{"id":121,"date":"2015-09-13T11:51:58","date_gmt":"2015-09-13T01:51:58","guid":{"rendered":"http:\/\/stevewatson.info\/blog\/?p=121"},"modified":"2015-09-13T20:35:17","modified_gmt":"2015-09-13T10:35:17","slug":"ensembles","status":"publish","type":"post","link":"https:\/\/stevewatson.info\/blog\/2015\/09\/13\/ensembles\/","title":{"rendered":"Ensembles"},"content":{"rendered":"<p><span style=\"color: #000000;\">Research indicates that certain kinds of collections of agents below the level of species are sociologically significant. \u2018Class\u2019, \u2018group\u2019, and \u2018organization\u2019, for example, are all terms used to name such collections \u2013 each of them with a slightly different intention and occupying a slightly different role in some sociological theory. To speak generally \u2013 but not too generally \u2013 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.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">Begin with the preliminary definition of a <u>Category Characterization<\/u> (CX) as a set of properties that may belong to agents. We write a CX as <em>?<\/em>\u00a0= {<em>P<sub>1<\/sub><\/em>, <em>P<sub>2<\/sub><\/em>, \u2026, <em>P<sub>n<\/sub><\/em>}.<\/span><\/p>\n<p><span style=\"color: #000000;\">The set of agents <em>X<\/em> = {<em>x<\/em>: (?<em>P<sub>i<\/sub><\/em>?<em>?<\/em>)[<em>P<sub>i<\/sub>x<\/em>]} is the <u>Category<\/u> characterized by <em>x<\/em>.<\/span><\/p>\n<p><span style=\"color: #000000;\">Let <em>X<\/em> be a set of agents and ?\u00a0a CX. If <em>?<\/em>(<em>X<\/em>) (by which we abbreviate (?<em>x?<\/em><em>X<\/em>)(?<em>P<sub>i<\/sub><\/em>?<em>?<\/em>)[<em>P<sub>i<\/sub>x<\/em>]) then <em>X<\/em> is a <u>Category Set<\/u> characterized by <em>?<\/em>.<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">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. \u2018Rational bipeds,\u2019 for example, characterizes many category sets, while \u2018rational bipeds residing at this address\u2019 has fewer possibilities, and \u2018rational bipeds living here called by my name\u2019 has just one.<\/span><\/li>\n<li><span style=\"color: #000000;\">Let <em>X<\/em> be a category set characterized by the CX <em>?<sub>1<\/sub><\/em>, <em>?<sub>2<\/sub><\/em> a CX such that <em>?<sub>2<\/sub><\/em>\u00a0? <em>?<sub>1<\/sub><\/em>, <em>Y<\/em> a category set characterized by <em>?<sub>2<\/sub><\/em>, and <em>Y<\/em> ? <em>X<\/em>; then <em>Y<\/em> is a <u>Category Subset<\/u> of <em>X<\/em>.<\/span><\/li>\n<\/ul>\n<p><span style=\"color: #000000;\">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.<\/span><\/p>\n<p><span style=\"color: #000000;\">In order to make this notion more precise it is necessary that some preliminary items be defined.<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">Let <em>?<sub>x<\/sub><\/em> = {<em>P<\/em>: <em>Px<\/em>} be the <u>Complete Description<\/u> of <em>x<\/em>; the set of all properties which may be applied truly to <em>x<\/em>. If <em>?<\/em>\u00a0is a CX and <em>X<\/em> is a set of agents for which <em>?<\/em>(<em>X<\/em>) (i.e. <em>X<\/em> is a category set characterized by <em>?<\/em>), and <em>?<\/em>?<em>X<\/em>, then <em>?<\/em>\u00a0? <em>?<sub>x<\/sub><\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\">Now define the set of maximal possible descriptions of <em>x<\/em> in the case that <em>x <\/em>is just as before but without the characteristics described by <em>?<\/em>. Thus<\/span><\/li>\n<\/ol>\n<p><span style=\"color: #000000;\"><em>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E<\/em> = {<em>?<sub>j<\/sub><\/em>: <em>?<sub>j<\/sub><\/em> = {?<em>e<sub>i<\/sub><\/em>}},<\/span><\/p>\n<p><span style=\"color: #000000;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 where for each <em>?