Commentary Aristotle Posterior Analytics
Lecture 35
That there is not an infinite process upward or downward in predicates is shown analytically a8. But an analytic process will show more briefly that neither the ascent nor the descent of predication can be infinite in the demonstrative sciences which are the object of our investigation.
a10. Demonstration proves the inherence of essential attributes in things.
a11. Now, attributes may be essential for two reasons: either because they are elements in the essential nature of their subjects, or because their subjects are elements in their essential nature. An example of the latter is odd as an attribute of number—though it is number’s attribute, yet number itself is an element in the definition of odd; of the former, multiplicity or the indivisible, which are elements in the definition of number.
a18. In neither kind of attribution can the terms be infinite. They are not infinite where each is related to the term below it as odd is to number, for this would mean the inherence in odd of another attribute of odd in whose nature odd was an essential element: but then number will be an ultimate subject of the whole infinite chain of attributes, and be an element in the definition of each of them. Hence, since an infinity of attributes such as contain their subject in their definition cannot inhere in a single thing, the ascending series is equally finite.
a23. Note, moreover, that all such attributes must so inhere in the ultimate subject—e.g., its attributes in number and number in them—as to be commensurate with the subject and not of wider extent.
a25. Attributes which are essential elements in the nature of their subjects are equally finite: otherwise definition would be impossible. Hence, if all the attributes predicated are essential and these cannot be infinite, the ascending series will terminate, and consequently the descending series too.
a28. If this is so, it follows that the intermediates between any two terms are also always limited in number.
a29. An immediately obvious consequence of this is that demonstrations necessarily involve basic truths, and that the contention of some — referred to at the outset — that all truths are demonstrable is mistaken. For if there are basic truths, (a) not all truths are demonstrable, and (b) an infinite regress is impossible; since if either (a) or (b) were not a fact, it would mean that no interval was immediate and indivisible, but that all intervals were divisible. This is true because a conclusion is demonstrated by the interposition, not the apposition, of a fresh term. If such interposition could continue to infinity there might be an infinite number of terms between any two terms; but this is impossible if both the ascending and descending series of predication terminate; and of this fact, which before was shown dialectically, analytic proof has now been given.
But an analytic process. After showing logically that there is no infinite process upwards or downwards in predicates, the Philosopher now shows the same thing analytically.
And his treatment falls into two parts. In the first he shows the principal proposition. In the second he infers certain corollaries from the aforesaid, at: it is an evident corollary (84b3) [L. 36].
Concerning the first he does two things. First, he proposes what he intends. Second, he proves his proposition, at: demonstration proves the inherence (84a10).
He says therefore first (84a8) that the fact an infinite process upwards or downwards does not occur in the demonstrative sciences with which we are concerned can be more briefly and quickly manifested analytically than it was logically.
Here we might note that analytic, that is, demonstrative, science which is called judicative, because it resolves to self-evident principles, is a part of logic which even contains dialectics under it. However, it pertains to logic in general to consider predication universally, that is, as containing under it predication which is per se and predication which is not per se. But predication per se is proper to demonstrative science. Therefore, above he proves his proposition logically, because he showed universally in every genus of predication that there is no infinite process. But here he intends to show it analytically, because he proves it only in things which are predicated per se. And this is a more efficient way; furthermore, it suffices for our purpose, because that is the only mode of predication we use in demonstration.
Then, at: demonstration proves the inherence (84a10) he shows his proposition.
About this, he does three things.
First, he proposes which predication analytic, that is, demonstrative, science employs, for it uses per se predication.
Second, he recalls how many modes there are of such predication, at: now, attributes may be essential (84a11; [301]).
Third, he shows that there cannot be an infinite process in any mode of per se predication, at: in neither kind of attribution (84a18; [302]).
He says therefore first (84a10) that demonstration is concerned exclusively with items that are per se in things. For such are its conclusions and from such does it demonstrate, as was established above.
Then, at: now, attributes may be essential (84a11) he lays down two modes of predicating per se. For in the first place those things are predicated per se which are present in their subjects as constituting their essence, namely, when predicates are placed in the definition of a subject. Second, when the subjects themselves are in the essence of the predicate, that is, when the subjects are placed in the definition of the predicate. And he gives examples of each of these ways: for odd is predicated per se of number in the second way, because “number” is placed in the definition of odd. For the odd is a number not divisible by two. Multitude or divisible, however, are predicated of number and are present in its definition; hence these are predicated per se of number in the first way. The other ways, which he mentioned previously, are reduced to these.
Then, at: in neither kind of attribution (84a18) he shows that there must be a stop in each of these modes of per se predication.
