle lies between both extremes. For in the second figure the middle lies outside the extremes, as being predicated of them; in the third figure [the middle] is below each of the extremes, as being subjected to them.
Then, at: just as an attribute A (79a33) he teaches how a negative proposition can be immediate.
First (79a33), he states his intention, saying that just as it is possible for every A to be in B atomically, that is, immediately, so too not to be, that is, so too it is possible for a proposition signifying that A is not in B to be immediate. Then he goes on to explain what it is to be or not to be atomically, namely, when the affirmation or negation does not have a middle through which it might be proved.
Second, at: it follows that if either A or B (79a36), he elucidates his proposition.
First, he shows when a negative proposition is mediate.
Second, when immediate, at: if, on the other hand (79b12; [220]).
Concerning the first he does two things.
First, he clarifies his proposition.
Second, he proves something he had presupposed, at: that the genus of A need not (79b5; [219]).
He says therefore first (79a36), that when A, that is, the major term, or B, that is, the minor term, is in some whole as a species in its genus, or both are under diverse genera or predicaments, it does not occur that A is not in B first, that is, it does not occur that this proposition, “No B is A,” is immediate. First, then, he manifests this when A is in some whole, say in C, and B is in no whole. For example, if A is 'man', C is 'substance', and B is 'quantity', it is possible to form a syllogism to prove that A is in no B on the ground that C is in every A and in no B. Thus we get a syllogism in the 318second figure:
“Every man is a substance;
No quantity is a substance:
Therefore, no quantity is a man”. In like manner, if B, that is, the minor term, is in some whole, say D, but A is not in that whole, it is possible to syllogize that A is in no B. For example if A is 'substance', B 'line', and D 'quantity', we get a syllogism in the 320first figure:
“No quantity is a substance;
Every line is a quantity;
Therefore, no line is a substance.” In the same way, a negative conclusion could be demonstrated if either is in some whole. For example, if A is 'line', C 'quantity', B 'whiteness' and D 'quality', it is possible to form a syllogism in the first and in the second figure. In the 322second figure thus:
“Every line is a quantity;
No whiteness is a quantity:
Therefore, no whiteness is a line.”
And in the first figure thus:
“No quality is a line;
Every whiteness is a quality:
Therefore, no whiteness is a line.”
We should understand, however, that a negative proposition is mediate, when both terms exist in some whole which is not the same but different for each. For if they are in the same whole, the proposition will be immediate, as “No rational being is irrational,” or “No biped is a quadruped.”
Then, at: that the genus of A (79b5) he explains something he had presupposed, namely, “on condition that one of the extremes exist in some whole and that the other be not in the same,” saying that it is clear from the orderings of the various predicaments, which are not mutually interchangeable. In other words, because that which is in one predicament is not in another, it is plain that B happens not to be in the whole in which A is, or vice versa, because one of the terms happens to be taken from one predicament in which the other is not found. Thus, let one ordering of the predicaments be ACD, say the predicament of substance, and another ordering be BEF, say the predicament of quantity. Then if none of those in the ordering ACD be predicated of none in the ordering BEF, while A is in P as in that most general item which is the principle of the whole first ordering, it is plain that B is not in P, because then the orderings, that is, the predicaments, would be interchanged. Similarly, if B is in some whole, say in E, it is plain that A is not in E.
Then, at: if, on the other hand (79b12) he indicates how a negative proposition may be immediate, saying that if neither is in some whole, that is, neither A nor B, and A is not in B, it is necessary that this proposition, “No B is A,” be immediate. Because if a middle were taken to syllogize it, then one of them would have to be in some whole, for the syllogism would have to be made either in the first figure or in the second, (since in the third figure a universal negative cannot be concluded, as is required for an immediate proposition). However, if it is made in the first, B would have to be in some whole, because B is the minor extreme, and in the first figure the minor proposition must always be affirmative. For a syllogism with the major affirmative and the minor negative cannot be formed in the first figure. But if it be in the second figure, either, that is, A or B, may be in a whole, because in the second figure the first proposition may be negative in some moods and the minor in other moods. Of course it is never permitted, neither in the first nor the second, to have both propositions negative. And so it is required that when either proposition is affirmative, one of the extremes must be in some whole. Thus it is clear that a negative proposition is immediate, when neither of its terms is in some whole.
This does not mean, however, that although neither is in some whole, a middle could be found to conclude it, namely, if one were to take a convertible middle, because it is necessary that such a middle be prior and better known. And this is either the genus itself or the definition, which is not without a genus.
Then, at: hence it is clear that one thing (79b21) he concludes and summarizes what has been said. Here the text is sufficiently clear.