Nested Hiearchy's Defining Property- Summativity

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Summativity- The sum of all entities at one level of organization is equal to the sum of all entities at some other level- Knox "The use of hierarchies as organizational models of systematics" Biological Journal of the Linnean Society (1998), 63:1-49, page 8

For example in Linnean taxonomy*, ie a nested hierarchy, the Animal Kingdom consists of and contains all of the levels and entities below it. It is the sum of its parts.

Linnean Classification:

The standard system of classification in which every organism is assigned a kingdom, phylum, class, order, family, genus, and species. This system groups organisms into ever smaller and smaller groups (like a series of boxes within boxes, called a nested hierarchy).

Looking closer at the nested hierarchy of living organisms we have the animal kingdom. To be placed in the animal kingdom an organism must have all of the criteria of an animal:

All members of the Animalia are multicellular (eukaryotes), and all are heterotrophs (that is, they rely directly or indirectly on other organisms for their nourishment). Most ingest food and digest it in an internal cavity.

Animal cells lack the rigid cell walls that characterize plant cells. The bodies of most animals (all except sponges) are made up of cells organized into tissues, each tissue specialized to some degree to perform specific functions.

The next level (after kingdom) contain the phyla. Phyla have all the characteristics of the kingdom PLUS other criteria.

For example one phylum under the Kingdom Animalia, is Chordata.

Chordates have all the characteristics of the Kingdom PLUS the following:

Chordates are defined as organisms that possess a structure called a notochord, at least during some part of their development. The notochord is a rod that extends most of the length of the body when it is fully developed. Lying dorsal to the gut but ventral to the central nervous system, it stiffens the body and acts as support during locomotion. Other characteristics shared by chordates include the following (from Hickman and Roberts, 1994):

bilateral symmetry
segmented body, including segmented muscles
three germ layers and a well-developed coelom.
single, dorsal, hollow nerve cord, usually with an enlarged anterior end (brain)
tail projecting beyond (posterior to) the anus at some stage of development
pharyngeal pouches present at some stage of development
ventral heart, with dorsal and ventral blood vessels and a closed blood system
complete digestive system
bony or cartilaginous endoskeleton usually present.

The next level is the class. All classes have the criteria of the kingdom, plus all the criteria of its phylum PLUS the criteria of its class.

This is important because it shows there is a direction- one of additive characteristics. That is how containment is kept and summativity is met.

(NOTE: evolution does NOT have a direction. Characteristics can be lost as well as gained. And characteristics can remain stable.)


An Army can also be put into a nested hierarchy- with the Army example we would be classifying the US Army which is broken up into Field Armies, which contain and consist of Corps, which contain and consist of Divisions, which contain and consist of Brigades, which contain and consist of Battalions, which contain and consist of Companies, which contain and consist of Platoons, which contain and consist of Squads & Sections. Squads and sections contain and consist of soldiers. Each level, down to the soldier, has a well defined role and place in the scheme.

The Army consists of and contains, soldiers- it exhibits summativity. Andy Schueler didn't even know what summativity was.

See also the summary of the principles of hierarchy theory:

The Hierarchy theory is a dialect of general systems theory. It has emerged as part of a movement toward a general science of complexity. Rooted in the work of economist, Herbert Simon, chemist, Ilya Prigogine, and psychologist, Jean Piaget, hierarchy theory focuses upon levels of organization and issues of scale. There is significant emphasis upon the observer in the system.

Hierarchies occur in social systems, biological structures, and in the biological taxonomies. Since scholars and laypersons use hierarchy and hierarchical concepts commonly, it would seem reasonable to have a theory of hierarchies. Hierarchy theory uses a relatively small set of principles to keep track of the complex structure and a behavior of systems with multiple levels. A set of definitions and principles follows immediately:

Hierarchy: in mathematical terms, it is a partially ordered set. In less austere terms, a hierarchy is a collection of parts with ordered asymmetric relationships inside a whole. That is to say, upper levels are above lower levels, and the relationship upwards is asymmetric with the relationships downwards.

Hierarchical levels: levels are populated by entities whose properties characterize the level in question. A given entity may belong to any number of levels, depending on the criteria used to link levels above and below. For example, an individual human being may be a member of the level i) human, ii) primate, iii) organism or iv) host of a parasite, depending on the relationship of the level in question to those above and below.

Level of organization: this type of level fits into its hierarchy by virtue of set of definitions that lock the level in question to those above and below. For example, a biological population level is an aggregate of entities from the organism level of organization, but it is only so by definition. There is no particular scale involved in the population level of organization, in that some organisms are larger than some populations, as in the case of skin parasites.

Level of observation: this type of level fits into its hierarchy by virtue of relative scaling considerations. For example, the host of a skin parasite represents the context for the population of parasites; it is a landscape, even though the host may be seen as belonging to a level of organization, organism, that is lower than the collection of parasites, a population.

The criterion for observation: when a system is observed, there are two separate considerations. One is the spatiotemporal scale at which the observations are made. The other is the criterion for observation, which defines the system in the foreground away from all the rest in the background. The criterion for observation uses the types of parts and their relationships to each other to characterize the system in the foreground. If criteria for observation are linked together in an asymmetric fashion, then the criteria lead to levels of organization. Otherwise, criteria for observation merely generate isolated classes.

The ordering of levels: there are several criteria whereby other levels reside above lower levels. These criteria often run in parallel, but sometimes only one or a few of them apply. Upper levels are above lower levels by virtue of: 1) being the context of, 2) offering constraint to, 3) behaving more slowly at a lower frequency than, 4) being populated by entities with greater integrity and higher bond strength than, and 5), containing and being made of - lower levels.

Nested and non-nested hierarchies: nested hierarchies involve levels which consist of, and contain, lower levels. Non-nested hierarchies are more general in that the requirement of containment of lower levels is relaxed. For example, an army consists of a collection of soldiers and is made up of them. Thus an army is a nested hierarchy. On the other hand, the general at the top of a military command does not consist of his soldiers and so the military command is a non-nested hierarchy with regard to the soldiers in the army. Pecking orders and a food chains are also non-nested hierarchies.

To achieve summativity the criteria "consist of and contain" must be met. NOTE- A parent population does NOT consist of nor contain it's daughter populations. That is why any tree of life is not a nested hierarchy.