Commentary Aristotle Posterior Analytics
b6. Further, that the synthesis of them constitutes the substance of triad is shown by the following argument. If it is not identical with the being of triad, it must be related to triad as a genus named or nameless. It will then be of wider extent than triad — assuming that wider potential extent is the character of a genus. If on the other hand this synthesis is applicable to no subject other than the individual triads, it will be identical with the being of triad, because we make the further assumption that the substance of each subject is the predication of elements in its essential nature down to the last differentia characterizing the individuals. It follows that any other synthesis thus exhibited will likewise be identical with the being of the subject.
So let us now discuss the method. After showing how the 'quod quid' and 'propter quid' are related to demonstration, the Philosopher now shows how they can be investigated.
First, how the 'quod quid' should be investigated.
Second, how the 'propter quid' ought to be investigated, at: in order to formulate (99a1; [560]).
Concerning the first he does two things.
First, he states his intention.
Second, he pursues it, at: now, of the attributes (97a24; [528]).
He says therefore first (97a22) that after stating how 'quod quid' is recognized and how the 'quod quid' or 'propter quid' is taken as a middle in demonstration, we must now point out how to investigate those items which are predicated in 'quod quid'.
Then, at: now, of the attributes (97a24) he states his proposal.
First, he indicates what characteristics should be present in things which are accepted as constituting the 'quod quid'.
Second, how to investigate them, at: the author of a hand-book (97b15; [536]).
Concerning the first he does three things.
First, he presents a certain division.
Second, he proposes what should be the characteristics of things which are taken as constituting 'quod quid', at: it is such attributes (97a32; [530]).
Third, he proves it, at: now, since we have shown (97b2; [534]).
In regard to the first (97a22) it should be noted that things predicated in quod quid must be such that they are predicated always and universally, as has been established above. Then taking those things which are predicated of each thing in the sense of always, he says that we find among them certain ones which apply to more than that in which they are present, but not to the extent of being found outside that genus. (He explains what is meant by apply to more, saying that those things are said to apply to more which are indeed universally present in something, but are not solely in it but in other things also). This would imply that there is another and opposite member which applies to more, but is outside the genus. He gives an example of the first one, saying that there is something which is present in every three, as well as in non-three’s, as is obviously true of being, which is universally present not only in three but in other things; and not only in the genus of number, but even in things outside the genus of number. Odd, however, is found in every three and in more things, because it is found also in five’s; however, it is not found outside the genus of three, namely, number, because even five is in the genus of number. For nothing outside the genus of number can be called odd.
Then, at: it is such attributes (97a32) he shows what should be the characteristics of things that are taken as constituting the quod quid.
First, he proposes his intention.
Second, he clarifies it with examples, at: for example, every triad (97a34; [531]).
He says therefore first (97a32) that in order to manifest the 'quod quid', we must take items which are both always and applicable to more (but not outside the genus), until the term is reached. And they should be so selected that each item when first taken should be applicable to more, but when all are taken together the combination does not apply to more, but is converted with the thing whose 'quod quid' is sought. For the 'quod quid' of a thing must signify its essence.
Then, at: for example, every triad (97a34) he manifests what he had said with an example.
Let us, therefore, take those four, namely, number, odd, and prime in both its senses. For there are two senses in which a number is said to be prime:
in one way because it is not divided by any other number, as opposed to four, which is not a prime number, since it can be divided by two; three, however, is a prime number, because it is not divided by any other number except one.
In another way a number is called prime because it is not composed of other numbers, as opposed to seven, which is prime in the first way, because it is not divided by any other number except one, but is not prime in the second way, for it is composed of three and four.
Three, however, is not composed of several numbers, but only of the number two, and one. And so it is obvious that each of the four aforesaid notions belongs universally to every three, although each of them is also found in other things in the genus of number. For number and odd are found in all odd numbers; but the last, that is, being prime in both ways, belongs also to two, which is neither divided by any other number nor composed of numbers, but only of units; hence, when they are all assembled they signify the 'quod quid' of three.
But the requirement that each particle of the definition apply to more than the definition seems superfluous.
For the Philosopher says in Metaphysics VII that when the ultimate differences are reached, those differences will be equal to the species; therefore, it is not required that the difference apply to more things than the species does.
This can also be proved with an argument. For the Philosopher says in Metaphysics VIII that a formality which is based on differences seems to be of the species and of the act, that is, of the form; because, as he says in the same place, the difference corresponds to the form. But each species has its own appropriate form which belongs to no other species. Therefore, it seems that the ultimate difference does not exceed the species.
Furthermore, the Philosopher in Metaphysics VII says that there is no more in a definition than genus and differences, and that it is possible for a definition to be formed of two things, one of which is a genus and the other a difference. But a difference cannot be found outside its appropriate genus; otherwise it would not divide the genus per se but per accidens. Therefore, it seems that the difference does not exceed the species.
But it should be answered that if one were able to discover the difference which would make known the substantial form of the species, then, as the arguments prove, the ultimate difference would not apply to more things than the species does. But because the essential forms are not known to us per se, they must be disclosed through certain accidents which are signs of that form, as is stated in Metaphysics VIII. However, one should not take the proper accidens of that species, because they are the ones that will be demonstrated by the definition of the species; rather the form of the species must be made known by certain accidents that are more common. Hence according to this, the differences which are used are indeed called substantial, inasmuch as they are adduced in order to declare the essential form; but they are more common than the species, inasmuch as they are taken from signs which follow upon higher genera.
Then, at: now, since we have shown (97b2) he explains what he had said above.
First, the statement that the items mentioned above should be predicated universally and necessarily of three.
Second, that the very essence of three is constituted by those items, at: further, that the synthesis (97b6; [535]).
He says therefore first (97b2) that since it has been established above that items which are predicated in 'quod quid' are present of necessity, and whatever is present of necessity is predicated universally, it follows that whether those items which are taken in the manner above indicated be predicated as quod quid of three or of any other thing, it is necessary that they be predicated necessarily and universally.
Then, at: further, that the synthesis (97b6) he shows that from things which are taken in the abovementioned way the essence of three or of anything else is constituted;
for it is necessary, if the items mentioned above are not the substance of three, that they be its genus, either named or unnamed, since they are predicated in quod quid. (For not every formality has a name. That is why there are many genera and species without names. However, the reason why the above formality is the genus of three, if it does not signify its essence, is that whatever is predicated in quid is either the genus or the definition signifying the essence.) Yet it cannot be the genus; otherwise it would apply to more things than three’s: for we assume a genus to be something which potentially contains several species under it. But we have established that the aforesaid formality applies only to the atoms, that is, to the individuals contained under three. What remains, therefore, is that the formality in question is the definition signifying the essence of three. For the essence of a thing is supposed to be that which is found in the individuals of that species ultimately, according to the manner of predication described above. And what has been said of three is also understood of any other things regarding which something is demonstrated to be the same in the manner indicated above.
Lecture 14
Dividing the genus to investigate which items should be put in a definition
b15. The author of a hand-book on a subject that is a generic whole should divide the genus into its first infimae species — number e.g., into triad and dyad — and then endeavor to seize their definitions by the method we have described — the definition, for example, of straight line or circle or right angle.
b18. After that, having established what the category is to which the subaltern genus belongs — quantity or quality, for instance — he should examine the properties ‘peculiar’ to the species, working through the proximate common differentiae. He should proceed thus because the attributes of the genera compounded of the infimae species will be clearly given by the definitions of the species; since the basic element of them all is the definition, that is, the simple infimae species, and the attributes inhere essentially in the simple infimae species, in the genera only in virtue of these.