Commentary Aristotle Posterior Analytics
b25. Divisions according to differentiae are a useful accessory to this method. What force they have as proofs we did, indeed, explain above, but that merely towards collecting the essential nature they may be of use we will proceed to show. They might, indeed, seem to be of no use at all, but rather to assume everything at the start and to be no better than an initial assumption made without division.
b30. But, in fact, the order in which the attributes are predicated does make a difference—it matters whether we say animal-tame-biped, or biped-animal-tame. For if every definable thing consists of two elements and ‘animal-tame’ forms a unity, and again out of this and the further differentia man (or whatever else is the unity under construction) is constituted, then the elements we assume have necessarily been reached by division.
b35. Again, division is the only possible method of avoiding the omission of any element of the essential nature. Thus, if the primary genus is assumed and we then take one of the lower divisions, the dividendum will not fall whole into this division: e.g., it is not all animal which is either whole-winged or split-winged but all winged animal, for it is winged animal to which this differentiation belongs. The primary differentiation of animal is that within which all animal falls. The like is true of every other genus, whether outside animal or a subaltern genus of animal; e.g., the primary differentiation of bird is that within which falls every bird, of fish that within which falls every fish. So, if we proceed in this way, we can be sure that nothing has been omitted: by any other method one is bound to omit something without knowing it.
The author of a hand-book. After showing what should be the characteristics of the items which constitute the definition signifying the essence of a thing, the Philosopher now shows how they should be investigated.
Concerning this he does two things.
First, he proposes the most suitable method of investigating the items to be put in the definition, namely, by division of the genus.
Second, he sets forth another method, namely, by similarities and differences, at: to resume our account (98b7; [552]).
Concerning the first he does two things.
First, he shows that one should employ division of the genus for investigating the particles of the definition.
Second, he shows what to look for in such an investigation, at: in establishing a definition (98a23; [547]).
First, he shows the truth.
Second, he excludes an error, at: to define and divide (98a6; [542]).
Concerning the first he does three things.
First, he shows how the particles of a definition are investigated by the method of dividing the genus.
Second, how the process of division is useful for this task, at: divisions according to differentiae (97b25; [539]).
Third, how to avoid pitfalls which can invalidate this process, at: but, in fact, the order (97b30; [540]).
Concerning the first he does two things.
First, he shows that one should employ division of the genus for defining.
Second, how one should take the differences, at: after that, having established (97b18; [538]).
He says therefore first (97b15) that when someone wishes to deal with some whole, that is, a universal, in order to define it, it is recommended that he first divide the genus into the first parts of that genus, that is, those that are not further divisible into species; for example, he should divide number into two and three. Having accomplished this division through which the genus is known, he should then try to obtain the definition of each species as is done in other matters, say in the matter of straight line and circle and right angle. For all these are fittingly defined after one has divided the genus.
Then, at: after that, having established (97b18) he shows how to obtain the differences.
He says that after we have learned what the genus is by dividing the genus into its species, for example, whether it is in the genus of quality or quantity, the next step is to investigate the differences by considering the proper attributes which, as has been said, are signs manifesting the forms proper to the species. And this should first be done by means of certain common items. For if we assemble the accidents from the more common genera (which he here calls indivisibles, because they are not resolved into prior genera), then from their definitions the things we are searching for will be immediately obvious. For the basic element of all definitions must be something simple, that is, a common genus; furthermore, it is only in such simple things that the accidents inhere per se which are commonly found in many, but they are found in all other things in virtue of those simple genera.
For example, black and white belong per se to terminated body; and in virtue of this common characteristic they belong also to man and horse and other things. Hence if one is to obtain the definition of something to which white belongs universally, say the definition of snow, he must have recourse to the more common genus, such as terminated body, and obtain from it the cause of whiteness; and according to this we would learn why snow is white universally. And that cause could pertain to the 'quod quid' of snow: for example, the solidifying of a moist element which makes it be terminated, some light being preserved.
