Commentary Aristotle Posterior Analytics
Lecture 32
Solution of some of these doubts hinges upon solution of others a21. Now, it is clear that if the predications terminate in both the upward and the downward direction (by ‘upward’ I mean the ascent to the more universal, by ‘downward’ the descent to the more particular), the middle terms cannot be infinite in number.
a24. For suppose that A is predicated of Z, and that the intermediates — call them B, B’, B”…—are infinite, then clearly you might descend from and find one term predicated of another ad infinitum, since you have an infinity of terms between you and Z; and equally, if you ascend from Z, there are infinite terms between you and A. It follows that if these processes are impossible there cannot be an infinity of intermediates between A and Z.
a30. Nor is it of any effect to urge that some terms of the series AB...F are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series B...I take, the number of intermediates in the direction either of A or of Z must be finite or infinite: where the infinite series starts, whether from the first term or from a later one, is of no moment, for the succeeding terms in any case are infinite in number.
a37. Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by ‘ultimate term’ I mean a term such as Z, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by ‘primary term’ I mean a term predicable of a subject but not itself a subject). If this assumption is justified, the series will also terminate in the case of negation.
b4. For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no B is A, all C is B. In packing the interval B-C we must reach immediate propositions—as is always the case with the minor premise—since B-C is affirmative. As regards the other premise it is plain that if the major term is denied of a term D prior to B, D will have to be predicable of all B, and if the major is denied of yet another term prior to D, this term must be predicable of all D. Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which A is primarily non-predicable.
b13. In the second figure the syllogism is, all A is B, no C is B, no C is A. If proof of this is required, plainly it may be shown either in the first figure as above, in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all B is D, no C is D..., since it is required that B should be a subject of which a predicate is affirmed. Next, since D is to be proved not to belong to C, then D has a further predicate which is denied of C. Therefore, since the succession of predicates affirmed of an ever higher universal terminates, the succession of predicates denied terminates too.
b23. The third figure shows it as follows: all B is A, some B is not C. Therefore, some A is not C. This premise, that is, C-B, will be proved either in the same figure or in one of the two figures discussed above. In the first and second figures the series terminates. If we use the third figure, we shall take as premises, all E is B, some E is not C, and this premise again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also.
b28. Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third—even so the regress will terminate, for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite. Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration.
Now, it is clear that if the predications. After raising the questions, the Philosopher here begins to settle them.
And his treatment is divided into two parts. In the first he shows that the solution of some of the doubts is reduced to the solution of others. In the second he settles the doubt as to those items in which the difficulty lies per se as in its source, at: that in fact the regress (82b34) [L. 33].
Concerning the first he does two things. First, he shows that the doubt bearing on the middles is reduced to the one which is concerned with the extremes and is solved by the solution of the latter. Second, he shows that the doubt bearing on negative demonstrations is reduced to the one which is concerned with affirmative demonstrations, at: further, if in affirmative demonstration (82a37).
In regard to the first he does three things. First, he states his intended proposition. Second, he proves this proposition, at: for suppose that A is predicated of Z (82a24). Third, he excludes a subterfuge, at: nor is it of any effect to urge (82a30).
He says therefore first (82a21) that it will be plain to anyone who considers the following reason that an infinity of middles does not occur, if the predications both upwards and downwards stop at certain terms, namely, at the highest predicate and the lowest subject. And he explains what upward and downward predication consists in, saying that one proceeds upwards when there is movement to the more universal, one of whose marks it is that it be predicated; but one proceeds downwards when there is movement to the most particular, one of whose marks it is that it functions as a subject.
Then, at: for suppose that A is predicated of Z (82a24) he shows what he has proposed in the following way: Let the case be that A is the highest predicate and Z the lowest subject, and that there is an infinitude of middles, each of which we shall call B. Now, since A was the first predicate, it will be predicated of some middle near it, and that middle of another middle below it. Since the middles are infinite, it follows that the predication will proceed downwards to infinity—which is contrary to what we are assuming. For it was assumed that the predications do not proceed downwards to infinity. The result is the same if we start at Z, which is the lowest subject, and proceed upwards to infinity before A is reached—this too would be against our assumption. Therefore, if these are impossible, namely, that one may proceed to infinity by ascending or by descending, it will be impossible for the middles to be infinite. Thus it is clear that the question of an infinity of middles is reduced to a question of the infinity of the extremes.
