Changes

: <math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq P_j \times Q_j\ \mbox{for all}\ j \in J .</math>

: <math>G = \{ G_j \} = \{ G_j : j \in J \}\ \mbox{with}\ G_j \subseteq P_j \times Q_j\ \mbox{for all}\ j \in J .</math>

Here, ''J'' is a set of actual (not formal) parameters used to index the OG, while ''P''_''j'' and ''Q''_''j'' are domains of objects (initially in the informal sense) that enter into the dyadic relations ''G''_''j''.

Here, ''J'' is a set of actual (not formal) parameters used to index the OG, while ''P''_''j'' and ''Q''_''j'' are domains of objects (initially in the informal sense) that enter into the dyadic relations ''G''_''j''&nbsp;.

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'', in this way distinguishing it from all other members of the indexed family ''G''.

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''.

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:

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation.  Generally speaking, it is always possible in principle to form the union required by the RUI, or without loss of generality to assume the equality imposed by the RUE.  The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context.  Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation.  Generally speaking, it is always possible in principle to form the union required by the RUI, or without loss of generality to assume the equality imposed by the RUE.  The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context.  Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

But an overall purpose of this formalism is to represent the objects and constituencies ''known to'' specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity.  Accordingly, keeping individual tabs on the relational domains Pj and Qj, though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents.  In short, whether or not domains appear with explicit subscripts, one should always be ready to answer ''Who subscribes to these domains?''

But an overall purpose of this formalism is to represent the objects and constituencies ''known to'' specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity.  Accordingly, keeping individual tabs on the relational domains ''P''_''j'' and ''Q''_''j''&nbsp;, though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents.  In short, whether or not domains appear with explicit subscripts, one should always be ready to answer ''Who subscribes to these domains?''

It is important to emphasize that the index set J and the particular attachments of indices to dyadic relations are part and parcel to G, befitting the concrete character intended for the concept of an OG, which is expected to realistically embody in the character of each Gj both ''a local habitation and a name''.  For this reason, among others, the Gj can safely be referred to as ''individual dyadic relations'' (IDR's).  Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' (TIP's) that a concrete index j adds to its thematic object Gj, whether parenthetically or paraphatically.

It is important to emphasize that the index set ''J'' and the particular attachments of indices to dyadic relations are part and parcel to ''G'', befitting the concrete character intended for the concept of an OG, which is expected to realistically embody in the character of each ''G''_''j'' both ''a local habitation and a name''.  For this reason, among others, the ''G''_''j'' can safely be referred to as ''individual dyadic relations'' (IDR's).  Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' (TIP's) that a concrete index ''j'' adds to its thematic object ''G''_''j''&nbsp;, whether parenthetically or paraphatically.

Depending on the prevailing direction of interest in the genre G, "<" or ">", the same symbol is used equivocally for all the relations Gj.  The Gj can be regarded as formalizing the OM's that make up the genre G, provided it is understood that the information corresponding to the parameter j constitutes an integral part of the ''motive'' or ''motif'' of Gj.

Depending on the prevailing direction of interest in the genre ''G'', "<font face="system"><s>&lt;</s></font>" or "<font face="system"><s>&gt;</s></font>", the same symbol is used equivocally for all the relations ''G''_''j''&nbsp;.  The ''G''_''j'' can be regarded as formalizing the OM's that make up the genre ''G'', provided it is understood that the information corresponding to the parameter ''j'' constitutes an integral part of the ''motive'' or ''motif'' of ''G''_''j''&nbsp;.

In this formulation, G constitutes an ''ontological hierarchy'' (OH) of a plenary and potentiating type, one that determines the complete array of objects and relationships that are conceivably available and describably ''effable'' within a given discussion.  Operating with reference to the global field of possibilities presented by G, each Gj corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.

In this formulation, ''G'' constitutes an ''ontological hierarchy'' (OH) of a plenary and potentiating type, one that determines the complete array of objects and relationships that are conceivably available and describably ''effable'' within a given discussion.  Operating with reference to the global field of possibilities presented by ''G'', each ''G''_''j'' corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.

Another way to formalize the defining structure of an OG can be posed in terms of a ''relative membership relation'' or a notion of ''relative elementhood''.  The constitutional structure of a particular OG can be set up in a flexible manner by taking it in two stages, starting from the level of finer detail and working up to the big picture:

Another way to formalize the defining structure of an OG can be posed in terms of a ''relative membership relation'' or a notion of ''relative elementhood''.  The constitutional structure of a particular OG can be set up in a flexible manner by taking it in two stages, starting from the level of finer detail and working up to the big picture: