Changes

One way to approach the formalization of an objective genre ''G'' is through an indexed collection of dyadic relations:

One way to approach the formalization of an objective genre ''G'' is through an indexed collection of dyadic relations:

: <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>

:<p><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></p>

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;.

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;.

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:

: <math>\begin{matrix}

:<p><math>\begin{matrix}

X_j = P_j \cup Q_j ,

X_j = P_j \cup Q_j ,

& P = \bigcup_j P_j ,

& P = \bigcup_j P_j ,

& Q = \bigcup_j Q_j .

& Q = \bigcup_j Q_j .

\end{matrix}</math>

\end{matrix}</math></p>

Ultimately, all of these ''totalitarian'' strategies end the same way, at first, by envisioning a domain ''X'' that is big enough to encompass all the objects of thought that might demand entry into a given discussion, and then, by invoking one of the following conventions:

Ultimately, all of these ''totalitarian'' strategies end the same way, at first, by envisioning a domain ''X'' that is big enough to encompass all the objects of thought that might demand entry into a given discussion, and then, by invoking one of the following conventions:

: Rubric of Universal Inclusion (RUI):  <math>X = \bigcup_j (P_j \cup Q_j) .</math>

: Rubric of Universal Inclusion (RUI):  <math>X = \textstyle \bigcup_j (P_j \cup Q_j) .</math>

: Rubric of Universal Equality (RUE):  <math>X = P_j = Q_j\ \mbox{for all}\ j \in J .</math>

: Rubric of Universal Equality (RUE):  <math>X = P_j = Q_j\ \mbox{for all}\ j \in J .</math>

Working under either of these assumptions, ''G'' can be provided with a simplified form of presentation:

Working under either of these assumptions, ''G'' can be provided with a simplified form of presentation:

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

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

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.