Book 05
Chapter 03
How to split things fairly, using geometrical proportion.

[Author's Note: In this chapter, and in several chapters to follow, we are going to talk about how to split things up fairly. Please stay with us as we go through a bit of math. Justice is a very important thing, and we have made extra efforts to make things clear for you. You have come this far, so please do stick with it.]

We talked before about notions of fair and unfair. Now let us get into the details. What we are going to do is talk about how to make a fair division between two people, a fair division when those two people are not equally deserving (i.e. when a 50/50 split will not be fair).

Now, whenever we make a split between two people there is just one way it can be done fairly, and there are two ways in which it can be done unfairly. In the first case, the first person gets too much and the second person gets too little. That is unfair to the second person. The situation gets reversed in the second case: the first person gets too little and the second person gets too much. Now, that is unfair to the first person. So, obviously, what we want is for each person to get a just right amount, and, when they do, the split will be fair.

This 'just right' amount is a mean. For whenever we have a 'too much' and a 'too little' then we can also have a 'just right', a mean. What is 'just right' here is a mean between two extremes (viz. the first person gets it all or the second person gets it all); a 'just right' is what makes for a fair division, which means both people get some amount that is fair to them. Any division other than the 'just right' one is an unfair one.

So far this looks like the other virtues we have been talking about, with an excess, and a deficiency, and a mean in between them both. But while justice is a virtue like the other virtues, it also has some special differences. In the case of the other virtues, we have one person, and cannot tell in advance what a person should do to be virtuous. Yet here, with the virtue of justice, we have multiple people and we do set down rules in advance as to what makes for a fair split. The people involved and what each one gets all have to go together in just the right way in order to make for a fair division. This means that unlike our discussions of the other virtues, for justice, we will have to get into more detail.

Going into more detail especially for justice will make more sense if you think about it like this: when it comes to taxes, a tax law first takes a kind of thing (e.g. property) and puts it in terms of numbers of one kind or another (viz. measured in terms of acres). Once the thing is down to a number, we can do math on it (and the government can assess the tax on it). For instance, different plots of land may have different shapes, but the law will still say they are measured in some number of acres. So likewise, when we are talking about two individuals doing a split, regardless of what they are splitting or how we figure out how much each deserves, it is still going to come down to some number, no matter what. Once we have it down to a number, it is all the same then as far as doing the split goes. It is all the same, and yet we still have to get it right in order to make for a just outcome.

Let us work through this slowly, so that we can understand it well.

When we divide something up between two people, each person gets a share. That share is for some amount. How much that amount is depends on where the dividing point is. For instance, the dividing point for a loaf of bread is where it is cut into two pieces.

Well, let us say that each person agrees, beforehand, that the dividing point should be at the mean. This mean is in between the two extremes of one or the other person getting the whole thing. Now, the mean for the other virtues (e.g. temperance or liberality) is something that is just right (e.g. how much to drink or how much to give), and it is likewise here with picking just the right point at which to divide. Note that this is not necessarily in the exact middle: sometimes one person is more deserving than the other.

So, for this scenario, we will say that one person will make the split and the other person will pick which piece they get. Let us imagine that it is the first person that divides it up, and they do so at the mean, and so they make a fair split. What the first person does is a just action, a virtuous action, and they hit the mean. When the second person takes their share, they are taking just the right amount, and they also hit the mean, and so what they do is virtuous as well. The main point here is that justice is a virtue, like the other virtues, but it also involves both people doing just the right thing (viz. making a fair split and taking a fair share).

Now, it could be argued that the second person only has to pick whichever share they think is best (and if the first person made a bad split, that is too bad for them). We do not agree. The truly just action is to notice that the split is not fair, and to help correct it. The just person wants both themselves and others to get just the right amount. As we said before, justice is another's good. You can see this more clearly if you think of when a merchant gives you back too much change for your purchase. The just thing to do is to refuse the extra money, and give it back. Perhaps this may go against one's instincts, but, like we have said, virtue is hard.

