Commentary Aristotle Posterior Analytics
Lecture 31
Three questions about proceeding to infinity in confirming demonstrations b10. Every syllogism is effected by means of three terms. One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C; the other is negative and one of its premises asserts one term of another, while the other denies one term of another.
b14. It is clear, then, that these are the principles and so-called hypotheses of syllogism. Assume them as they have been stated, and proof is bound to follow—proof that A inheres in C through B, and again that A inheres in B through some other middle term, and similarly that B inheres in C.
b18. If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premises as credible as possible: so that if a middle term between A and B is credible though not real, one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes.
b24. Thus, since there are attributes which are predicated of a subject essentially or naturally and not coincidentally—not, that is, in the sense in which we say ‘That white (thing) is a man’, which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one substratum—therefore there are terms such as are naturally subjects of predicates.
b30. Suppose, then, C is such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute B—that is, so that B-C is immediate; suppose further E related immediately to F, and F to B. The first question is, must this series terminate, or can it proceed to infinity?
b34. The second question is as follows: Suppose nothing is essentially predicated of A, but A is predicated primarily of H and of no intermediate prior term, and suppose H similarly related to G and G to B; then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates, and descend to infinity?
a2. A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that A inheres in C and B is intermediate between them, but between B and A there are other middles, and between these again fresh middles; can these proceed to infinity or can they not?
a7. This is the equivalent of inquiring, do demonstrations proceed to infinity, that is, is everything demonstrable? Or do ultimate subject and primary attribute limit one another?
a9. I hold that the same questions arise with regard to negative conclusions and premises: viz. if A is attributable to no B, then either this predication will be primary, or there will be an intermediate term prior to B to which A is not attributable—G, let us say, which is attributable to all B—and there may still be another term H prior to G, which is attributable to all G.
The same questions arise, I say, because in these cases too either the series of prior terms to which A is not attributable is infinite or it terminates.
a15. One cannot ask the same questions in the case of reciprocating terms, since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals qua subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes—and we raised the question in both cases—are infinite in number. These questions then cannot be asked—unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other.
Every syllogism is effected. After the Philosopher has determined concerning the demonstrative syllogism by showing from what and what sort of things it proceeds and in which figure demonstrations can be formed, he now inquires whether demonstrations can proceed to infinity.
First, he raises the question.
Second, he settles it, at: now, it is clear that if the predications (82a21).
Concerning the first he does two things.
First, he sets down certain prefatory remarks needed for understanding the question.
Second, he raises the question, at: suppose, then, C is such a term (81b30).
Concerning the first he does two things.
First, he prefaces something about the syllogistic form one must observe in demonstrations.
Second, he reviews what the matter of demonstration should be, at: it is clear, then, that these (81b14).
With respect to the first (81b10) he touches on three things, the first of which is common to every syllogism, namely, that every syllogism is formed in three terms, as is indicated in Prior Analytics I. The second however, pertains to an affirmative syllogism whose form is such that it concludes A to be in C, because A is in B and B in C. And this is the form of a syllogism in the first figure in which alone can be concluded a universal affirmative, the chief quest in demonstration. The third pertains to a negative syllogism which of necessity has one affirmative proposition and one negative, but differently in the first figure and in the second, as is clear from what has been shown in the Prior Analytics.
Then, at: it is clear, then, that these (81b14) he reviews what the matter of demonstration should be.
In regard to this he does three things.
First, he states what this matter is.
Second, he shows the difference between this matter and the matter of a dialectical syllogism, at: if our reasoning aims (81b18).
Third, he clarifies this difference, at: thus, since there are attributes (81b24).
He says therefore first (81b14) that since a syllogism has three terms from which are formed the two propositions which conclude the third, it is clear that these propositions, from which one proceeds in a demonstrative syllogism according to the aforesaid form, are the principles and suppositions we discussed earlier. For one who accepts such principles thus demonstrates through them in the syllogistic form we have mentioned, namely, that A is in C is proved through B; and if the proposition AB is again mediate, another middle is used to demonstrate that A is in B. The like is done if the minor proposition, BC, is mediate.
Then, at: if our reasoning aims (81b18), apropos of what has been said, he shows the difference between a demonstrative and a dialectical syllogism.
For since the latter aims at producing opinion, the sole intent of a dialectician is to proceed from things that are most probable, and these are things that appear to the majority or to the very wise. Hence if a dialectician in syllogizing happens upon a proposition which really has a middle through which it could be proved, but it seems not to have a middle because it appears to be per se known on account of its probability, this is enough for the dialectician: he does not search for a middle, even though the proposition is mediate. Rather he syllogizes from it and completes the dialectical syllogism satisfactorily. The demonstrative syllogism, on the other hand, is ordained to scientific knowledge of the truth; accordingly, it pertains to the demonstrator to proceed from truths which are really immediate. Hence if he happens upon a mediate proposition, he must prove it through its proper middle until he reaches something immediate, because he is not content with the probability of a proposition.
Then, at: thus, since there are attributes (81b24) he elucidates what he has said, asserting that his claim that the demonstrator proceeds to the truth from things that are per se is supported by the fact that it is possible to find something which is not predicated of a thing accidentally. What this means he explains by showing how in the case of affirmative statements something is predicated accidentally. For something is predicated accidentally in two ways: in one way, when the subject is predicated of an accident, as when we say, “The white thing is a man”; in another way, when the accident is predicated of the subject, as when we say, “The man is white.” Now, this way differs from the first, because when the accident is predicated of the subject, it is stated that the man is white not because something else is white but because the man himself is white. Yet it is a per accidens proposition, because “white” does not belong to man according to the specific nature of man. For neither is placed in the definition of the other. But when it is stated that the white thing is a man, this is not so stated because being a man is in the whiteness, but because being a man is in the subject of whiteness, which subject happens to be white. Hence this way is further removed from per se predication than the first. But there are certain things which are not predicated per accidens in either of these ways: these are said to be per se. Such are the things from which the demonstrator proceeds. But the dialectician is not so demanding; consequently, the question concerning such things as are predicated per se is not relevant to the dialectical syllogism but only to the demonstrative syllogism.
