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INTEGRAL WORLD: EXPLORING THEORIES OF EVERYTHING
An independent forum for a critical discussion of the integral philosophy of Ken Wilber
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Publication dates of essays (month/year) can be found under "Essays".
Michael Lamport Commons is an American complex systems scientist, who developed the Model of hierarchical complexity, and is founder of the Journal of Adult Development, and coeditor of the journal Terrorism Research. Sara Nora Ross is founder of ARINA, and serves on the editorial boards of several refereed journals, including Integral Review. Jonas Gensaku Miller is research assistant at Dare Institute. This essay is a response to Mark Edwards' essay "Meyerhoff, Wilber and the Postformal Stages", a reflection on Jeff Meyerhoff's book chapter "Vision Logic". See also Meyerhoff's response to Edwards: "What's Worthy of Inclusion?" and "An 'Intellectual Tragedy'".
Why Postformal Stages of

Order/Stage Ordinal and Name  General descriptions of tasks performed 

9 Abstract 
Discriminate variables such as stereotypes; use logical quantification; form variables out of finite classes based on an abstract feature. Make and quantify propositions; use variable time, place, act, actor, state, type; uses quantifiers (all, none, some); make categorical assertions (e.g., We all die.). Task: All the forms of five in the five rows in the example are equivalent in value, x = 5. 
10 Formal 
Argue using empirical or logical evidence; logic is linear, onedimensional; use Boolean logic s connectives (not, and, or, if, if and only if); solve problems with one unknown using algebra, logic, and empiricism; form relationships out of variables; use terms such as if…then, thus, therefore, because; favor correct scientific solutions. Task: The general left hand distributive relation is x * (y + z) = (x * y) + (x * z) 
11 Systematic 
Construct multivariate systems and matrices, coordinate more than one variable as input; situate events and ideas in a larger context, i.e., considers relationships in contexts; form or conceive systems out of relations: legal, societal, corporate, economic, national. Task: The right hand distribution law is not true for numbers but is true for proportions and sets. x + (y * z) = (x * y) + (x * z); x ∪ (y ∩ z) = (x ∩ y) ∪ (x ∩ z) Symbols: ∪ = union (total elements); ∩ = intersection (elements in common) 
To indicate why postformal actions cannot be done at the formal order, that is, that they are not a horizontal extension of formal order action, it is only necessary to show that no next, systematic order task can be reduced to a chain of formal order actions. To show this we state that the systematic order actions are sets of actions from the formal order. Central to confusion to date may be set relationships. Sets cannot be equal to their members. Thus, A does not equal a, or b when A = {a, b}.
To show this, consider the empty set φ. Note that φ = { } has no members. Nothing means there are no members. How can φ = nothing when φ is a set and nothing is nothing? Something cannot equal nothing.
Two examples of this concept from set theory are given below. The first uses narrative from an established instrument called the HelperPerson Problem, and the second uses algebra.
The HelperPerson Problem begins with a vignette that relates the generic situation of a client or patient seeking assistance from a professional named Allen. After Allen speaks with the Person to assess the problem, the following sequence of actions is given:
 Allen offers to provide guidance and assistance.
 This form of guidance and assistance is seen as the most effective in treating this problem.
 Allen also presents other forms of guidance and assistance as well
 Allen discusses the benefits and risks of each as well, including doing nothing.
 Allen tries to understand the Person's needs and concerns.
 Allen asks and answers many questions.
 Allen also observes the Person's body language.
 Allen wants to know whether their body language matches their statements
 Allen asks if the Person is ready to make a choice.
 Allen tells the Person to base their decision on their previous discussion
 The Person feels Allen knows best
 The Person accepts the guidance and assistance.
Formal statement 1 consists of
 Allen tries to understand the Person's needs and concerns.
This leads to  Allen then asks and answers many questions.
Together these form a formal order statement
Allen trying to understand the Person's needs and concerns “causes” Allen to ask and answer many questions.
Formal statement 2 consists of
 Allen also observes the Person's body language.
This leads to  Allen wants to know whether their body language matches their statements
Together these form a formal order statement:
Allen observing the Person's body language “causes” Allen to want to know if their body language matches their statements.
Together, these formal order statements form a system at the systematic order. In the first part of the system, Formal statement 1 is defined in terms of abstract statements 5 and 6. It organizes them into a causal sequence. Formal statement 2 is defined in terms of abstract statements 7 and 8. It organizes them in a causal sequence. Thus, there are two causal statements. Causal statements are defined as formal statements, that is, they rest on linear logic that uses one causal input. Each formal statement is a set formed out of two abstract statements. Each formal statement is independent of the other.
The systematic order coordination task is to form a set containing the two formal statements as elements. The systematic order coordination is reflected by statements 9 and 10: Allen asks if the Person is ready to make a choice. Allen tells the Person to base their decision on their previous discussion. This results in a system that coordinates the previous formal relations without the formal relations being repeated in the system. It forms a set containing the elements by forming a system that could not be formed without them as its elements.
To underscore the relation of set theory to the foregoing discussion, the system corresponds to a set. The formal relations that are not repeated in the system correspond to the elements of lower rank elements that comprise the set. That is, a set is not at the same rank as its elements, the elements are at a lower rank than the set, and therefore the set is not equal to its elements.
