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 "&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 <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:

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 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, <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 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 <math>G\!</math> is through an indexed collection of dyadic relations:

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

| <math>G = \{ G_j \} = \{ G_j : j \in J \}\ \text{with}\ G_j \subseteq P_j \times Q_j\ (\forall 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''&nbsp;.

Here, <math>J\!</math> is a set of actual (not formal) parameters used to index the OG, while <math>P_j\!</math> and <math>Q_j\!</math> are domains of objects (initially in the informal sense) that enter into the dyadic relations <math>G_j.\!</math>

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''.

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''.