Commentary Aristotle Posterior Analytics
Lecture 12
How error occurs in taking the universal
a4. We must not fail to observe that we often fall into error because our conclusion is not in fact primary and commensurately universal in the sense in which we think we prove it so.
a6. We make this mistake (1) when the subject is an individual or individuals above which there is no universal to be found:
(2) when the subjects belong to different species and there is a higher universal, but it has no name:
(3) when the subject which the demonstrator takes as a whole is really only a part of a larger whole; for then the demonstration will be true of the individual instances within the part and will hold in every instance of it, yet the demonstration will not be true of this subject primarily and commensurately and universally. When a demonstration is true of a subject primarily and commensurately and universally, that is to be taken to mean that it is true of a given subject primarily and as such.
a13. Case (3) may be thus exemplified. If a proof were given that perpendiculars to the same line are parallel, it might be supposed that lines thus perpendicular were the proper subject of the demonstration because being parallel is true of every instance of them. But it is not so, for the parallelism depends not on these angles being equal to one another because each is a right angle, but simply on their being equal to one another.
a17. An example of (1) would be as follows: if isosceles were the only triangle, it would be thought to have its angles equal to two right angles qua isosceles.
a18. An instance of (2) would be the law that proportionals alternate. Alternation used to be demonstrated separately of numbers, lines, solids, and durations, though it could have been proved of them all by a single demonstration. Because there was no single name to denote that in which numbers, lengths, durations, and solids are identical, and because they differed specifically from one another, this property was proved of each of them separately.
Today, however, the proof is commensurately universal, for they do not possess this attribute qua lines or qua numbers, but qua manifesting this generic character which they are postulated as possessing universally.
a25. Hence, even if one prove of each kind of triangle that its angles are equal to two right angles, whether by means of the same or different proofs; still, as long as one treats separately equilateral, scalene, and isosceles, one does not yet know, except sophistically, that triangle has its angles equal to two right angles, nor does one yet know that triangle has this property commensurately and universally, even if there is no other species of triangle but these. For one does not know that triangle as such has this property, nor even that ‘all’ triangles have it — unless ‘all’ means ‘each taken singly’: if ‘all’ means ‘as a whole class’, then, though there be none in which one does not recognize this property, one does not know it of ‘all triangles’.
a33. When, then, does our knowledge fail of commensurate universality, and when it is unqualified knowledge? If triangle be identical in essence with equilateral, that is, with each or all equilaterals, then clearly we have unqualified knowledge: if on the other hand it be not, and the attribute belongs to equilateral qua triangle; then our knowledge fails of commensurate universality.
a35. ‘But’, it will be asked, ‘does this attribute belong to the subject of which it has been demonstrated qua triangle or qua isosceles? What is the point at which the subject to which it belongs is primary? (that is, to what subject can it be demonstrated as belonging commensurately and universally?) Clearly this point is the first term in which it is found to inhere as the elimination of inferior differentiae proceeds. Thus the angles of a brazen isosceles triangle are equal to two right angles: but eliminate brazen and isosceles and the attribute remains. ‘But’ — you may say — ‘eliminate figure or limit, and the attribute vanishes.’ True, but figure and limit are not the first differentiae whose elimination destroys the attribute. ‘Then what is the first?’ If it is triangle, it will be in virtue of triangle that the attribute belongs to all the other subjects of which it is predicable, and triangle is the subject to which it can be demonstrated as belonging commensurately and universally.
We must not fail to observe. After specifying what the universal is, the Philosopher here shows how one might err in understanding the universal.
In regard to this he does three things.
First, he says that sometimes one might err in this matter.
Second, he tells in how many ways, at: we make this mistake (74a6; [100]).
Third, he gives the criterion for knowing whether the universal is being employed correctly, at: “but,” it will be asked, “does this attribute” (74a35; [108]).
He says therefore first (74a4), that in order to avoid mistakes in demonstrating, one should be aware of the fact that quite often something universal seems to be demonstrated, which is not being demonstrated.
