Lecture 7 — Finding the Mean of Commutative Justice

It is as though the judge were dealing with a line divided into unequal sections and took from the greater section the length exceeding the half and added it to the smaller section. When a whole belonging to two men is divided by the dicha or measure, then it is said that each has what is his inasmuch as each receives an equal portion—the equal portion being a mean between something greater and something less according to arithmetic proportionality. [956]

Therefore, this mean is called dicheon—since it is a dicha (measure)—in the way they say dicheon (just thing) and dichastes (just man) and dichaste (justice). (W. D. Ross translates it: “It is for this reason also that it is called dikaion, because it is a division into two equal parts (dicha), just as if one were to call it dichaion; and the judge (dicastes) is one who bisects (dichastes).” –p. 1009.) [958]

If there are two equals, and the half of one is taken from it and added to the other, the other will exceed it by two. But, if what was taken away was not added to the other, the other would exceed the half by one. Therefore, the half taken is equal to one, and the half from which subtraction was made is equal to one. From this, we know both what must be taken from the person with too much and what must be added to the one with too little. The amount exceeding the mean must be awarded to the man with too little and taken from the one with too much. [959]

Let us take three equal lines and mark them by the terms AA, BB, and GG. Subtract AE (the half of A) from AA, add it to GG, and call it GD. Therefore, the whole line DGG exceeds the line AE by that which is GD and by that which is GB (the half of G), but it exceeds line BB by that which is GD. [961]

This is true also in other arts, for they would be destroyed if the craftsman doing the quality and quantity of work that he should is not supported accordingly. [962]

The names “gain” and “loss” have their origin in voluntary transactions. When a man [now] owns more than he [formerly] did own, he is said to have gain, but when less, he is said to have loss, as in buying, selling, and other exchanges permitted by law. However, when men have neither more nor less after their transactions, but rather the same, they are said to have what is theirs, neither gaining profit nor suffering loss. Therefore, justice is a mean between some kind of gain and loss arising in involuntary transactions; it is having an equal amount of these both before and after the transaction. [963]

956. It is as though the judge were dealing. After the Philosopher has shown the difference between the mean of justice regulating transactions and the mean of justice regulating distributions, now he shows how the mean of that justice that regulates transactions is understood. He handles this point in a twofold fashion.

First, he discloses his proposition.

Second, at the names (1132b11; [963]), he explains the origin of the names “gain” and “loss” he has used.

He discusses the initial point from two aspects.

First, he shows how we may discover the mean of commutative justice in these things;

second, at this is true (1132b9; [962]), how we may discover it in the matter of the different arts.

He treats the first point in two ways.

First, he introduces an example to show how the mean is applied in commutative justice.

Second, at if there are (1132a32; [959]), he clarifies what he has said.

In regard to the initial point, he does two things.

First, he gives the example to explain his proposition.

Second, at therefore, this mean (1132a30; [958]), he shows the appropriateness of the example from the very manner of speaking.

957. Aristotle says that this is the way a judge expresses a reduction to equality. If he wishes to reduce to equality a line divided into unequal parts, he takes away from the larger part that portion by which it exceeds the half of the whole line and adds it to the smaller part such that the half of the whole line is a certain dicha, that is, rule or measure for reducing unequal portions to an equality. So, when a whole thing belonging to two men is divided by such a dicha or measure, then it is said that each one has what is his inasmuch as he receives equality—which is the mean between more and less—according to arithmetic proportionality. The reason is that the mean of justice is exceeded by the one with more to the extent that it exceeds the person with less—this pertains to arithmetic proportionality, as we pointed out before [944, 950].

958. Then, at therefore, this mean (1132a30), he shows that the preceding example is suitable according to Greek usage. He says that, since the mean of this justice is a certain dicha, the just thing is called dicheon by the Greeks, as if a person, wanting to vary the names, should say that dicheon is the just thing, dichastes the just man, and dichaste justice.

959. Next, at if there are two equals (1132a32), he makes clear what he has said, namely, that it is necessary to take from one with more in the amount exceeding the mean and to give to one with less.

First, he explains his statement;

second, at let us take (1132b6; [961]), he expresses it by terminals.

First he says to let us take two equal lines both of which are two measures long, for example, two palms breadth or two feet; let us subtract half from one line and add it to the other. Obviously, the line receiving the addition exceeds the other by two units, because the line from which the subtraction was made has only one unit remaining, and the line to which the addition has been made has three units. But if the section subtracted from one line is not added to the other, there will be an excess of only one unit. By that line to which nothing is added or from which nothing is subtracted, we understand the mean of justice, having as it does neither more nor less than what belongs to it. By the line to which addition has been made, we understand the person who has too much. By the line from which subtraction has been made, we understand the person who has too little.

960. In this way, then, it is evident that the man who has too much exceeds the mean by one unit, which has been added to it over and above, but the mean exceeds by one—which has been taken from it—that from which subtraction has been made. Therefore, we will know by this mean what we ought to take from him who has more and give to him who has less. Besides, we will know that we ought to take from the greater, namely, from him who has more, the amount by which he exceeds the mean, because we ought to give him who has less in the amount the mean exceeds him.

961. At let us take (1132b6), he sets forth in figure what was said. Let us take three equal lines and mark the terminations of one AA, of another BB, of the third GG. Then let BB remain undivided, but divide AA in half at the point E, and divide GG in half at the point Z. Next, take away from line AA a section AE, add it to the line GG and call the addition GD. It is clear then that the whole line DG exceeds the line AE by two units (by that which is GG and by that which is GD), but it exceeds the line BB by one unit only (GD). Therefore, obviously, that which is longest exceeds the mean by one unit and the shortest by two units, after the manner of arithmetic proportionality.

962. Then, at this is true (1132b9), he shows that what has been said must be observed in transactions having to do with the different arts. The arts would be destroyed if the craftsman who works at some handicraft would not be supported, that is, would not receive for his workmanship according to the quantity and quality of what he produced. For that reason, the work of one craftsman must be commensurate with the work of another to the extent that there is a just transaction.

963. Next, at the names (1132b11), he explains the origin of the names “gain” and “loss,” saying that they come from voluntary transactions in which names of this kind were first used. When a man owned more than he previously had owned, he was said to have gained; but when less, he was said to have suffered loss, as in buying, selling, and all other transactions permitted by law. However, when men have neither more nor less than they had in the beginning, but bring back in equal quantity the same as they had taken by their transactions, then they are said to have what belongs to them, neither gaining nor losing.

964. He draws the final inference that he had principally intended. It is evident from the premises that the justice we are now discussing is a mean between gain and loss, that justice is simply the possession of an equal amount before and after a transaction, even an involuntary one, as we see in the person who, when constrained by a judge, restores to another what he had in excess.