Then, at: divisions according to differentiae (97b25) he shows how the above method is useful in definitions.
And he says that when someone seeks to define according to the above method, that is, by dividing the genus into species, the benefit it confers is that it accomplishes the division of the genus through differences. How this method reveals the 'quod quid' has been indicated above. Furthermore, these divisions are useful for achieving the 'quod quid' solely in the manner described above; they seem to contribute nothing to the syllogizing of the quod quid, as we said earlier. Rather, it seems that upon dividing, one immediately obtains everything without syllogizing, as though he knew them from the beginning before he divided.
Then, at: but, in fact, the order (97b30) he shows what to avoid if this method is not to fail.
Concerning this he does two things.
First, he shows that one must avoid improper order.
Second, one must avoid diminution, at: again, division is the only possible method (97b35; [541]).
He says therefore first (97b30) that it makes a great difference, when arranging the items present in a definition, which items are mentioned first and which are mentioned later. For it is possible to state that man is a gentle animal which is two-legged, or to state that man is a two-legged animal which is gentle. That the order of terms does make a difference in defining is clear from the fact that anything which is defined should be composed of two things, namely, a genus and a difference. Therefore, if 'gentle' is taken as a difference of animal, it is required that 'gentle animal' be some one thing which can be taken as a genus from which, with the addition of the difference “two-legged,” man is constituted. And the same reason holds for anything else which is formed from several things into a unit which is per se and not per accidens.
Therefore, just as it does make a difference whether this or that be taken for the genus or the difference, or whether something is taken as a difference constituting the genus or as dividing it, so it makes a difference in defining just how the parts of the definition are ordered. For if I say that man is a two-legged gentle animal, 'animal' will be taken as the genus, 'gentle' as the difference constituting a genus, and 'two-legged' as the difference dividing it. It will be the opposite if I say that man is a gentle two-legged animal.
Therefore, since a variation in order makes a difference in 'quod quid', the consequence is that one who divides should not only suppose the things which are taken for defining, but should take care about the ordering of those things. And so it is clear that a definition does not syllogize the 'quod quid'.
Then, at: again, division is the only possible method (97b35) he teaches that one should avoid diminution, pointing out how to make certain that nothing required for the 'quod quid' be omitted.
And he says that the only way this can be avoided is by following the method he will indicate. To understand this it should be noted that all the differences of higher genera pertain to the 'quod quid' of some species. For the lower genus is constituted by the difference which divides the higher genus. Therefore, to avoid diminution it is necessary that none of these differences be overlooked. But they are overlooked if someone, after taking the supreme genus, were to take a difference which divides not that supreme genus but some lower genus. But this can be recognized in the following way: when animal is taken as a supreme genus, if someone then takes the division of something pertaining to lower genera, not everything which is contained under the higher genus will fall into that division. As an example of this he says: for example, not every animal is either whole-winged or possessed of divided wings. (An animal is said to be 'whole-winged' if its wings are each a continuous whole, as in a bat; but an animal is said to be possessed of divided wings if its wings are composed of distinct feathers, as in a hawk or raven). But neither of these belongs to a non-winged animal. However, every animal that flies is contained under one or other of these differences, because it is according to the above differences that this genus, 'flying animal', is divided. But the first and immediate difference of animal is such that every animal falls under the division. And the same applies to all other genera, whether we are dealing with genera extrinsic to animal, such as stone and plant, or those which are contained under animal, such as bird and fish. Yet the first difference of bird is such that every bird is included; and the same is true of fish. He concludes, therefore, that if someone proceeds to divide according to this method, namely, that the totality which is divided is contained under the parts of the division, he will be able to know that nothing necessary for defining has been omitted. But if he proceeds some other way, he is bound to omit something; and he will not be sure that he has defined integrally.
Lecture 15
Two errors are excluded: what is really required for constituting a definition according to the method of division