Changes

The staging relations divide into two orientations, <math>\lessdot</math> versus <math>\gtrdot,</math> indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:

The staging relations divide into two orientations, <math>\lessdot</math> versus <math>\gtrdot,</math> indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:

: The ''standing relations'', indicated by <math>\lessdot,</math> are analogous to the ''element of'' or membership relation <math>\in.\!</math>  Another interpretation of <math>\lessdot</math> is the ''instance of'' relation.  At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.

: The ''standing relations'', indicated by <math>\lessdot</math>, are analogous to the ''element of'' or membership relation <math>\in\!</math>. Another interpretation of <math>\lessdot</math> is the ''instance of'' relation.  At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.

: The ''propping relations'', indicated by <math>\gtrdot,</math> are analogous to the ''class of'' relation or converse of the membership relation.  An alternate meaning for <math>\gtrdot</math> is the ''property of'' relation.  Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.

: The ''propping relations'', indicated by <math>\gtrdot</math>, are analogous to the ''class of'' relation or converse of the membership relation.  An alternate meaning for <math>\gtrdot</math> is the ''property of'' relation.  Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.

Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, <math>\lessdot</math> and <math>\gtrdot,</math> and to maintain a formal calculus that treats analogous pairs of relations on an equal footing.  Extra measures of convenience come into play when the relations are used as assignment operations or ''field promotions'', in other words, to create titles, define terms, and establish offices of objects in the active contexts of given relations.  Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.

Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, <math>\lessdot</math> and <math>\gtrdot</math>, and to maintain a formal calculus that treats analogous pairs of relations on an equal footing.  Extra measures of convenience come into play when the relations are used as assignment operations to create titles, define terms, and establish offices of objects in the active contexts of given relations.  Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.

Next, I present several different ways of formalizing OG's and OM's.  The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.

Next, I present several different ways of formalizing OG's and OM's.  The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.

|}

|}

Here, <math>J\!</math> is a set of actual (not formal) parameters used to index the OG, while <math>P_j\!</math> and <math>Q_j\!</math> are domains of objects (initially in the informal sense) that enter into the dyadic relations <math>G_j.\!</math>

Here, <math>J\!</math> is a set of actual (not formal) parameters used to index the OG, while <math>P_j\!</math> and <math>Q_j\!</math> are domains of objects (initially in the informal sense) that enter into the dyadic relations <math>G_j\!</math>.

Aside from their indices, many of the ''G''_''j'' in ''G'' can be abstractly identical to each other.  This would earn ''G'' the designation of a ''multi-family'' or a ''multi-set'' according to some usages, but I prefer to treat the index ''j'' as a concrete part of the indexed relation ''G''_''j''&nbsp;, in this way distinguishing it from all other members of the indexed family ''G''.

Aside from their indices, many of the <math>G_j\!</math> in <math>G\!</math> can be abstractly identical to each other.  This would earn <math>G\!</math> the designation of a ''multi-family'' or a ''multi-set'', but I prefer to treat the index <math>j\!</math> as a concrete part of the indexed relation <math>G_j\!</math>, in this way distinguishing it from all other members of the indexed family <math>G\!</math>.

Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, ''P''_''j'' and ''Q''_''j'' for all ''j'' &isin; ''J''.  Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts.  Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'' (RUM's), can be defined as follows:

Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, ''P''_''j'' and ''Q''_''j'' for all ''j'' &isin; ''J''.  Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts.  Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'' (RUM's), can be defined as follows: