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=====1.3.4.13.  Formalization of OF : Objective Levels=====

=====1.3.4.13.  Formalization of OF : Objective Levels=====

The three levels of objective detail to be discussed are referred to as the objective "framework", "genre", and "motive" that one finds actively involved in organizing, guiding, and regulating a particular inquiry.

The three levels of objective detail to be discussed are referred to as the objective ''framework'', ''genre'', and ''motive'' that one finds actively involved in organizing, guiding, and regulating a particular inquiry.

1. An "objective framework" (OF) consists of one or more "objective genres" (OG's), also called "forms of analysis" (FOA's), "forms of synthesis" (FOS's), or "ontological hierarchies (OH's).  Typically, these span a diverse spectrum of formal characteristics and intended interpretations.

# An ''objective framework'' (OF) consists of one or more ''objective genres'' (OG's), also called ''forms of analysis'' (FOA's), ''forms of synthesis'' (FOS's), or ''ontological hierarchies'' (OH's).  Typically, these span a diverse spectrum of formal characteristics and intended interpretations.

 

# An OG is made up of one or more ''objective motives'' or ''objective motifs'' (OM's), sometimes regarded as particular ''instances of analysis'' (IOA's) or ''instances of synthesis'' (IOS's).  All of the OM's governed by a particular OG exhibit a kinship of structures and intentions, and each OM roughly fits the pattern or ''follows in the footsteps'' of its guiding OG.

2. An OG is made up of one or more "objective motives" or "objective motifs" (OM's), sometimes regarded as particular "instances of analysis" (IOA's) or "instances of synthesis" (IOS's).  All of the OM's governed by a particular OG exhibit a kinship of structures and intentions, and each OM roughly fits the pattern or "follows in the footsteps" of its guiding OG.

# An OM can be identified with a certain moment of interpretation, one in which a particular dyadic relation appears to govern all the objects in its purview.  Initially presented as an abstraction, an individual OM is commonly fleshed out by identifying it with its interpretive agent.  As this practice amounts to a very loose form of personification, it is subject to all the dangers of its type and is bound eventually to engender a multitude of misunderstandings.  In contexts where more precision is needed it is best to recognize that the application of an OM is restricted to special instants and limited intervals of time.  This means that an individual OM must look to the ''interpretive moment'' (IM) of its immediate activity to find the materials available for both its concrete instantiation and its real implementation.  Finally, having come round to the picture of an objective motive realized in an interpretive moment, this discussion has made a discrete advance toward the desired forms of dynamically realistic models, providing itself with what begins to look like the elemental states and dispositions needed to build fully actualized systems of interpretation.

 

3. An OM can be identified with a certain moment of interpretation, one in which a particular dyadic relation appears to govern all the objects in its purview.  Initially presented as an abstraction, an individual OM is commonly fleshed out by identifying it with its interpretive agent.  As this practice amounts to a very loose form of personification, it is subject to all the dangers of its type and is bound eventually to engender a multitude of misunderstandings.  In contexts where more precision is needed it is best to recognize that the application of an OM is restricted to special instants and limited intervals of time.  This means that an individual OM must look to the "interpretive moment" (IM) of its immediate activity to find the materials available for both its concrete instantiation and its real implementation.  Finally, having come round to the picture of an objective motive realized in an interpretive moment, this discussion has made a discrete advance toward the desired forms of dynamically realistic models, providing itself with what begins to look like the elemental states and dispositions needed to build fully actualized systems of interpretation.

A major theoretical task that remains outstanding for this project is to discover a minimally adequate basis for defining the state of uncertainty that an interpretive system has with respect to the questions it is able to formulate about the state of an object system.  Achieving this would permit a measure of definiteness to be brought to the question of inquiry's nature, since it can be grasped intuitively that the gist of inquiry is to reduce an agent's level of uncertainty about its object, objective, or objectivity through appropriate changes of state.

A major theoretical task that remains outstanding for this project is to discover a minimally adequate basis for defining the state of uncertainty that an interpretive system has with respect to the questions it is able to formulate about the state of an object system.  Achieving this would permit a measure of definiteness to be brought to the question of inquiry's nature, since it can be grasped intuitively that the gist of inquiry is to reduce an agent's level of uncertainty about its object, objective, or objectivity through appropriate changes of state.

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 "<" and ">".  At the more generic levels of OF's and OG's the "staging operations" associated with the generators "<" and ">" involve the application of dyadic relations analogous to class membership "?" and its converse, 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 "<" and ">".  At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators "<" and ">" involve the application of dyadic relations analogous to class membership "&isin;" and its converse, 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, "<" versus ">", indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:

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

1. The "standing relations", indicated by "<", are analogous to the "element of" or membership relation "?".  Another interpretation of "<" 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, "<" and ">", 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.

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.

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:

G = {Gj} = {Gj : j ? J} with Gj ? PjxQj for all j ? J.

G = {Gj} = {Gj : j ? J} with Gj ? PjxQj for all j ? J.

 

Here, J is a set of actual (not formal) parameters used to index the OG, while Pj and Qj are domains of objects (initially in the informal sense) that enter into the dyadic relations Gj.

Here, J is a set of actual (not formal) parameters used to index the OG, while Pj and Qj are domains of objects (initially in the informal sense) that enter into the dyadic relations Gj.

Aside from their indices, many of the Gj 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 Gj, in this way distinguishing it from all other members of the indexed family G.

Aside from their indices, many of the Gj 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 Gj, 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, Pj and Qj for all j ? 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, Pj and Qj for all j ? 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:

Xj = Pj U Qj, P = Uj Pj, Q = Uj Qj.

Xj = Pj U Qj, P = Uj Pj, Q = Uj Qj.

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): X = Uj (Pj U Qj).

Rubric of Universal Inclusion (RUI): X = Uj (Pj U Qj).

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:

G  =  {Gj}  =  {Gj : j ? J}  with  Gj ? XxX for all j ? J.

G  =  {Gj}  =  {Gj : j ? J}  with  Gj ? XxX for all j ? J.

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 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?"

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 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.