An example from Algebra may demonstrate the same distinction. Take the simple formal order equation,
x = ½ y  1
There is a very simple solution at the formal order to solve for y.
But consider the pairs of equations
Equation 1: x = ½ y  2z
Equation 2: 2x = y + 2z
There are no formal order actions that tell one how to work with two equations. Each of the equations belong to the set of actions at the systematic order, for reasons explained next.
At the formal order, solving a linear equation is straight forward. One puts the variable one wants to solve for on the left side, divides out its coefficient, and moves any other variables to the right hand side, remembering to multiply them by minus one. (This is the same as subtracting the term from both sides.)
Demands of solving two equations with two unknowns requires some way of combining equations. The only way to do this is to have some way of eliminating one of the variables. The only way to eliminate one variable is to make the same variable in the two equations have the same coefficient and then to combine the equations.
The systematic order task will be the coordination performed by adding the equations. There are other coordinations possible for solving these two equations. In each case, the goal is to eliminate one of the variables. Adding the equations is what is necessary but not available with just formal actions. One has to see that y is codetermined by both x and z, two input variables. This task does not exist at formal order, which can operate on only one input variable, i.e., solving for one unknown.
Therefore, the following is as far as one can go in solving each of the equations at the formal order:
Equation 1:
½ y  2z = x
½ y = 2z + x,
y = 4z + 2x
y = 2x + 4z
Equation 2:
2x = y +2z
y = 2x  2z
y = 2x + 2z
The formal order task enables one to get the y unknown on the left side of each equation. Think about it. What is the formal action that tells you what to do? But, beyond the step of getting the y unknown on the left, step, there is no formal order action to inform one about the next step. Adding equations is not a task available in formal order actions. Adding equations is more complex in this algebra example because of the higher order task that algebraic solving for multiple unknowns involves. However, merely adding things in other cases is merely adding, which is horizontal, not vertical, complexity.
Hence there is no formal action that tells one how to combine two formal relations. Formal order actions include relations between variables. They do not include actions about how to combine two or more relationships among formal relations. Note that y is a function of x and z and is not a relation between variables. It is a relation among relations of variables. Such function relationships are systematic order tasks to conceive and operate upon. A set of relations is different and not equal to a member relation. Hence the action of adding equations is not a relation between variables but a relation among relations, so that a systematic order relation is the result of the sum of equations.
Theoretical Summary
Concepts from set theory were applied here to clarify why formal stage tasks can be coordinated only at the next stage, systematic. Consistent with Piaget and the Model of Hierarchical Complexity, the concepts apply to all stages that precede the formal stage, as well (and in the case of the Model), those that follow the formal stage). A central premise in these theories is that each next stage of performance coordinates the actions performed at the preceding order of complexity. To apply the premise successfully, the actions of each stage must be unambiguously specified. The stage generator concept successfully eliminates ambiguity about makes a stage a stage by precise specification.
The Model of Hierarchical Complexity specifies how these relationships of sets and their elements relate in the development of increasingly complex actions. The theory's axioms may be used to test if an action is performed at a higher order of hierarchical complexity or not, i.e., if it is at the same or a higher stage. We supply simple sample material below to indicate how to do this, which supplements the higher stage HelperPerson items used earlier. There are three axioms, which can be used as follows to test this on content where there are two or more adjacent tasks or behaviors in a sequence. Although the first axiom was introduced above, it is repeated below in the set of all three axioms along with questions that can be used to apply them.
The informal statement of axioms below are next applied to the following examples. These examples supply content comprised of two or more adjacent tasks or behaviors in a sequence. The question to be addressed are the sequence of actions just a chain of behaviors or do they form a hierarchically ordered sequence?
 Axiom 1. Higher order actions are defined in terms of two or more lower order ones.
Question to apply to each example: Is the last action in the sequence defined in terms of those that precede it? This is usually enough to reject that the sequence of actions under examination is a hierarchical sequence rather than just a chain of actions sequence.  Axiom 2. The higher order actions organize or transform the lower order ones.
Question to apply to each example: Does the last item in the sequence organize or transform,
Organization may been putting the action is some temporal or spacial sequence of execution.  Axiom 3. The organization is not arbitrary.
Questions to apply to each example: Is the organization from the application of axiom 2 nonarbitrary? Could it be other than it is? Is it necessarily so, for the action under consideration to match some real world, logical, or mathematical constraint?
Marchand (Personal communication, January 2010), also suggests that the MHC conception of stage may be more functionalist than structuralist (i.e., stage as performance of tasks of a given order. But there is probably both functionalism and structuralism in the MHC. The functionalism is that stages are based on performance on tasks. The structuralist part is that sequences that are generated using the MHC are ordinal structures. Each order is qualitatively different and irreducible to any of the lower orders.
Piaget also has studied the functional aspects seeing development not only as succession of stages of equilibrium but also as moments of preparation and construction and of conclusiveness. He identified these two moments in formal stage (FA and FB). Theoretically, for Marchand, the systematic stage could be FB. But Kohlberg (1990) argued that it was FC. FA is abstract, FB is formal. Also Rasch Analyses (Commons, Goodheart, Pekker, Dawson, Draney, & Adams, 2008) validated the sequence from concrete, through abstract, then formal, systematic and then metasystematic.
We are hopeful that this presentation is instructive and helps to lay to rest the confusions about the existence of stages that follow the formal stage and are not extensions of it or reducible to it.
References
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