We are ready, now, to use some numbers, and do some very easy math, to show how to make a fair split. We will start out easy, with a 50/50 split. Let us say that each person deserves the same amount. They have $100 to split up. They make a 50/50 split. How much money does each person get? It is $50 each. And if we had $200 to split? Then it is $100 each ($100 + $100 = $200). Finally, if we had $50 to split, it would be $25 each ($50 = $25 + $25).

Next, let us say they are not equally deserving. The first person did three times as much work as the second. That means that in order for the split to be fair, the first person must get three times as much as the second. So, again, let us say we have $100. The split is now what? 75/25. Here that means the first person gets $75, and the second gets $25. And if we had $200 to split? $150 goes to the first person, $50 goes to the second. Notice that we double the amount, and each person gets twice as much, but the first person still gets three times as much as the second. Finally, if we had $60 to split, it would be $45 to the first person and $15 to the second ($15 * 3 = $45).

So, you can see that there is a bit of math, but it is not really that difficult. You probably already have a feeling for this kind of fairness. So we can now set down the equation we have been using all along:

first person's worth / first person's share = second person's worth / second person's share

The two sides in this equation must match in order for the split to be fair. So for the cases where it was a 50/50 split, if we plug in then we get: (1) 50 / 50 = 50 / 50, (2) 50 / 100 = 50 / 100, and (3) 50 / 25 = 50 / 25. You can see that in each of the cases the numbers change, but the left and right sides always match.

For the 75 / 25 split, we have (1) 75 / 75 = 25 / 25, (2) 75 / 150 = 25 / 50, and (3) 75 / 45 = 25 / 15. If you reduce those fractions, you will find the equations all balance and the divisions are fair.

Note that that equation has to hold in order for the distribution (i.e. what each person gets) to be proportionate, and in order for the division to be fair. But if you put numbers in there and the equation does not hold, then the distribution is disproportionate and the division is unfair.

In a fair split, you always get something that is proportionate to what you are worth.

And now, let us go more into this notion of 'proportionate', because we will need it for the following chapters. The just is one kind, or species of, proportion. (And there are other, different kinds of proportion which we will get into later.) Proportion is not a property of numbers which stand for things in the abstract ("1" by itself), but of numbers in general (i.e. amounts of money, like "$1"). Proportion is equality between ratios; saying "1/2 = 2/4" is the same as saying "1/2 and 2/4 are in proportion with one another". If the 1/2 and 2/4 stand for people and their shares (like we have been saying), then they are in proportion with one another, and all is just.

(In mathematics, this kind of proportion is called 'geometrical'. Think of it like this: Whole is to whole as part is to matching part. Everything is drawn to scale. So, on a piece of paper, which has plans for a house, if the kitchen takes up 1/5th of the first floor, and, in the actual house, the actual kitchen also takes up 1/5th of the first floor, then the house drawn on the paper and the actual house are in geometrical proportion with one another. There is also a funny way to think of this and fair division. Imagine a child's drawing where some people are drawn bigger than others. Imagine a person that is more deserving as being twice as tall as the second, and the first person holds in their hands a gold bar that is twice as big as the second. Again, the people and their shares are drawn in a kind of geometrical proportion.)

So, anyhow, we can see that what is proportional is what is just, and what is unjust is what violates the proportion. In the unjust case the amount on one side gets to be too much and the other side does not have enough, and the person who takes too much is acting unjustly, and the person who gets too little is unjustly treated.

If we are talking about dividing up what is bad (e.g. how much each person has to pay for repairs) then the reverse is true. A lesser bad (paying less) is more a good than a greater bad (paying more), since the lesser is rather to be chosen than the greater; what is worthy of choice is a good, and what is more worthy of choice is a greater good.

So here we have finished one species, or kind, of the just. Now, let us go on to the next chapter and talk about the next species.


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