On hierarchies

We often make an appeal to hierarchical relations, in social and political discourse, in religion, in metaphysics (or that odd part of it called mereology) and more recently in social behaviour in animals, called ethology. But what we don’t do much is discuss what it is that a hierarchy is, in general terms. So I aim to do that now.
The term “hierarchy” has a venerable history – it was first employed as the arrangement of angels by Pseudo-Dionysius the Areopagite from the Greek word for “bishop”, ????????, which came from the Greek for “sacred rule”. It was applied also to the structure of the Catholic hierarchy, in which priests and elders reported to a bishop, the bishop to an archbishop, and the archbishop to the cardinals and the pope (and the pope to God, of course. More on that later). Army hierarchies also follow a similar pattern. It entered the English language in 1643, in Milton, according to the Oxford English Dictionary, as a general concept:

A body of persons or things ranked in grades, orders, or classes, one above another; spec. in Natural Science and Logic, a system or series of terms of successive rank (as classes, orders, genera, species, etc.), used in classification.

Hierarchies are usually expressed in elevational terms: one entity or level is “higher” than another, and the inverse, one is “lower” than another. What this relative ranking consists of depends a lot on what kind of a hierarchy it is. The term also has an extensive role in mathematics, which is largely a matter of set inclusion, and so we might generalise to say that some hierarchies are inclusion hierarchies (in mathematics, a “containment” hierarchy).

Inclusion hierarchies are the meat and drink of taxonomy, where they are expressed as trees:

The topology of the tree doesn’t matter, so long as it represents the group-within-group relations of the taxa; by which I mean that the representation of the topology doesn’t matter – the order could be CBAEDFHG, so long as the nodes, the branching points that represent the inclusive group, are in the right order. Inclusive hierarchies are useful for representing evolutionary history, although they only constrain rather than unambiguously set out that history.

But there are other kinds of hierarchies than containment or inclusion hierarchies. I think of these as control hierarchies. They represent power relations in social contexts; they represent literal control systems in cybernetics, and they represent, often, things that are better seen as inclusion hierarchies rather than control hierarchies.

Here’s your basic garden-variety control hierarchy:

I have chosen the term “dominate” (which also has religiopolitical connotations, coming as it does from the Latin dominus, “lord” or “master”, or, in the case of slaves and peons and worshippers, “owner”) as the transitive relation. Adam dominates Betty who dominates Charles. Hence, Adam dominates Charles. The Great Chain of Being looks like a simple control hierarchy, and was often used as such, but it is really an inverted inclusion hierarchy, based on the properties of entities at each grade: rocks exist, plants both exist and live, animals exist, live and sense, and so on up to God who has all properties of those things He created; the control relations are justified in terms of the inclusion hierarchy, so it is in effect a mixed hierarchy.

Hierarchies are usually evaluative. “Higher” entities are “better”. A good illustration is this fifteenth century woodcut:

On the left side, you have the great chain as based on Aristotle’s theory of motivating souls from De Anima: rocks merely exist, plants live, animals sense, and humans think. On the right you have a moral scale: studious scholars think, hedonistic humans merely sense, gluttons merely live, and those suffering from acedia, are unable to even live; they merely exist. Notice that scholars are virtus. I wonder who drew that scale up?

Let us now look at a “standard” social hierarchy. Please make nothing of the gender of the names – it’s enough they start with a letter that indicates their notional grade, class or level. In ethology, such ranks are given Greek names: ?, alpha; ?, beta; ?, gamma; ?, delta; and ?, epsilon. One rarely needs more than five such ranks in a social hierarchy, for reasons I’ll discuss later.

Here the dominance relation forms layers, which are usually referred to as ranks. Each relation is binary: Adam dominates Betty. He also dominates Bob, and Brigitte, each individually. Since these individual relations are also transitive, Adam also dominates Carl, in virtue of dominating Betty, and so on for all members of the hierarchy. This is a well formed hierarchy, since there are always clear lines of dominance, and no member of the group is undominated but Adam, who is the alpha individual. But what about Betty and, say, Cindy? Cross hierarchical dominance occurs as well, in virtue of equality of rankings:

This leads us to think that what is actually going on here is not about individual dominance relations, but about class relations. I mean here, not the social notion of class, which is only one instance of class hierarchy, but what logicians call equivalence class hierarchy. Another way to express this, one which is familiar to biologists, is that these form ranks or grades. Thus, a rank is dominant to the lower ones.

On this account, any member of a class is dominant to any member of another class below it. The relations that count are those that pertain between the classes, not the individuals. This is a view that the classes are entities, and the members of those classes inherit their properties from the entities, a view often called holism. The alternative view, called individualism in political and social theory, holds that the properties of the wholes are the sum (possibly the vector sum) of the properties of the components or parts or members.

Why this is worth discussing in the context of hierarchies, is that in ethology, the ranks are in effect the outcomes of numerous individual interactions. As the dominance hierarchy that results sorts out, individuals end up at or about the same rank, such that they have equal status with some, and are dominant or subordinate to others.

There is no presumption that ranks should develop, and in some ways the ethological notion of a rank is an abstraction. Ethology is not only individualistic, it is nominalistic, in that it treats the dominance relationship between individuals as constitutive of the social order, and not vice versa. That will matter when I post later on dominance psychology in humans.

Now well ordered hierarchies are just rare. In fact dominance hierarchies are fairly settled in the top three layers or so, but are often confused and messy at lower levels. This is, in part, because the hierarchy is a dynamic thing, with members constantly under challenge for status, and so the relations of larger numbers will tend to be unresolved, while the few at the top will tend to be settled, at least for most of the time. It is also because the characteristics that set up dominance relations shift as you become more generalised – it might be size of the bull male at the alpha level, but at the delta level it might be size, territory, geographical position of one’s nest, and so on. Hence, we will tend to see hierarchies that are not well formed, and are intransitive or partially ordered:

And this is indeed what we see. Pinnipeds, which have a bull male/harem social structure, show just this sort of noisy structure, particularly in elephant seals (see Riedman 1991).

In the next in this series, if I ever get to write it, I will discuss human social dominance. Then I will use that to approach the phenomena of religion.

Riedman, Marianne. 1991. The pinnipeds: seals, sea lions, and walruses. Berkeley: University of California Press.

©2009 John Wilkins. All rights reserved.