Inequality

1. There is nothing intrinsically bad in heaven.
2. There is inequality in heaven.
3. So, inequality is not intrinsically bad.

Some reasons for accepting (2):

• God is in heaven and humans are in heaven, and there is infinite inequality there.
• There are angels in heaven and humans are in heaven, and there is inequality there.
• Heavenly bliss is proportional to merit (though of course the merit is the work of God's grace), and merit is unequal.

Hierarchy and unity

Vatican II gives a very hierarchical account of unity in the Church:

This collegial union is apparent also in the mutual relations of the individual bishops with particular churches and with the universal Church. The Roman Pontiff, as the successor of Peter, is the perpetual and visible principle and foundation [principium et fundamentum] of unity of both the bishops and of the faithful. The individual bishops, however, are the visible principle and foundation of unity in their particular churches, fashioned after the model of the universal Church, in and from which churches comes into being the one and only Catholic Church. For this reason the individual bishops represent each his own church, but all of them together and with the Pope represent the entire Church in the bond of peace, love and unity.(Lumen Gentium 23)

The unity of each local Church is grounded in the one local bishop, and the unity of the bishops is grounded in the one pope. Unity at each level comes not from mutual agreement, but from a subordination to a single individual who serves as the principle (principium; recall the archai of Greek thought) of unity. This principle of unity has authority, as the preceding section of the text tells us. In the case of the bishops, this is an authority dependent on union with the pope. (The Council is speaking synchronically. One might also add a diachronic element whereby the popes are unified by Christ, whose vicars they are.)

A hierarchical model of unity is perhaps not fashionable, but it neatly avoids circularity problems. Suppose, for instance, we talk of the unity of a non-hierarchical group in terms of the mutual agreement of the members on some goals. But for this to be a genuine unity, the agreement of the members cannot simply be coincidental. Many people have discovered for themselves that cutting across a corner can save walking time (a consequence of Pythagoras' theorem and the inequality a²+b²<(a+b)² for positive a and b), but their agreement is merely coincidental and they do not form a genuine unified group. For mutual agreement to constitute people into a genuine group, people must agree in pursuing the group's goals at least in part because they are the goals of the group. But that, obviously, presents a vicious regress: for the group must already eist for people to pursue its goals.

The problem is alleviated in the case of a hierarchical unity. A simple case is where one person offers to be an authority, and others agree to be under her authority. They are united not by their mutual agreement, but by all subordinating themselves to the authority of the founder. A somewhat more complex case is where several people come together and agree to select a leader by some procedure. In that case, they are still united, but now by a potential subordination rather than an actual one. This is like the case of the Church after a pope has died and another has yet to be elected. And of course one may have more complex hierarchies, with multiple persons owed obedience, either collectively or in different respects.

This, I think, helps shed some light on Paul's need to add a call for a special asymmetrical submission in the family—"Wives, be subject to your husbands, as to the Lord" (Eph. 5:22)—right after his call for symmetrical submission among Christians: "Be subject to one another out of reverence for Christ" (Eph. 5:21). Symmetrical submission is insufficient for genuine group unity. And while, of course, everyone in a family is subject to Christ, that subjection does not suffice to unite the family as a family, since subjection to Christ equally unites two members of one Christian family as it does members of different Christian families. The need for asymmetrical authority is not just there for the sake of practical coordination, but helps unite the family as one.

In these kinds of cases, it is not that those under authority are there for the benefit of the one in authority. That is the pagan model of authority that Jesus condemns in Matthew 20:25. Rather, the principle of unity fulfills a need for unity among those who are unified, serves by unifying.

There is a variety of patterns here. In some cases, the individual in authority is replaceable. In others, there is no such replaceability. In most of the cases I can think of there is in some important respect an equality between the one in authority and those falling under the authority—this is true even in the case of Christ's lordship over the Church, since Christ did indeed become one of us. But in all cases there is an asymmetry.

Here is an interesting case. The "standard view" among orthodox Catholic bioethicists (and I think among most pro-life bioethicists in general) is that:

1. Humans begin to live significantly before their brains come into existence.
2. Humans no longer live when their brains have ceased all function (though their souls continue to exist).

There is an apparent tension between these two claims. Claim (2) suggests that brains are central to our identity as living animals. Claim (1) suggests otherwise. But there is a way of seeing the rest of the human body as hierarchically subject to the brain that allows one to defend both (1) and (2). For there is a crucial difference between the state of the embryonic body prior to the brain's formation and the state of the adult body after the brain's destruction. In the embryonic case, there is a developmental striving for the production of a brain to be subject to. This is like a group that has come together to select a leader, and they are already unified by their disposition to be subject to the leader once selected. In the case of an adult all of whose brain function has ceased, even if there is heartbeat and respiration (say, because the news that the brain has ceased to function hasn't reached the rest of the body, or because of electrical stimulation), there is no striving towards the production of a brain to be subject to. This is like a bunch of people whose leader has died and where there is neither disposition nor obligation to select another: the social group has effectively been dissolved.

