Hierarchy comes in different forms. In the literature, two classical approaches dominate: 1. the 'is a part of' approach of 'meronomy', 2. the 'is a kind of' approach of taxonomy. A third point of view that recently receives growing attention is 3. the use of 'cooperation' or 'integration' as the basis for a constructional hierarchy. This page discusses relationships between the three methods.


For more references and information about this topic:
General laws and centripetal science.
Jagers op Akkerhuis G.A.J.M. (2014). European Review 22: 113-144
The operator hierarchy. A chain of closures linking matter, life and artificial intelligence.
Jagers op Akkerhuis G.A.J.M. (2010). Alterra Scientific Contributions 34: 17-31

Introduction to hierarchy

Hierarchical approaches are frequently based on a 'tripartite approach': lower level, focal level, higher level. If a higher level contains the middle level,and this contains the lower level, the system shows a 'nested hierarchy'.There are several viewpoints on hierarchy in the literature, of which we discuss meronomy, taxonomy and construction.

Hierarchy

What exactly is hierarchy? Hierarchy is generally regarded as a relationship that follows two simple rules, transitivity and irreflexivity (Bunge, 1969; Simon, 1973; Webster, 1979) as follows:
  • Transitivity implies that as long as A has a lower hierarchical position than B, which, in turn, has a lower position than C, the hierarchical position of A is automatically lower than C.
  • Irreflexivity means that A can never hold a hierarchical position below or above itself.
From these rules antisymmetry follows because if we suppose a symmetrical relationship in which A is lower than B, and B lower than A, then, by transitivity, A would be lower than A, which is forbidden because of irreflexivity.The difference between a transitive and non-transitive relationship is comparable to the difference between using ancestry or parenthood as a ranking rule. Using ancestry, both the grandparent C and the parent B are ancestors of A. Using parenthood (and excluding the possibility of incest), the situation is different because when A is child of B, and B is child of C, then A cannot be a child of C. When defined as above, a hierarchy must always be transitive.

A ranking can at the same time be hierarchical and non-hierarchical

Interestingly, the same objects can be placed in a sequence that is hierarchical (and transitive) for one property, whilst it is non-transitive for another property. As an example one can analyse the ranking: atom -> molecule -> cell. If we look at the 'complexity' of these objects, the atom, molecule and cell show increasing complexity. Because the cell is more complex than the atom, the complexity ranking is transitive. If we look instead at how atoms form a molecule, and molecules form a cell, this ranking is not transitive, because atoms alone cannot form a cell.

Meronomy

Meronomy represents the ‘is-a-part-of’ relationship. This implies that one starts with a higher level entity, and that other entities are viewed as parts of it. For this reason one can view meronomy as to work from the top down. In principle, everything can be viewed as ‘a-part-of’ the universe. As big things are composed of smaller things, meronomy is also named ‘‘compositional hierarchy’. As an example, one can think of a specific car, which has a specific chair, which has a specific handle bar, a specific screw, etc. In abstract/conceptual language, one could say that ‘cars’, have ‘chairs’, which have ‘handle bars’, ‘screws’, etc. Meronomy is in principle limited to (kinds of) 'objects' and their (kinds of) 'parts'. When using 'object' in a general sense, however, meronomy is sometimes applied without taking into account the difference between whether the top level is a physical object, such as a car, or whether the top level is an abstract selection of objects, such as a group (e.g. a herd of deer, or a population). It is interesting to realise that the application of meronomy presumes a methodology which offers the names for the different kinds of parts. Without such a pre-existing kind-system, the fact that a car, has a seat, has a srew, can only be addressed as an 'entity', has an 'entity', has an 'entity'. In principle, such a kind-system must be constructed from the ground up. Moreover, the use of meronomy presumes the existence of rules for the identification of the borderline between a larger object, and its smaller 'part', for all levels of objects and parts. The question is, where can one find a theory for such borderlines?

Taxonomy

Taxonomy stands for the ranking of abstractions according to an ‘is-a-kind-of’ relationship. This implies that one starts with a large selection of objects or concepts, and defines a more limited selection using more specific criteria. The objects of this smaller selection are viewed as a 'kind' within the larger selection. A taxonomy is also named a ‘specification hierarchy’ or a ‘subsumption hierarchy’. A well-known example of taxonomy in biology is the linnean taxonomy, where the group of 'animals' has a specific subgroup of 'mammals', which has a specific subgroup of 'primates', and a specific subgroup of 'homonids'. In turn, homonids are a kind of primate, which are a kind of mammal, etc. The use of taxonomy this way presumes that one has first defined from the ground up the more abstract levels, such as those of primates or mammals. Only after this has been done does it become possible to apply taxonomy from the top down.

Constructional hierarchy

This third approach to hierarchy and levels is defined from the ground up. It uses 'integration' or 'cooperation'. Roughly speaking one can say that entities at one level show interactions due to which they form a group. Sometimes such a group can be viewed as an entity at the next level, and so on. To distinguish next level entities, on needs criteria for what is an entity and what is a group.