Then, at: suppose, then, C is such a term (81b30) he raises the questions he intended.
Concerning this he does two things.
First, he raises the questions in regard to things to which they are relevant.
Second, he shows the cases in which they are not relevant, at: the same questions arise (82a15).
Concerning the first he does two things.
First, he raises questions in affirmative demonstrations.
Second, he shows that these questions also have relevance in negative demonstrations, at: I hold that the same questions (82a9).
First, he raises the questions.
Second, he shows where such questions are relevant, at: this is the equivalent of inquiring (82a7).
In regard to the first (81b30) he raises three questions corresponding to the three terms of a syllogism.
First, he raises a question concerning the major extreme, namely, whether one can go to infinity in ascending order. In this question an ultimate subject is supposed which is not predicated of any other, but other things are predicated of it. Let this subject be C, and let B be in C first and immediately, and let E be in B, as universally predicated of B; furthermore, let Z be in E as universally predicated of it. The question is this: Should this ascending process come to a halt somewhere, so that something is reached which is predicated universally of other things but nothing else is predicated of it, or is that not necessary but a process to infinity occurs?
Second, at: the second question is as follows (81b34) he raises the question on the part of the minor term, namely, whether one can go to infinity in descending. In this question some first universal predicate is supposed which is predicated of other things and nothing is more universal than it so as to be predicated of it. Thus let A be such that nothing is predicated of it as a universal whole of a part, but A is predicated of H both first and immediately, and H of G, and G of B. The question then is this: Is it necessary to come to a halt in this descending process, or may it proceed to infinity?
Then he shows the difference between these two questions. For in the first one we asked: If someone begins from a most particular subject which is in nothing else the way a universal whole is in a part but other things are in it, does an infinite ascending process occur? But in the second we are asking: If someone begins with a most universal predicate, which is predicated of other things as a universal whole of its parts but nothing is predicated of it in this way, does an infinite descending process occur?
Third, at: a third question is (82a2) he raises the third question on the part of the middle term. In this question two determinate extremes are supposed, namely, a most universal predicate and a most particular subject. The question is whether under these conditions there can be an infinity of middles. Thus, if A is the most universal predicate and C the most particular subject, and if between A and C there is the middle, B, and again between A and B another middle, and likewise between B and C; furthermore, if there are other middles of these middles between them and the extremes both in ascending and in descending order. The question then is this: May these processes go on to infinity or is that impossible?
Then, at: this is the equivalent of inquiring (82a7) he shows what is the tenor of these questions. And he says that these questions pertain to the matter now under discussion, namely, to demonstrations.
He says, therefore, that the attempt to reach true answers to these questions is the same as trying to settle the question whether demonstrations proceed to infinity by ascending or descending. By ascending, that is, so that each proposition from which a demonstration proceeds would be demonstrable by another prior proposition. (This is what he means when he asks: is there a demonstration of everything, that is, of every proposition? By so thinking, some have erred in regard to principles, as is stated in Metaphysics IV.) And by descending, that is, whether it is possible from any demonstrated proposition to proceed again to another demonstration subsequent to it. Thus, one element of the doubt is whether demonstrations proceed to infinity either by ascending or by descending. The other element is whether demonstrations are mutually limiting, so that one demonstration may be confirmed by another in an ascending process, and from one demonstration another may proceed by a descending process: and this until a limit is reached.
Then, at: I hold that the same questions arise (82a9) he shows that these questions are also relevant to negative demonstrations, because a negative demonstration must employ an affirmative proposition in which the subject of the conclusion is contained under the middle and from which the predicate of the conclusion is removed. Therefore, to the extent that there is ascent and descent in affirmative, there must be ascent and descent in negative syllogisms and propositions. For example, if the conclusion of a demonstrative syllogism is, “No C is A,” and the middle taken is B, from which A is removed: the first thing to be considered, therefore, is whether A is removed from B first and immediately, or whether there is another middle I to be taken, from which A would be removed before it would be removed from B. In that case it would again be necessary to consider whether A would be removed from something else before I, namely, from T which is predicated universally of I. Therefore, in these also the question arises whether one can proceed to infinity in removing (so that something would always remain from which it would have to be removed), or must one stop somewhere.
Then, at: the same questions arise (82a15) he shows the cases in which these questions have no relevance.
In cases in which there is mutual predication and mutual conversion there is no prior and subsequent to be taken in the sense in which the prior [notion] is that with which a subsequent [notion] is not convertible, as universals are prior; because no matter whether the predicates be infinite, so that one might proceed to infinity in predicating, or whether there be infinity on both sides, that is, on the side of the predicate as well as of the subject, all such infinites bear a like relationship to all, because any of them could be predicated of any other and be the subject of any of the convertibles. However, there can be this difference: one of them might be predicated as an accident and another as a predicament, that is, as a substantial predicate.
And this is the difference between a property and a definition: although the two are convertible with the subject, nevertheless the definition is an essential predicate and therefore naturally prior to the property, which is an accidental predicate. That is why in demonstrations we use the definition as the middle to demonstrate a proper attribute of the subject.