<sub>j<\/sub><\/em><\/span><\/p>\n<p><span style=\"color: #000000;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <em>e<sub>0<\/sub><\/em> = ?<em>e<sub>i-1<\/sub><\/em><br \/>\n<\/span><\/p>\n<p><span style=\"color: #000000;\"><em>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 e<sub>i<\/sub><\/em> = <em>e<sub>i-1<\/sub><\/em>\u00a0?\u00a0<em>P<sub>i<\/sub><\/em> for some <em>P<sub>i<\/sub><\/em> ? <em>?<sub>x<\/sub><\/em>\\<em>e<sub>i-1<\/sub><\/em> for which (?<em>P??<\/em>)[~(<em>e<\/em><sub>i-1<\/sub> ?\u00a0<em>P<sub>i<\/sub><\/em> |\u2013<em> P<\/em>)]<\/span><\/p>\n<ol start=\"3\">\n<li><span style=\"color: #000000;\"><em>E<\/em> above is the set of <u>Proximal Possible Descriptions of <em>x<\/em> Without <em>?<\/em><\/u>, written <em>?<\/em>(<em>x<\/em>,<em>?<\/em>)<\/span><\/li>\n<\/ol>\n<p><span style=\"color: #000000;\">An <u>Ensemble Characterization (EX)<\/u>, <em>?<\/em>, is a category characterization for which:<\/span><\/p>\n<p><span style=\"color: #000000;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <em>?<\/em>(<em>X<\/em>) ? (?<em>x?<\/em><em>X<\/em>)(?<em>C<sub>x,applies<\/sub><\/em>?<em>?<sub>x<\/sub><\/em>)(?<em>c<sub>x<\/sub><\/em>?<em>C<sub>x,applies<\/sub><\/em>)(?<em>y<\/em>)(?<em>?<\/em>?<em>?<\/em>(<em>x<\/em>,<em>?<\/em>))[<em>?<\/em>(<em>y<\/em>) ? (<em>A<sub>x<\/sub><\/em>(<em>c<sub>x<\/sub><\/em>, <em>q<sub>x<\/sub><\/em>) ? <em>A<sub>y<\/sub><\/em>(<em>c<sub>x<\/sub><\/em>, <em>q<sub>y<\/sub><\/em>))]<\/span><\/p>\n<p><span style=\"color: #000000;\">Let <em>X<\/em> be a set of agents and <em>?<\/em>\u00a0an EX. If <em>?<\/em>(<em>X<\/em>) then <em>X<\/em> is an <u>Ensemble<\/u> characterized by <em>?<\/em>.<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">There may be many ensembles identically characterized<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>A<sub>x<\/sub><\/em> and <em>A<sub>y<\/sub><\/em> are assumed to be identical in all respects other than those actually dependent upon <em>x<\/em>, which is justifiable if we claim that the definition of <em>A<sub>x<\/sub><\/em> is included in <em>?<sub>x<\/sub><\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\">Let <em>X<\/em> be an ensemble characterized by the EX <em>?<sub>1<\/sub><\/em>, <em>?<sub>2<\/sub><\/em> an EX such that <em>?<sub>2<\/sub><\/em> ?\u00a0<em>?<sub>1<\/sub><\/em>, <em>Y<\/em> an ensemble characterized by <em>?<sub>2<\/sub><\/em>, and <em>Y<\/em> ? <em>X<\/em>; then <em>Y<\/em> is a <u>Subensemble<\/u> of <em>X<\/em>.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">Using the notation presented just above, and by analogy with norm-governed actions, we say that for all <em>c<sub>x<\/sub> <\/em>?\u00a0<em>C<sub>x,applies<\/sub><\/em>, <em>c<sub>x<\/sub><\/em> is a <u>Context Governed By <em>?<\/em><\/u>\u00a0and write <em>g<\/em>(<em>?<\/em>, <em>c<sub>x<\/sub><\/em>)<\/span><\/p>\n<p><span style=\"color: #000000;\">We say <em>a<sub>x1<\/sub><\/em> is an <u>Action Governed By <em>?<\/em><\/u>\u00a0for the EX <em>?<\/em>\u00a0and write <em>g<\/em>(<em>?<\/em>, <em>a<sub>x1<\/sub><\/em>) iff<\/span><\/p>\n<p><span style=\"color: #000000;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <em>c<sub>x,0<\/sub><\/em> ?\u00a0<em>C<sub>x,applies<\/sub><\/em> &amp; <em>a<sub>x1<\/sub> <\/em>= <em>A<sub>x<\/sub><\/em>(<em>q<sub>x,0<\/sub><\/em>, <em>c<sub>x,0<\/sub><\/em>)<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">The context is governed by <em>?<\/em>\u00a0and the action is produced in that context.<\/span><\/li>\n<li><span style=\"color: #000000;\">Note that the contexts of several EX may govern any action<\/span><\/li>\n<\/ul>\n<p><span style=\"color: #000000;\">Several observations may be made here.