It sometimes leads to confusion that construction may lead to a nontransitive hierarchy (A produces B, B produces C, but A does not produce C) while at the same time the complexity of the objects that are produced can be analysed as a transitive hierarchy (A is less complex than B, B is less complex than C, and A is less complex than C).

Advantages and disadvantages of the possibility to skip a level in meronomy and taxonomy

In meronomy and in taxonomy, one can skip levels. For example if a screw is part of a chair, and this chair is part of a car, then the screw is also a part of the car. Similarly, because homonids are a kind of primates, which are a kind of mammals, it follows from taxonomy that homonids are a kind of mammals. When levels can be skipped, the methodology is viewed as to be 'transitive'. Transitivity has the advantage that it makes the ranking flexible, because one does not have to worry very much about the insertion or deletion of a particular level. This flexibility comes at a price, however, because as a result there are no clear criteria to decide at precisely what level an entity resides. Many things, from chairs to screws can be 'part of the car'. And in a different car, the same part may reside at a different level.

Advantages and disadvantages of constructional hierarchy

A hierarchy which is based on 'integration' or 'cooperation' assumes that entities at one level integrate into groups which are viewed as the larger entities at the next level, and so on. When used in a generic way, this reasoning has the advantage that one does not have to critically examine the mechanisms involved. Our brain readily imagines that things can form groups, and that these can form even larger groups. What the brain 'sees' is an abstract process in which circles/dots 'integrate' to form a larger circle/dot, while several of the larger circles/dots become unified to even larger circles/dots, etc, etc. (as if in figure 1). If we accept this reasoning without reservations, this may lead to imprecise resoning. I like to demonstrate this by the following thought experiment.

Abstract hierarchy.jpg
Figure 1: A generic and abstract way of depicting how interactions can lead to higher levels of integration. The simplicity of a figure like this can be misleading, because no details can be derived about the integration processes or of the kinds of entities that are involved.

Thought experiment about constructional hierarchy

For this thought experiment we choose a cell with endosymbionts, which is also called a "eukaryotic" cell, as the initial level. Next, several endosymbiont cells 'integrate' to a 'multicellular life form'. And after this, these multicellulars integrate to form a 'population'. This thought experiment aims at illustrating that if one accepts this ranking, the resoning is based on a mixture of integration rules, a mixture of types of entities, and a mixture of presumptions about what represents a 'level'. If we first take a look at the kinds of entities involved, an endosymbiont cell is a 'physical object' or 'token'. A token exists in a specific physically united an limited form at a certain place and moment. If several of such cells 'integrate' they can form a structure of attached cells, a 'multicellular', which represents a physical object/token as well. If at the next step several multicellular beings 'integrate' to form a 'population', the result is a group of objects/tokens. The population itself is not a token. The concept of a populaton, represents a group of tokens. Physically speaking, the population only exists as an abstraction, as a virtual circle that our mind has drawn around a group of objects which we have selected by using certain interactive criteria. By including a population in the hierarchy endosymbiont cell->multicellular->population, our thought experiment thus mixes tokens and abstract concepts. Furthermore, our thought experiment also uses different rules for 'next levels'. From cell to plant we accepted an (unspecified) form of 'physical attatchment'. And from plant to population we use the 'potential to mate'. Mating does not imply physical bonding. These criteria are not only different, but they are of a very different kind! In our thought experiment it is furthermore not obvious when exactly a 'next level entity' starts to exist. When we think about the cooperation/integration of cells, does this mean that it is sufficient when they are glued together by a gell? Or must the cell mebranes be physically attached? What kind of physical attachment must be involved? Must the plasma's be connected between the cells? Are all these examples acceptable to our thought experiment, or do we have to identify a certain level of 'integration' that identifies the moment when we can speak about a multicellular organism?

The impact of circularity on hierarchy

What happens with a hierarchy in the case of a circular relationship? In fact, a circular relationship no longer represents a hierarchy because it has lost transitivity, non-transitivity and irreflexivity. Consequently, there is no way of determining which interaction is first or last. Heylighen (2010) used this aspect to indicate that a circular system must also be regarded as a new unit for defining time. Therefore, when circularity occurs, a refocusing of hierarchy analysis is implied. Before circularity, the analysis of hierarchy focused on relationships between the elements that create the circularity. When the circularity is present, a new focus is required that ranks the holistic properties of the circularity in comparison to the holistic properties of other circularities.

Is there an answer to the above questions?

In the following chapters a novel approach, based on combinations of specific circular, self-referring processes, and specific circular, self-referring structures, will be offered as a solution to many of the above questions and challenges. The novel approach has the advantage that it works strictly from the bottom up (it is 'axiomatic') and offers a stringent rule for the recognition/definition of what is a level in the hierarchy. Importantly, every level in the resulting hierarchy has a fixed position (the method is non-transitive).