Changes

Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems.  The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.

Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems.  The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.

The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs <math>\lessdot</math> and <math>\gtrdot.</math>  At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators <math>\lessdot</math> and <math>\gtrdot</math> involve the application of dyadic relations analogous to class membership, <math>\in,\!</math> and its converse, <math>\ni,\!</math> but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OM's the full power of triadic relations.  Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context.  In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.

The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs <math>\lessdot</math> and <math>\gtrdot</math>. At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators <math>\lessdot</math> and <math>\gtrdot</math> involve the application of dyadic relations analogous to class membership <math>\in\!</math> and its converse <math>\ni\!</math>, but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OM's the full power of triadic relations.  Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context.  In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.

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.

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.

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 objective genres and motives.  The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.

One way to approach the formalization of an objective genre <math>G\!</math> is through an indexed collection of dyadic relations:

One way to approach the formalization of an objective genre <math>G\!</math> is through an indexed collection of dyadic relations:

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>.

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, <math>P_j\!</math> and <math>Q_j\!</math> for all <math>j\!</math> in <math>J\!</math>.  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'', can be defined as follows:

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

X_j = P_j \cup Q_j ,

| <math>X_j = P_j \cup Q_j,</math>

& P = \bigcup_j P_j ,

| <math>P = \textstyle \bigcup_j P_j,</math>

& Q = \bigcup_j Q_j .

| <math>Q = \textstyle \bigcup_j Q_j.</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 <math>X\!</math> 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 = \textstyle \bigcup_j (P_j \cup Q_j) .</math>

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

: Rubric of Universal Equality (RUE):  <math>X = P_j = Q_j\ (\forall j \in J) .</math>

: Rubric of Universal Equality:  <math>X = P_j = Q_j\ (\forall 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, <math>G\!</math> can be provided with a simplified form of presentation:

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

| <math>G = \{ G_j \} = \{ G_j : j \in J \}\ \text{with}\ G_j \subseteq X \times X\ (\forall j \in J).</math>

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.