<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\">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\u2019t 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.<\/span><\/li>\n<li><span style=\"color: #000000;\">The most obvious content for <em>?<\/em>\u00a0is (?<em>N<\/em>)(?<em>n?<\/em><em>N<\/em>)<em>K<\/em>(<em>n<\/em>), and we have seen how norms change the behaviour of an agent in the circumstances governed by those norms.<\/span><\/li>\n<li><span style=\"color: #000000;\">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 \u2013 and the oppression is likely to do that.<\/span><\/li>\n<\/ul>\n<table style=\"height: 737px;\" width=\"755\">\n<tbody>\n<tr>\n<td width=\"616\"><span style=\"color: #000000;\"><strong>Example:<\/strong><\/span><\/p>\n<p><span style=\"color: #000000;\"><strong>\u00a0<\/strong><\/span><span style=\"color: #000000;\">An important example of an ensemble is \u2018class\u2019 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 &amp; C. Wittich) <em>Economy and Society<\/em>, Berkeley: University of California Press, p. 927:)<\/span><\/p>\n<p><span style=\"color: #000000;\">We may speak of a \u201cclass\u201d 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<\/span><\/p>\n<p><span style=\"color: #000000;\">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 <em>Capital: A Critique of Political Economy<\/em> Vol. 3, Moscow, p. 886) that<\/span><\/p>\n<p><span style=\"color: #000000;\">There are three great social groups, whose members&#8230; live on wages, profit and ground rent respectively.<\/span><\/p>\n<p><span style=\"color: #000000;\">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.<\/span><\/p>\n<p><span style=\"color: #000000;\">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<sup>th<\/sup> C England are uniform enough \u2013 if considered in sufficient generality \u2013 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 \u2013 there are few non-trivial regularities of action to be found in \u2018wage-earners\u2019 in XXI<sup>st<\/sup> C Australia, for example.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #000000;\"><em><u>Characteristics<\/u><\/em><\/span><\/p>\n<p><span style=\"color: #000000;\">The following characteristics of an ensemble that are of sociological interest can be defined in terms of the material presented above.<\/span><\/p>\n<ul>\n<li><span style=\"color: #000000;\"><em>N<sub>O<\/sub><\/em>(<em>X<\/em>) = {<em>n<\/em>: <em>X<\/em>\u00a0? <em>K<\/em>(<em>n<\/em>)} \u2013 the <u>Outer Norm Formation<\/u> of<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>N<sub>I<\/sub><\/em>(<em>X<\/em>) = {<em>n<\/em>: <em>K<\/em>(<em>n<\/em>) ? <em>X<\/em> } \u2013 the <u>Inner Norm Formation<\/u> of<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>Z<\/em>(<em>X<\/em><sup>2<\/sup>, <em>t<\/em>) \u2013 the <u>Action Diagram for <em>X<\/em> With Probability ?<\/u><u>\u00a0<em>t<\/em><\/u><\/span><\/li>\n<li><span style=\"color: #000000;\"><em>Z<sub>NI<\/sub><\/em><sub>(<em>X<\/em>)<\/sub> (<em>X<\/em><sup>2<\/sup>, <em>t<\/em>) \u2013 the <u>Inner Normative <\/u><u>Action Diagram for <em>X<\/em> With Probability ?<\/u><u>\u00a0<em>t<\/em><\/u><\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ZZ<\/em>(<em>X<\/em><sup>2<\/sup>, <em>t<\/em>) \u2013 the <u>Communication Diagram for <em>X<\/em> With Probability ?<\/u><u>\u00a0<em>t<\/em><\/u><\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ZZ<sub>NI<\/sub><\/em><sub>(<em>X<\/em>)<\/sub> (<em>X<\/em><sup>2<\/sup>, <em>t<\/em>) \u2013 the <u>Inner Normative <\/u><u>Communication Diagram for <em>X<\/em> With Probability ?