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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.

The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs "<font face="system"><s>&lt;</s></font>" and "<font face="system"><s>&gt;</s></font>".  At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators "<font face="system"><s>&lt;</s></font>" and "<font face="system"><s>&gt;</s></font>" involve the application of dyadic relations analogous to class membership "&isin;" and its converse "&ni;", 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 "<font face="system"><s>&lt;</s></font>" and "<font face="system"><s>&gt;</s></font>".  At the more generic levels of OF's and OG's the ''staging operations'' associated with the generators "<font face="system"><s>&lt;</s></font>" and "<font face="system"><s>&gt;</s></font>" involve the application of dyadic relations analogous to class membership "&isin;" and its converse "&ni;", 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, "<font face="system"><s>&lt;</s></font>" versus "<font face="system"><s>&gt;</s></font>", indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:

# The ''standing relations'', indicated by "<", are analogous to the ''element of'' or membership relation "&isin;".  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.

# The ''standing relations'', indicated by "<font face="system"><s>&lt;</s></font>", are analogous to the ''element of'' or membership relation "&isin;".  Another interpretation of "<font face="system"><s>&lt;</s></font>" 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 ">", are analogous to the ''class of'' relation or converse of the membership relation.  An alternate meaning for ">" 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 "<font face="system"><s>&gt;</s></font>", are analogous to the ''class of'' relation or converse of the membership relation.  An alternate meaning for "<font face="system"><s>&gt;</s></font>" 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, "<" 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.

Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, "<font face="system"><s>&lt;</s></font>" and "<font face="system"><s>&gt;</s></font>", 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'' = {''G''_''j''} = {''G''_''j'' : ''j'' &isin; ''J''} with ''G''_''j'' &sube; ''P''_''j''&nbsp;&times;&nbsp;''Q''_''j'' for all ''j'' &isin; ''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.