The Phaedo equality argument

I've never quite got the Phaedo 75 "equality" argument. The point is made that whenever we have two equal things in the physical world, they are never simply equal, but are always only equal in some respect. From this we are supposed to infer that we do not get the concept of equality from the two things. Here are two readings that build arguments out of the text. Whether they're faithful to what Plato is saying is a different question.

Reading 1: Take two sticks. They are related in many respects. In one respect, they are equal. In another, they are not. Their may be equal length, but not in their width. Moreover, the length of the one is certainly not equal to the width of the other. (I include that remark in case one is tempted to say: "Why not just consider the same stick twice over, and then it'll surely be equal to itself?" But no, it, too, will only be equal to itself secundum quid—its length will be equal to its length, but not to its width, say.) The two sticks are related in all kinds of ways other than equality. Among these many relations that they stand in (such as inequality in width, difference in color, similarity in value, etc.), there is equality, in repect of length. To recognize the equality, in respect of length, among the many relations that they stand in, requires that we already have the concept of equality so that seeing it in the crowd of relations will pick it out from that crowd.

Objection: We can't do it just with two sticks, but if we have enough items, we can abstract equality from them. For it might be that a₁ and b₁ stand in a multitudinous set M₁ of relations including equality, and a₂ and b₂ stand in a multitudinous set M₂ of relations, and so on. But maybe the intersection of M₁,M₂,...,M_n contains only equality.

Response: There are so many relations that things stand in, that it is very unlikely that the intersection will be a mere singleton.

Objection: We can get to equality as long as we specify "in respect of length". So we do get the concept of equality from the sticks—"the relation in which their lengths stand to one another."

Response: First, the lengths of the sticks stand in infinitely many relations, equality being but one of those relations. (To give a non-Platonic example, the two lengths stand in the relation of being equal or the same color. Or the two lengths stand in the relation of being observed by the same observer.) So the problem reappears. Second, "length" must be defined in some respect—from which exact point on one end of the stick do we measure to which exact point on the other end do we measure? And, note, that almost surely we cannot really exactly specify points—the Cartesian coordinates are triples of real numbers, and almost no real numbers can be exactly specified (there are uncountably many real numbers, but only countably many can be exactly specified by us), so almost surely the ones here cannot be.

Reading 2: The two sticks are only equal in some respect R. But even the claim "the two sticks are equal in respect R" only holds in some further respect. And so on. Hence, we never get to equality itself. Concretely, let's start with: they are equal in respect of length. But that only holds in respect of one time—at some later time, one of the sticks will slightly oscillate and they won't be equal. So, they're equal in respect of length in respect of some time. But now, their length has one value in respect of one way of defining lengths, and another value in respect of another way of defining lengths. (There are probably little whiskers of wood fiber sticking out both ends. Do we measure them, or not? Which ones do we measure? Where in the atoms do we start measuring? And of course we have the uncertainty principles to contend with.) Moreover, in what way do we compare the lengths? Do we take a measuring stick to the one, and then to the other? But equality then only holds in respect of measuring sticks that don't change their lengths. And how do we define the measurement with the measuring sticks? Let's say they have tick marks. Where in the tick mark is the relevant point? It is not impossible that these questions go on ad infinitum. But even if they don't, they go further than we can answer them—and so we didn't get the concept of equality from the sticks.

Raz on egalitarianism

Raz has criticized egalitarianism as follows: equality is not in and of itself the sort of thing that can be good for anybody, but anything that is of intrinsic value "can benefit people."

I'm going to argue that either (a) Raz's critique also applies to act utilitarianism, or (b) there are views that are appropriately characterized as egalitarian that escape Raz's critique. Both options are interesting. Which one we take depends on how we read the phrase "benefit people".

Consider the following family CONSEQ of consequentialist views: We can describe the utility-state of the world as a vector U=(u₁,...,u_n) of utilities, where n is the number of persons in history and u_i is a real number. There is a combination function C that assigns a real number C(U) to each such vector U (it's a multigrade function) and that satisfies two formal conditions:

1. If U' is any permutation of U, then C(U')=C(U).
2. If U and U' are n-dimensional vectors, and every component of U is less than or equal to the corresponding component of U', and for at least one of the components equality does not hold, then C(U)<C(U').

Then, according to the view, the right thing to do on any occasion is to act so as to maximize C₁(U) where U is the utility-state of the world.

Standard act utilitarianism is a member of CONSEQ—just let C(U)=u₁+...+u_n. But there are other views taht are members of CONSEQ. For instance, suppose that we believe that utilities cannot be negative (life is always worth living). Then we can define p-utilitarianism, for any positive real number p, by means of the combination function C_p(U)=(u₁^p+...+u_n^p)^1/p (this is just the L^p norm on Rⁿ, assuming the u_i are non-negative).