Second, at: we make this mistake (74a6), he indicates the ways in which this mistake can occur.
And in regard to this he does two things.
First, he enumerates these ways and says that there are three possible errors in understanding a universal.
The first is likely to occur when under some common genus there is nothing else to take as the thing to which the universal initially applies than this singular, to which it is incorrectly applied. For example, if man were the only animal existing, and “sensible,” which is initially and per se in animal, were to be assigned as a primary universal to man.
(It should be noted that 'singular' is being used here in a wide sense for any inferior, in the way that a species might be called a 'singular' contained under a genus.)
Or we might say that it is not possible to find a genus with only one species: for a genus is divided into species through opposing differences. But if one contrary is found in nature, so must the other, as the Philosopher explains in On the Heavens II. Therefore, if one species is found, another will be found. However, one species is divided into distinct individuals by the division of matter. But it sometimes happens that all the matter proportionate to a given species is comprehended under one individual, so that in that case there is only one individual under one species. Hence it is significant that he did say, 'singular'.
The second way is when it is possible to take several inferiors under something common which is verified in things that differ in species, but that common item has no name. For example, if “animality” had no name, and “sensibility,” which is proper to animal, were to be assigned to the inferiors of animal (either collectively or distributively) as their primary universal.
The third way is when that of which something is demonstrated to be its primary universal is related to what is demonstrated of it, as a whole is related to a part. For example, if the power to see were assigned as a primary universal to animal: for not every animal can see. In this case the demonstration, that is, what should have been demonstrated, is “in those things which are in part,” that is, in some but not all of the things included under the subject; furthermore, it will be a demonstration “of all,” but not of all that the demonstration mentions. For the power to see can indeed be demonstrated universally of something, but not of animal universally, as of that to which it belongs primarily. And he explains why he says, “primarily,” namely, because a demonstration bears on what is both universal and first.
Second, at: if a proof were given (74a13), he gives examples of each of these ways.
First, of the third way, saying that if someone were to demonstrate of two straight lines that they do not intersect, that is, that they do not meet, it might seem that we have a demonstration of this sort, that is, one that bears on a primary universal, on the ground that “not to meet” is true of certain straight lines, and not that this happens only because the straight lines are equal, that is, equally distant. But if the lines should be equal, that is, equidistant, then “not to meet” belongs to any and all of them, because it is universally true that lines which are straight and equally distant, even should they be lengthened ad infinitum, will not meet.
Second, at: if isosceles were the only triangle (74a17), he gives an example of the first way, saying that if there were no triangle but the isosceles which is a triangle having two equal sides, it might seem that what is true of triangle as triangle should be true of isosceles as isosceles. But this would not be so.
Third, at: the law that proportionals alternate (74a18), he gives an example of the second way. Now, it seems that he saved this for last because he wished to spend more time on it.
Concerning it he does three things.
First, he gives the example.
Second, he draws a corollary from what he has said, at: hence, even if one prove (74a25; [106]).
Third, he gives a reason for the aforesaid, at: when, then, does our knowledge (74a33; [107]).
Concerning the first it should be noted that a ratio is a relation of one quantity to another, as 6 is related to 3 in the ratio of 2 to 1. The comparing of one ratio to another is a proportion, which, if it is disjoint, has four terms: for example, "as 4 is to 2, so 6 is to 3". But if it is joint, it has three terms, one of which is used twice: for example, "as 8 is to 4, so 4 is to 2". Now, it is obvious that in a proportion two terms are antecedents and two are consequents. For example, "in the proportion that 6 is to 3 as 4 is to 2", 6 and 4 are the antecedents, 3 and 2 the consequents. Again, a proportion is alternated by bringing the antecedents together and the consequents together. For example, when I say: "As 4 is to 2, so 6 is to 3"; therefore, "as 4 is to 6, so 2 is to 3".
