Commentary Aristotle Posterior Analytics
Lecture 11
How something is said to be predicated as commensurately universal b27. I term “commensurately universal” an attribute which belongs to every instance of its subject, and to every instance per se and as such;
b28. from which it clearly follows that all commensurate universals inhere necessarily in their subjects.
b29. The essential attribute, and the attribute that belongs to its subject as such, are identical. e.g., point and straight belong to line per se, for they belong to line as such; and triangle as such has two right angles, for it is per se equal to two right angles.
b32. An attribute belongs commensurately and universally to a subject when it can be shown to belong to any random instance of that subject and when the subject is the first thing to which it can be shown to belong.
b34. Thus, e.g., (1) the equality of its angles to two right angles is not a commensurately universal attribute of figure. For though it is possible to show that a figure has its angles equal to two right angles, this attribute cannot be demonstrated of any figure selected at haphazard, nor in demonstrating does one take a figure at random — a square is a figure but its angles are not equal to two right angles. On the other hand, any isosceles triangle has its angles equal to two right angles, yet isosceles triangle is not the primary subject of this attribute but triangle is prior. So whatever can be shown to have its angles equal to two right angles, or to possess any other attribute, in any random instance of itself and primarily — that is the first subject to which the predicate in question belongs commensurately and universally, a1. and the demonstration, in the essential sense, of any predicate is the proof of it as belonging to this first subject commensurately and universally: while the proof of it as belonging to the other subjects to which it attaches is demonstration only in a secondary and unessential sense. Nor again (2) is equality to two right angles a commensurately universal attribute of isosceles; it is of wider application.
I term “commensurately universal." After determining about 'said of all' and 'said per se', the Philosopher here determines concerning the “universal.”
This treatment falls into two parts.
In the first he shows what the universal is.
Second, how error occurs in our understanding of it, at: we must not fail to observe (74a4; [98]).
Concerning the first he does two things.
First, he shows what the universal is.
Second, how the demonstrator uses the universal, at: and the demonstration (74a1; [97]).
Concerning the first he does two things.
First, he shows that the universal contains within itself the attributes of 'being said of all' and of being said per se'.
Second, he shows what the universal adds to them, at: an attribute belongs commensurately (73b33; [95]).
To understand what is being said here it should be noted that “universal” is not to be taken here in the sense that anything predicated of several is a universal, as when Porphyry treats of the five universals; rather “universal” is taken here according to a certain correspondence or commensurateness of the subject with the predicate, so that the predicate is not found outside the subject nor is the subject without the predicate.
With this in mind, it should be noted that he does three things with respect to the first point.
First (73b27), he says that the universal, namely, the predicate, is both verified of all, that is, is predicated universally of its subject, and is said per se, that is, is in and belongs to the subject according to the essential nature of the subject. For many things are said universally of certain things to which they do not belong per se and as such. Thus, every stone is colored, but not precisely as stone, but as it has a surface.
Second, at: from which it clearly follows (73b28), he draws a corollary from this and says that since the universal is something which is per se in a thing, and since it has been shown that whatever things are in something per se are in it of necessity, it is obvious that universal predicates, as they are being taken here, are necessarily present in the things of which they are predicated.
Third, at: the essential attribute (73b29), lest anyone suppose that 'per se' and 'precisely as such', both of which were mentioned in the definition of the universal, are different, he shows that they are the same. Thus, point is per se in line in the first way, and straight in the second way. For each is in line precisely as it is a line. In like manner, two right angles belongs to triangle precisely as triangle, that is, its angles are equal to two right angles, which is per se in triangle.
Then, at: an attribute belongs commensurately (73b33), he shows what “universal” adds to the notions, 'being said of all' and 'being said per se'.
In regard to this he does two things:
First, he says that a predicate is universal, when it is not only in each thing of which it is asserted, but it is demonstrated to be first or primarily in the thing which receives that predicate.
Second, at: the equality of its angles (73b34), he clarifies this with an example and says that “having three angles equal to two right angles” is not found in just any figure in general, although this could be demonstrated of some figure, because it is demonstrated of triangle, which is a figure; yet it is not found in any random figure, nor is just any figure used when it is demonstrated. For a rhombus is a figure, but it does not have three angles equal to two right angles. But an isosceles, that is, a triangle with two equal sides, always has its three angles equal to two right angles. Nevertheless, isosceles is not the primary thing to which this belongs, for it belongs basically to triangle, and belongs to isosceles precisely as it is a triangle. Therefore, whatever is demonstrated basically to have its three angles equal to two right angles (or whatever else be thus demonstrated), the universal predicate is present in it primarily, as in triangle.
Then, at: and the demonstration (74a1), he shows how a demonstrator uses the “universal,” saying that demonstration is concerned per se with such a universal, but with other things qualifiedly and not per se. For a demonstrator demonstrates a proper attribute of its proper subject; and if he demonstrates it of anything else, he does so only insofar as it pertains to that subject. Thus, he proves that some property of triangle belongs to a figure and to an isosceles precisely as some figure is a triangle, and as the isosceles is a triangle. But the reason why “having three” is not in isosceles primarily is not because it is not predicated of it universally, but because it is found more frequently, that is, in more things than in isosceles, since this is common to every triangle.