## Ensembles

September 13, 2015 – 11:51 amResearch indicates that certain kinds of collections of agents below the level of species are sociologically significant. ‘Class’, ‘group’, and ‘organization’, for example, are all terms used to name such collections – each of them with a slightly different intention and occupying a slightly different role in some sociological theory. To speak generally – but not too generally – of agents in the plural without presupposing the nature of those collections we will introduce the notion of an ensemble, which is intended to be a minimally defined collection of agents that is sociologically significant.

Begin with the preliminary definition of a __Category Characterization__ (CX) as a set of properties that may belong to agents. We write a CX as *?* = {*P _{1}*,

*P*, …,

_{2}*P*}.

_{n}The set of agents *X* = {*x*: (?*P _{i}*?

*?*)[

*P*]} is the

_{i}x__Category__characterized by

*x*.

Let *X* be a set of agents and ? a CX. If *?*(*X*) (by which we abbreviate (?*x?**X*)(?*P _{i}*?

*?*)[

*P*]) then

_{i}x*X*is a

__Category Set__characterized by

*?*.

- There may be many category sets identically characterized. The point of talking about a category set is to make explicit the assumption that there is a set of properties common to the set of agents in question. What distinguishes that set of agents from another set with the same category characterization may be some set of properties that do not occur in every category characterization. ‘Rational bipeds,’ for example, characterizes many category sets, while ‘rational bipeds residing at this address’ has fewer possibilities, and ‘rational bipeds living here called by my name’ has just one.
- Let
*X*be a category set characterized by the CX*?*,_{1}*?*a CX such that_{2}*?*?_{2}*?*,_{1}*Y*a category set characterized by*?*, and_{2}*Y*?*X*; then*Y*is a__Category Subset__of*X*.

A category set, however, may be arbitrarily defined and is thus not necessarily sociologically significant. In order for a category set to be sociologically significant it must have some effect upon members of the society (agents) which is a consequence of actions of the members of the category set where those actions are a consequence of their membership in the category set.

In order to make this notion more precise it is necessary that some preliminary items be defined.

- Let
*?*= {_{x}*P*:*Px*} be the__Complete Description__of*x*; the set of all properties which may be applied truly to*x*. If*?*is a CX and*X*is a set of agents for which*?*(*X*) (i.e.*X*is a category set characterized by*?*), and*?*?*X*, then*?*?*?*._{x} - Now define the set of maximal possible descriptions of
*x*in the case that*x*is just as before but without the characteristics described by*?*. Thus

* E* = {*? _{j}*:

*?*= {?

_{j}*e*}},

_{i} where for each *? _{j}*

*e _{0}* = ?

*e*

_{i-1}* e _{i}* =

*e*?

_{i-1}*P*for some

_{i}*P*?

_{i}*?*\

_{x}*e*for which (?

_{i-1}*P??*)[~(

*e*

_{i-1}?

*P*|–

_{i}*P*)]

*E*above is the set of__Proximal Possible Descriptions of__, written*x*Without*?**?*(*x*,*?*)

An __Ensemble Characterization (EX)__, *?*, is a category characterization for which:

*?*(*X*) ? (?*x?**X*)(?*C _{x,applies}*?

*?*)(?

_{x}*c*?

_{x}*C*)(?

_{x,applies}*y*)(?

*?*?

*?*(

*x*,

*?*))[

*?*(

*y*) ? (

*A*(

_{x}*c*,

_{x}*q*) ?

_{x}*A*(

_{y}*c*,

_{x}*q*))]

_{y}Let *X* be a set of agents and *?* an EX. If *?*(*X*) then *X* is an __Ensemble__ characterized by *?*.

- There may be many ensembles identically characterized
*A*and_{x}*A*are assumed to be identical in all respects other than those actually dependent upon_{y}*x*, which is justifiable if we claim that the definition of*A*is included in_{x}*?*._{x}- Let
*X*be an ensemble characterized by the EX*?*,_{1}*?*an EX such that_{2}*?*?_{2}*?*,_{1}*Y*an ensemble characterized by*?*, and_{2}*Y*?*X*; then*Y*is a__Subensemble__of*X*.

Using the notation presented just above, and by analogy with norm-governed actions, we say that for all *c _{x} *?

*C*,

_{x,applies}*c*is a

_{x}__Context Governed By__and write

*?**g*(

*?*,

*c*)

_{x}We say *a _{x1}* is an

__Action Governed By__for the EX

*?**?*and write

*g*(

*?*,

*a*) iff

_{x1} *c _{x,0}* ?

*C*&

_{x,applies}*a*=

_{x1}*A*(

_{x}*q*,

_{x,0}*c*)

_{x,0}- The context is governed by
*?*and the action is produced in that context. - Note that the contexts of several EX may govern any action

Several observations may be made here.