Then, at: nor is it of any effect (82a30) he excludes an objection. For someone might object saying that the aforesaid proof would hold if ABC, that is, the middle and the extremes, were so related as to be had to one another, that is, so that there would be no middle between them: for this is the way the Philosopher defines 'had' in Physics V, namely, that it is next to something without anything between. And this seemed to be supposed in the above proof, namely, that A is predicated of some middle as though had to it, that is, following it immediately. But one who posits infinite middles will say that this cannot be supposed. For he will say that between any two terms that are taken there is a middle.
But the philosopher says that it makes no difference whether the infinitude be of middles that are “had” to one another, the way discrete things are (for example, in a city, house is consecutive to house, and in numbers, unity to unity), or whether something “had” cannot be found in the middles although between any two middles it is always possible to find another, as happens in continua, in which, between any two signs, that is, between two points, another can be found between them. That this makes no difference one way or the other to the matter at hand he manifests in the following way: Granted that between A and Z there is an infinitude of middles, each of which is called B, yet no matter which of these I employ, there is either an infinitude of middles between it and A and Z, or there is not an infinitude of them between it and one or the other of the extremes. (For example, let us suppose that the middles are mutually “had,” as happens in discrete things, and let us take a middle which is “had” to A; then it will be necessary that between that middle and Z there is still an infinitude of middles; and similarly, if we assume a certain finitude of middles between that middle and A. And the same reasoning holds if the middle which is taken be joined immediately to Z or is distant from it by a finitude of middles.) From the fact, therefore, that from any given middle one must take an infinitude of middles to one or other of the extremes, it makes no difference whether it is joined to either extreme immediately, that is, without a middle, or not immediately, that is, through other middles: because even if it be joined to one extreme without a middle, it will still be necessary later to find an infinitude of middles in relation to the other. Consequently, it will always be required, if there is an infinitude of middles, to proceed to infinity in predications either by ascending or by descending, as the above proof showed.
Then, at: further, if in affirmative demonstration (82a37) he shows that if there is no process to infinity in affirmative demonstrations, then neither in negative demonstrations: and thus the question of negative demonstrations is reduced to the question of affirmative ones.
He does three things in regard to this point. First, he proposes what he intends. Second, he proves what he proposed, at: for a negative conclusion can be proved (82b4). Third, he excludes an objection, at: even supposing that the proof (82b28).
He says therefore first (82a37) that it will be clear from what follows that if in the predicative, that is, in the affirmative demonstration, a stop is made at both, that is, upwards and downwards, it will be necessary that a stop be made in the negative demonstration. To elucidate what he is proposing he says: Let the case be such that from the ultimate, that is, from the lowest subject one cannot go in ascending order to infinity toward universal predicates. And he explains that 'ultimate' means that which is not in any other as in a less particular, but something else is in it, and let it be Z. And let the case also be that one does not go to infinity when proceeding from the first to the ultimate. And he explains that the 'first' means that which is predicated of others but nothing else is predicated of it as more universal than it. Thus the 'first' is understood to be the most universal, and the 'ultimate' the most particular. If, therefore, on both sides there be a stop in affirmative demonstrations, he says that as a consequence, there is also a stop in negative demonstrations.
Then, at: for a negative conclusion can be proved (82b4) he proves his proposition.
First, in the first figure. Second, in the second, at: in the second figure (82b13). Third, in the third, at: the third figure shows (82b23). For a negative can be concluded in three figures.
He says therefore first (82b4) that there are three ways of demonstrating a negative proposition through which something is signified not to be.