Now, let's ask whether Raz's critique applies to act utilitarianism (i.e., to 1-utilitarianism). Can it "benefit people" to maximize C₁(U)? I think there is at least a good prima facie case for a negative, and if so, we have a powerful critique of utilitarianism. Of course, if I am the ith person, it benefits me to maximize u_i. I am not benefited by maximizing C₁(U) in and of itself, unless this increase happens to be accompanied by an increase of u_i. And the same is true for every individual. Hence, maximizing C₁(U) is of no benefit to anybody.

On one interpretation of the "can benefit people" criterion, this has shown that utilitarianism fails the criterion, and hence (a) is true. I don't think this is the best reading of the "can benefit people". For instead, to release a virus that kills exactly one person "harm people" even if the concept of releasing such a virus does not entail of any specific person that she is killed.

Here seems to be the way to argue that utilitarianism escapes Raz's condition. That U' and U are such that C₁(U')>C₁(U) entails that at least one of the u'_i is greater than the corresponding u_i. Hence, it is better for somebody if we maximize C₁(U).

But now note that what I just said holds not just for 1-utilitarianism but for any theory in CONSEQ (the crucial condition is (2)). In particular, it holds for p-utilitarianism for all positive p. But p-utilitarianism is, in a precise sense, an egalitarian theory if 0<p<1. Why? Because the combination function C_p(U) favors more equal distributions when 0<p<1. Here is an easy exercise for the reader (just use the concavity of f(x)=x^p): if every component of U' is equal to the arithmetic mean of all the components of U, and equality does not reign among the components of U, then C_p(U')>C_p(U), assuming 0<p<1. Moreover, C_p(U) increases if we equalize any subset of the components of U (by equalizing a set of components, I mean replacing it by its arithmetic mean).

I suppose Raz could ask: What reason would we have for picking p-utilitarianism, other than our valuing equality in and of itself? And if we do value equality in and of itself, don't we fall prey to the "cannot benefit people" argument? But I think the response can simply be: We do not on this view separate out equality as a value to be added on—rather, we have a combination function which fits our intuitions better than C₁.

Do I think p-utilitarianism is plausible in the final analysis? Not at all. We would still have to answer the question of which value of p is the right one to choose, and the answer to that would seem to be ad hoc. One might think that clearly one should choose p=1 as that gives the simplest and most natural theory. But to a mathematician, p=2 (which is anti-egalitarian, in that it disfavors more equal distributions) is at least as natural, on the assumption that utilities can't be negative, since C₂ is just the Euclidean norm on a quadrant of Rⁿ. This, I think, is a fine argument against 1-utilitarianism (I blogged a version of it before).

Equality and creation

If inequality in respect of the possession of goods is intrinsically bad, then by creating the world, God necessarily produced something bad, since, necessarily, no creature can have all the goods that God does. But it is absurd that creation necessarily involves a bad. Hence, inequality in respect of the possession of goods is not intrinsically bad.

Equality and leveling down

A common popular criticism of valuing equality in and of itself is that one can achieve equality, say in utility, simply by bringing down everybody who is above the level of the least happy member of the community, which is plainly undesirable. I am ashamed to say that I've used this criticism myself in the past.

But the criticism only applies to a naive view where equality is considered in a binary way—you either have it or you don't, and there is a value in having it and a disvalue in not having it. But of course on any non-naive view, equality is valued along a continuum—a minor inequality has small disvalue, while a large inequality has large disvalue. If one takes this into account, it's easy to come up with ways of weighing the value of equality, or equivalently the disvalue of inequality, that are not subject to the above criticism.

For instance, suppose we have n persons, with utilities: u₁,...,u_n. Standard consequentialism calculates an overall value of u₁+...+u_n. But there are many ways of modifying this so that one (a) takes equality into account, and (b) avoids the popular criticism. Now, the intuition behind the popular criticism is, I think, based on the following intuition:

1. It is good if the utility of some is increased and the utility of none is decreased.

So what we need to do is to combine utilities in such a way that equality is valued, but (1) is maintained. Here is one very simple way of doing this. Just model the total good as u₁+...+u_n−c_n(|u₁−a|+...+|u_n−a|), where a is the arithmetic average (u₁+...+u_n)/n, and c_n is a constant such that 0<c_n<1/2. You get a different model, with different normative consequences, for different values of c_n. It's easy to check that increasing any one of the u_i increases the total good on this model[note 1] and so we have (1) and (b). It is never the case that on this valuation, decreasing the utility of some without increasing the utility of any will improve total good—thus, leveling down is not something to worry about. Moreover, equality is taken into account—a more equal distribution is, ceteris paribus, preferable even if it decreases u₁+...+u_n (but not preferable if it decreases each individual utility, or even some individual utilities without increasing others).

All that said, I think such numerical models are not something to take very seriously. Here's one reason. While we might think that there is an objective answer to the question: "What is the mass of the electron?", and that this might be some number, the idea that there should be an objective answer to the question: "What is the true value of c_n in the above total-good formula?" is very implausible to me. And all such additive formulae assume commensurability of goods between people, which I deny. But the models may still be useful, say for showing how an advocate of equality might avoid the leveling-down criticism.