What he is saying, therefore, is that alternate proportion is verified in lines, numbers, solids, (that is, bodies), and times. But just as this is established separately for each of them, namely, for numbers in arithmetic, for lines and solids in geometry, and for times in natural philosophy or astronomy, so it is possible to prove it of all of them in a single demonstration. But the reason why a separate demonstration was employed to prove alternation for each was that the one feature they had in common was unnamed.
For although quantity is a feature common to all of them, quantity includes other things besides them; for example, speech and other things that are accidentally quantitative.
Or, better still, it was because alternate proportion does not belong to quantity precisely as quantity, but as compared to another quantity according to a fixed ratio. That is why at the very start he spoke of alternate proportion. But there is no general name for the aforesaid things precisely as they are proportional.
Furthermore, when alternate proportion is demonstrated separately for each of them, it is not a universal that is being demonstrated. For “to be alternately proportional” is not in numbers and lines according to what each of them is, but according to something common. Besides, those who demonstrate alternate proportion of lines look upon this attribute as a universal predicate of line precisely as it is a line, or of number precisely as it is number.
Then, at: hence, even if one prove (74a25), he draws a corollary from the above and says that for the same reason that something universal is not being demonstrated when a common unnamed attribute is proved to be a universal predicate of each species, so too when something is demonstrated this way of a common attribute which does have a name.
For example, if someone uses the same demonstration for each species of triangle and proves that each has three angles equal to two right angles, or if he uses different demonstrations, say one for isosceles and another for scalene, even then he does not on that account know that a triangle has three angles equal to two right angles except in a sophistical way, that is, per accidens: for he does not know it of triangle as triangle, but as equilateral or as having two equal sides or as having three unequal sides. Furthermore, one who demonstrates in this way does not know the universal of triangle, that is, he does not have a knowledge of triangle in a universal way (even if there happens to be no other triangle besides those of which he knew this: and this because he had not the knowledge of triangle as triangle but under the aspect of its various species).
Again, strictly speaking, he did not know every triangle: for although he knew every triangle according to number (if there was none he did not know), yet according to species he did not know each one. For something is known universally according to species, when it is known according to the notion of the species, but according to number and not universally, when it is known according to the multifarious things contained under the species. And in this matter there is no difference whether we compare species to individuals or genera to species. For triangle is the genus of equilateral and isosceles.
Then, at: when, then, does our knowledge (74a33) he assigns the reason for the aforesaid, first asking when does one know universally and absolutely, if one who knew in the aforesaid way did not know universally.
And he answers that obviously if the essence of triangle in general and of each of its species (each taken separately or all taken together) were the same, then one would know universally and absolutely about triangle, when he knew about any one species of it separately or about all of them together. But if the essence is not the same, then it will not be the same but something different to know triangle and to know its several species. And by knowing something of the species one does not know it of triangle precisely as triangle.
Then, at: “but,” it will be asked, “does this attribute” (74a35), he gives the rule on how the universal can be properly understood, saying that whether something belongs to triangle precisely as triangle or to isosceles precisely as isosceles, and when it is that what is demonstrated is first and universal in the given subject precisely as this, that is, precisely according to the subject—all this will be clear from what I shall say. For whenever some item is removed from the subject and the original universal still applies to what remains, then it is not the first universal of that subject. For example, upon the removal of isosceles or of brazen there still remains the triangle which has three angles equal to two right angles; hence the possession of three angles equal to two right angles is not a first universal of isosceles or of brazen triangle. But if the item “figure” be removed, nothing which has three angles equal to two right angles remains, nor again if you remove “bounded” which is more universal than figure, since a figure is something enclosed by a bound or bounds. Nevertheless, the attribute in question [namely, having three angles equal to two right angles] does not belong first to figure or to bounded, because it does not belong to either of them universally.
To what then will it be first? Obviously to triangle, because it belongs to the others (both superiors and inferiors) precisely as they are triangles. For it belongs to figure to have three angles equal to two right angles only because a triangle is some figure, and similarly to isosceles, only because it is a triangle, and it is of triangle that “having three...” is universally demonstrated. Hence, it is to triangle that it belongs as a first universal.