In one way in the first figure according to the mode that B is universally in C in the universal affirmative minor, but A is in no B in the universal negative major. Now, since we are supposing that there is a stop in affirmatives, both upwards and downwards, it is necessary that the proposition, BC, which is affirmative, if it is not immediate but a space exists with middles between B and C, be reduced to immediates, because that space which exists between the middle term and the extreme is affirmative, in which a stop is supposed. But if we take the other space, which is between B and A, it is clear that if this proposition, “No B is A,” is not immediate, it is necessary that A be removed from something else before being removed from B. Let this be D. But if this D be taken as a middle between A and B, it is necessary that it be universally predicated of B, because the minor must be affirmative. And if this too is not immediate, that is, “No D is A” then A has to be denied of something prior to D, say E, which again will be predicated universally of D for the same reason. Therefore, since there is a stop in affirmatives when we ascend, as supposed, it follows that something is reached of which A should be denied first and immediately; otherwise one would go still further in affirmatives, as is clear from the foregoing.
Then, at: in the second figure (82b13) he proves the same thing for the negative which is concluded in the second figure. For let the case be that B, which is the middle, is predicated universally of A and denied universally of C, so that the conclusion is “No C is A.” Now, if the negative premise again needs to be demonstrated because it is mediate, it must be demonstrated either in the first figure in the mode of demonstrating concerning which we have shown that there is a stop, if there is a stop in affirmatives; or it must be demonstrated through this mode, that is, in the second figure, or through a third mode, that is, in the third figure. Now, it has been established that there is a stop in negatives of the first figure, if there is a stop in affirmatives. Consequently, the same will now be demonstrated as to the second figure. Therefore, let this proposition, “No C is B,” be demonstrated in such a way that D is universally predicated of B in the universal affirmative major and denied universally of C in the universal negative minor. Now, again, if the proposition, “No C is D,” is mediate, it will be required to take some other middle which will be predicated universally of D and universally removed from C. Continuing thus, it will be necessary to proceed in negative demonstrations just as we do in affirmatives, namely, B will be predicated of A and D of B, and something else of D, and so on to infinity in affirmatives. But because we are supposing an upward stop in the affirmatives, it is also necessary to come to a stop in the negatives according to this mode in which a negative is demonstrated in the second figure.
Then, at: the third figure shows it (82b23) he shows the same thing in the third figure. Therefore, let B be a middle of which A is universally predicated, but C is universally denied of it: the conclusion will be a particular negative, namely, C is denied of some A. Now, that there is a stop in the affirmative premise, “Every B is A,” is granted by our supposition. Furthermore, that there must be a stop in the negative, “No B is C,” which is the major, is evident, because if it had to be demonstrated, it would be done either through what was said above, that is, through the first and second figure or in the way that the conclusion was concluded, namely, through the third figure, in which case this minor is not affirmed as universal but as particular. But there is a stop in that way, that is, if one proceeds in the first and second figure. But if one proceeds in the third figure to conclude that “Some B is not C,” let a middle, E, be taken such that B is universally affirmed of it but C is particularly denied of it. Then this happens once more in like manner, that is, according to this, one will always proceed in the negative demonstration by accumulating affirmative predications in descending order, because B which was the first middle will be predicated of E, and E of something else, and so on to infinity. But since we are supposing that there is a stop in the descending order in affirmatives, it is clear that there will be a stop in the negatives on the part of C.
Then, at: even supposing that the proof (82b28) he excludes an objection. For someone could say that it is necessary to stop in negative propositions when there is a stop in the affirmatives, provided that one always syllogizes according to the same figure; but if one demonstrates now in one figure and now in another, one can go to infinity. And he says that it is obvious that if one does not limit himself to one figure in demonstrating but uses all, proceeding now in the first figure and now in the second and third, there must still be a stop in the negatives if there is one in the affirmatives. For these various ways of demonstrating are finite, and each of them will be enlarged not to infinity but finitely by ascending or descending, as was shown. Now, if finite things be taken a finite number of times, the result must be finite. Hence it remains that in all the modes there must be a stop in negative demonstrations, if there is a stop in the affirmatives.