- Very many category sets will be ensembles to some degree, but are unlikely to ever feature in sociological theories. For example, people who like ice cream will react differently from those who don’t like ice cream when asked whether they like ice cream. The fact of sociological significance for an ensemble (rather than its potential) depends upon whether these altered reactions have a wider significance, and in which HLST they are to appear.
- The most obvious content for
*?*is (?*N*)(?*n?**N*)*K*(*n*), and we have seen how norms change the behaviour of an agent in the circumstances governed by those norms. - It is possible to define also a collective in terms of it being the patient rather than the agent, so to speak. For example, red haired people do nothing as an ensemble, yet they are universally hated and affect the actions of those around them by inspiring aggression towards themselves. They might therefore be considered a sociologically significant category set. Nevertheless, we will make no such definition since it is not clear that any examples of patienthood really exist; or that if they do they have any real significance before they become agents. To extend the notion of oppression; red heads, who may be an oppressed category set in some society, are irrelevant considered collectively until they develop the characteristics of an ensemble – and the oppression is likely to do that.

Example:
We may speak of a “class” when (1) a number of people have in common a specific causal component of their life chances, insofar as (2) this component is represented exclusively by economic interests in the possession of goods and opportunities for income, and (3) is represented under the conditions of the commodity or labor markets Marx, on the other hand, has no very precise definition to offer for this most basic element of his system. In general, we may understand what he has in mind by the use he makes of the concept. The closest thing to a definition might be his statement (1971 There are three great social groups, whose members… live on wages, profit and ground rent respectively. In both cases, as can be seen, class is taken to be essentially determined by the economic status of the class members. However, in order for these to count as ensembles it has to be argued that the economic status gives rises to regularities in action. Only in the case that that is reasonably established would a CX rise to the status of an EX as defined. In both cases, it is reasonable to argue that such regularities may occur, and that the regularities may be distinctive for the category sets that are typically defined. The experiences of the industrial workers of XIX |

__Characteristics__

The following characteristics of an ensemble that are of sociological interest can be defined in terms of the material presented above.

*N*(_{O}*X*) = {*n*:*X*?*K*(*n*)} – the__Outer Norm Formation__of*N*(_{I}*X*) = {*n*:*K*(*n*) ?*X*} – the__Inner Norm Formation__of*Z*(*X*^{2},*t*) – the__Action Diagram for__*X*With Probability ?*t**Z*_{NI}_{(X)}(*X*^{2},*t*) – the__Inner Normative____Action Diagram for__*X*With Probability ?*t**ZZ*(*X*^{2},*t*) – the__Communication Diagram for__*X*With Probability ?*t**ZZ*_{NI}_{(X)}(*X*^{2},*t*) – the__Inner Normative____Communication Diagram for__*X*With Probability ?*t**ACD*(*X*) = {(*x*,*y*):*x*,*y*?*X*,*x*/*y*} – the__Agent Constraint Diagram__of*X*.*ADD*(*X*) = {(*x*,*y*):*x*,*y*?*X*,*x*>*y*} – the__Agent Dominance Diagram__of*X*.*ANCD*(*X*) = {(*x*,*y*):*x*,*y*?*X*,*x*/_{NI}_{(X) }*y*} – the__Agent Normative Constraint Diagram__of*X*.*ANDD*(*X*) = {(*x*,*y*):*x*,*y*?*X*,*x*>_{NI}_{(X) }*y*} – the__Agent Normative Dominance Diagram__of*X*.

__Partitions__

Let *X* be an ensemble. *P*(*X*) = {*p _{1}*, …,

*p*} ? 2

_{n}*is a*

^{X}__Partition__of

*X*, when

- (?
*x*?*X*)(?*p*?*P*(*X*)) [*x*?*p*] - (?
*x*?*X*)(?*p*?_{i}, p_{j}*P*(*X*)) [*x*?*p*&_{i}*x*?*p*?_{j}*p*=_{i}*p*]_{j}

- We call the elements of
*P*(*X*) the__Parts__of the partition of*X*. - According to 1, every member is in a part in
*P*(*X*). - According to 2, each member is in just one part in
*P*(*X*). - There may be a use for a subset of 2
^{2}^{exp}Consider the case where we wish to speak of the management of a company being the upper echelons of the financial, operational, etc. sectors of the company; or the heads of departments being a special organizational set within the company. For the purposes of simplicity, let us for now disregard this possibility.^{X}

## One Response to “Ensembles”

I spent an hour and a half putting in all the special characters, only to have the damn things eaten by the program. Perhaps I’ll fix it later, but perhaps I won’t.

By

SteveGWon Sep 13, 2015