In regard to this he does three things.
First, he shows that there must be a stop both upwards and downwards in both of these modes of per se predication.
Second, he concludes that there cannot be an infinitude of middles, at: if this is so, it follows that (84a28; [306]).
Third, he concludes that one cannot proceed to infinity in demonstrations, at: an immediately obvious (84a29; [307]).
Concerning the first he does two things.
First, he shows his proposition in regard to the second way of saying per se, namely, when the subject is placed in the definition of the predicate.
Second, in the first way, when the predicate is placed in the definition of the subject, at: attributes which are essential (84a25; [305]).
In regard to the first he gives two reasons.
The first of these he treats of in the following way. First (84a18) he states his proposition, namely, that in neither of these modes of saying per se does an infinite process occur. Then he proves this of the second mode, as when “odd” is predicated of number. For if one goes further and states that something else should be predicated per se of “odd” according to that mode of saying per se, it follows that “odd” is present in its definition. But “number” is placed in the definition of odd; hence it will follow that “number” is also present in the definition of that third thing which is present per se in “odd.” However, this cannot go on to infinity, so that an infinitude of things would be in the definition of something, as was established above. It remains, therefore, that in such per se predications an upward process to infinity does not occur, that is, on the side of the predicate.
He presents the second reason, at: note, moreover, that all (84a23).
He says that no matter how far one advances in these per se predications of the second mode, it will be required that all the predicates taken in order be in their first subject, say in number, as predicated of it, because if “odd” is predicated per se of number, it will be required that whatever is predicated of odd be also predicated of number. And it is further required that “number” be in all of them, because if “number” is placed in the definition of odd, it has to be placed in the definition of all those things which are defined by “odd.” And thus it follows that they are mutually in one another. Therefore, they will be convertible and none of wider extent than another: for this is the way proper attributes are related to their subjects. Hence, even though there be an infinitude of per se predicates in this way, it offers nothing to the purpose of one who intends to establish that there is an infinitude of predicates upwards or downwards.
Then, at: attributes which are essential (84a25) he proves his own point in regard to the first mode of saying per se, and he states that those things which are predicated in essence, that is, as pertaining to the definition of the subject, cannot be infinite; otherwise, definition would be impossible, as we showed above. From this, therefore, he concludes that if all the items predicated in demonstrations are predicated per se, and if there is no infinite upward process in per se predicates, it is necessary that the predicates in demonstrations stop in the upward movement. And from this it also follows that they must stop in the downward movement, because no matter on which side infinity is posited, science and definition are destroyed, as is evident from what has been said above.
Then, at: if this is so, it follows that (84a28) he concludes from the foregoing that if there is a stop upwards and downwards, the middles cannot be infinite. For it has been established above that if the extremes are determinate, there cannot be an infinitude of middles.
Then, at: an immediately obvious consequence (84a29) he further concludes that there is no infinite process in demonstrations.
And he says that if the above statements are true, it is necessary that there be certain first principles of demonstration that are not demonstrated; consequently, there will not be demonstration of everything, as some claim, as was stated in the beginning of this book. Then he goes on to show that his consequence follows. For if it is granted that there are certain principles of demonstrations, it is necessary that they be indemonstrable: for since every demonstration proceeds from things that are prior, as has been established above, then if the principles are demonstrated, it will follow that something would be prior to the principles, and this is contrary to the notion of a principle. And so, if not all things are demonstrable, it will follow that demonstrations do not proceed to infinity. But all these follow from what has been established, namely, from the fact that there is no infinite process in middles, because it is no different to say that any of the foregoing statements is true (namely, that demonstrations proceed to infinity, or that all things are demonstrable, or that there are no principles of demonstrations) than it is to say that no distance is immediate and indivisible, that is, to the position that the two terms of any affirmative or negative proposition belong together only in virtue of a middle. For if any proposition is immediate, it follows that it is indemonstrable; because when something is demonstrated, it is necessary to take a term by interposing, that is, by setting it between the subject and predicate, so that the predicate will be predicated of that term before being predicated of the subject—or removed from it. But the middle in demonstrations is not taken by assuming extraneously; for this would be to assume an extraneous middle and not a proper middle—which occurs in contentious and dialectical syllogisms. Therefore, if demonstrations were to proceed to infinity, it would follow that there is an infinitude of middles between two extremes. But this is impossible if, as has been established above, the predications stop in the upward and downward process. But as we have shown, first logically and then analytically, these predications do stop both upwards and downwards, as explained.
Therefore, in virtue of this conclusion finally induced, he manifests the intent of the entire chapter and why each proposition was introduced.