<\/u><u>\u00a0<em>t<\/em><\/u><\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ACD<\/em>(<em>X<\/em>) = {(<em>x<\/em>, <em>y<\/em>): <em>x<\/em>, <em>y<\/em>\u00a0? <em>X<\/em>, <em>x<\/em>\/<em>y<\/em>} \u2013 the <u>Agent Constraint Diagram<\/u> of <em>X<\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ADD<\/em>(<em>X<\/em>) = {(<em>x<\/em>, <em>y<\/em>): <em>x<\/em>, <em>y<\/em>\u00a0? <em>X<\/em>, <em>x<\/em>&gt;<em>y<\/em>} \u2013 the <u>Agent Dominance Diagram<\/u> of <em>X<\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ANCD<\/em>(<em>X<\/em>) = {(<em>x<\/em>, <em>y<\/em>): <em>x<\/em>, <em>y<\/em>\u00a0? <em>X<\/em>, <em>x <\/em>\/<em><sub>NI<\/sub><\/em><sub>(<em>X<\/em>) <\/sub><em>y<\/em>} \u2013 the <u>Agent Normative Constraint Diagram<\/u> of <em>X<\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\"><em>ANDD<\/em>(<em>X<\/em>) = {(<em>x<\/em>, <em>y<\/em>): <em>x<\/em>, <em>y<\/em>\u00a0? <em>X<\/em>, <em>x <\/em>&gt;<em><sub>NI<\/sub><\/em><sub>(<em>X<\/em>) <\/sub><em>y<\/em>} \u2013 the <u>Agent Normative Dominance Diagram<\/u> of <em>X<\/em>.<\/span><\/li>\n<\/ul>\n<p><span style=\"color: #000000;\"><em><u>Partitions<\/u><\/em><\/span><\/p>\n<p><span style=\"color: #000000;\">Let <em>X<\/em> be an ensemble. <em>P<\/em>(<em>X<\/em>) = {<em>p<sub>1<\/sub><\/em>, \u2026, <em>p<sub>n<\/sub><\/em>} ? 2<em><sup>X<\/sup><\/em> is a <u>Partition<\/u> of <em>X<\/em>, when<\/span><\/p>\n<ol>\n<li><span style=\"color: #000000;\">(?<em>x<\/em>\u00a0? <em>X<\/em>)(?<em>p<\/em>\u00a0? <em>P<\/em>(<em>X<\/em>)) [<em>x<\/em>\u00a0? <em>p<\/em>]<\/span><\/li>\n<li><span style=\"color: #000000;\">(?<em>x<\/em>\u00a0? <em>X<\/em>)(?<em>p<sub>i<\/sub>, p<sub>j<\/sub><\/em>\u00a0? <em>P<\/em>(<em>X<\/em>)) [<em>x<\/em>\u00a0? <em>p<sub>i<\/sub> <\/em>&amp; <em>x<\/em>\u00a0? <em>p<sub>j<\/sub><\/em>\u00a0? <em>p<sub>i<\/sub><\/em> = <em>p<sub>j<\/sub><\/em>]<\/span><\/li>\n<\/ol>\n<ul>\n<li><span style=\"color: #000000;\">We call the elements of <em>P<\/em>(<em>X<\/em>) the <u>Parts<\/u> of the partition of <em>X<\/em>.<\/span><\/li>\n<li><span style=\"color: #000000;\">According to 1, every member is in a part in <em>P<\/em>(<em>X<\/em>).<\/span><\/li>\n<li><span style=\"color: #000000;\">According to 2, each member is in just one part in <em>P<\/em>(<em>X<\/em>).<\/span><\/li>\n<li><span style=\"color: #000000;\">There may be a use for a subset of 2<em><sup>2<\/sup><\/em><sup>exp<\/sup><em><sup>X<\/sup><\/em> 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.<\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Research indicates that certain kinds of collections of agents below the level of species are sociologically significant. \u2018Class\u2019, \u2018group\u2019, and \u2018organization\u2019, for example, are all terms used to name such collections \u2013 each of them with a slightly different intention and occupying a slightly different role in some sociological theory. To speak generally \u2013 but [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28],"tags":[],"class_list":["post-121","post","type-post","status-publish","format-standard","hentry","category-sociology"],"_links":{"self":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts\/121","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/comments?post=121"}],"version-history":[{"count":3,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts\/121\/revisions"}],"predecessor-version":[{"id":128,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/posts\/121\/revisions\/128"}],"wp:attachment":[{"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/media?parent=121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/categories?post=121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/stevewatson.info\/blog\/wp-json\/wp\/v2\/tags?post=121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}