Ensembles

September 13, 2015 – 11:51 am

Research 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 ? = {P1, P2, …, Pn}.

The set of agents X = {x: (?Pi??)[Pix]} is the Category characterized by x.

Let X be a set of agents and ? a CX. If ?(X) (by which we abbreviate (?x?X)(?Pi??)[Pix]) then 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, ?2 a CX such that ?2 ? ?1, Y a category set characterized by ?2, and 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.

  1. 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.
  2. 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 = {?ei}},

          where for each ?j

                            e0 = ?ei-1

                            ei = ei-1 ? Pi for some Pi ? ?x\ei-1 for which (?P??)[~(ei-1Pi |– P)]

  1. E above is the set of Proximal Possible Descriptions of x Without ?, written ?(x,?)

An Ensemble Characterization (EX), ?, is a category characterization for which:

            ?(X) ? (?x?X)(?Cx,applies??x)(?cx?Cx,applies)(?y)(????(x,?))[?(y) ? (Ax(cx, qx) ? Ay(cx, qy))]

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
  • Ax and Ay are assumed to be identical in all respects other than those actually dependent upon x, which is justifiable if we claim that the definition of Ax is included in ?x.
  • Let X be an ensemble characterized by the EX ?1, ?2 an EX such that ?2?1, Y an ensemble characterized by ?2, and 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 cx Cx,applies, cx is a Context Governed By ? and write g(?, cx)

We say ax1 is an Action Governed By ? for the EX ? and write g(?, ax1) iff

            cx,0Cx,applies & ax1 = Ax(qx,0, cx,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:

 An important example of an ensemble is ‘class’ as it occurs in, for example, Marxist or Weberian sociological theories. It is common however to decry the lack of a well-accepted definition for class. Regard, for example, the definitions offered by those two theorists. Weber writes ([1924] 1978 (eds Roth, G & C. Wittich) Economy and Society, Berkeley: University of California Press, p. 927:)

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 Capital: A Critique of Political Economy Vol. 3, Moscow, p. 886) that

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 XIXth C England are uniform enough – if considered in sufficient generality – to distinguish their regularities of action from those of the landed gentry of the same time and place, for example. On the other hand, it is not obvious that all such definitions will give rise to appropriate EX – there are few non-trivial regularities of action to be found in ‘wage-earners’ in XXIst C Australia, for example.

Characteristics

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

  • NO(X) = {n: X ? K(n)} – the Outer Norm Formation of
  • NI(X) = {n: K(n) ? X } – the Inner Norm Formation of
  • Z(X2, t) – the Action Diagram for X With Probability ? t
  • ZNI(X) (X2, t) – the Inner Normative Action Diagram for X With Probability ? t
  • ZZ(X2, t) – the Communication Diagram for X With Probability ? t
  • ZZNI(X) (X2, 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) = {p1, …, pn} ? 2X is a Partition of X, when

  1. (?x ? X)(?p ? P(X)) [x ? p]
  2. (?x ? X)(?pi, pj ? P(X)) [x ? pi & x ? pj ? pi = pj]
  • 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 22expX 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.

Tags:
  1. One Response to “Ensembles”

  2. 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 SteveGW on Sep 13, 2015

Post a Comment