Changes

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.

# 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'' (OGs), also called ''forms of analysis'' (FOAs), ''forms of synthesis'' (FOSs), or ''ontological hierarchies'' (OHs).  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.

# An OG is made up of one or more ''objective motives'' or ''objective motifs'' (OMs), sometimes regarded as particular ''instances of analysis'' (IOAs) or ''instances of synthesis'' (IOSs).  All of the OMs 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.

# 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.

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 OFs and OGs 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 OMs 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:

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:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

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

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

|}

|}

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

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

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

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

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

| <math>Q = \textstyle \bigcup_j Q_j.</math>

| <math>Q = \textstyle \bigcup_j Q_j</math>.

|}

|}

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:

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:  <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:  <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, <math>G\!</math> 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:

{| align="center" cellpadding="8"

{| align="center" cellpadding="8"

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

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

|}

|}

</ol></ol>

</ol></ol>

<li>The various OM's of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM.  If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them ''relements'', in this way jostling the mind to ask:  ''Relement to what?''</li>

<li>The various OMs of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM.  If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them ''relements'', in this way jostling the mind to ask:  ''Relement to what?''</li>

</ol>

</ol>

The last and perhaps the best way to form an objective genre <math>G\!</math> is to present it as a triadic relation:

The last and perhaps the best way to form an objective genre <math>G\!</math> is to present it as a triadic relation:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q ,</math>

| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q</math>,

|}

|}

or:

or:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X .</math>

| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X</math>.

|}

|}

There is a ''partial converse'' of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the ''propping relation'' of the genre, and it can be taken as an alternate way of defining the genre.

There is a ''partial converse'' of the standing relation that transposes the order in which the two object domains are mentioned.  This is called the ''propping relation'' of the genre, and it can be taken as an alternate way of defining the genre.

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>G\!\uparrow \ = \ \{(j, q, p) \in J \times Q \times P : (j, p, q) \in G \},</math>

| <math>G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \}</math>,

|}

|}

or:

or:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>G\!\uparrow \ = \ \{(j, y, x) \in J \times X \times X : (j, x, y) \in G \}.</math>

| <math>G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}</math>.

|}

|}

The standing relation of a genre is denoted by the symbol <math>:\!\lessdot</math>, pronounced ''set-in'', with either of the following two type-markings:

The standing relation of a genre is denoted by the symbol <math>:\!\lessdot</math>, pronounced ''set-in'', with either of the following two type-markings:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>:\!\lessdot\ \subseteq\ J \times P \times Q,</math>

| <math>:\!\lessdot ~\subseteq~ J \times P \times Q</math>,

|-

|-

| <math>:\!\lessdot\ \subseteq\ J \times X \times X.</math>

| <math>:\!\lessdot ~\subseteq~ J \times X \times X</math>.

|}

|}

The propping relation of a genre is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:

The propping relation of a genre is denoted by the symbol <math>:\!\gtrdot</math>, pronounced ''set-on'', with either of the following two type-markings:

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>:\!\gtrdot\ \subseteq\ J \times Q \times P,</math>

| <math>:\!\gtrdot ~\subseteq~ J \times Q \times P</math>,

|-

|-

| <math>:\!\gtrdot\ \subseteq\ J \times X \times X.</math>

| <math>:\!\gtrdot ~\subseteq~ J \times X \times X</math>.

|}

|}

Often one's level of interest in a genre is ''purely generic''.  When the relevant genre is regarded as an indexed family of dyadic relations, <math>G = \{ G_j \}\!</math>, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

Often one's level of interest in a genre is ''purely generic''.  When the relevant genre is regarded as an indexed family of dyadic relations, <math>G = \{ G_j \}\!</math>, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j\ (\exists j \in J) \}.</math>

| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}</math>.

|}

|}

When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X</math>.

When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X</math>.

{| align="center" cellpadding="8"

{| align="center" cellpadding="10"

| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G\ (\exists j \in J) \}.</math>

| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}</math>.

|}

|}

On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}</math> can be indicated by any one of the following equivalent expressions:

On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}</math> can be indicated by any one of the following equivalent expressions:

{| align="center" cellpadding="8" style="text-align:center; width:75%"

{| align="center" cellpadding="10" style="text-align:center; width:75%"

| <math>G : x \lessdot y,</math>

| <math>G : x \lessdot y</math>,

| <math>x \lessdot_G y,</math>

| <math>x \lessdot_G y</math>,

| <math>x \lessdot y : G,</math>

| <math>x \lessdot y : G</math>,

|-

|-

| <math>G : y \gtrdot x,</math>

| <math>G : y \gtrdot x</math>,

| <math>y \gtrdot_G x,</math>

| <math>y \gtrdot_G x</math>,

| <math>y \gtrdot x : G.</math>

| <math>y \gtrdot x : G</math>.

|}

|}

At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects.  To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>(j, x, y) \in\ :\!\lessdot</math>, or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>(j, y, x) \in\ :\!\gtrdot</math>, all of the following notations are equivalent:

At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects.  To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>(j, x, y) \in ~ :\!\lessdot</math>, or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>(j, y, x) \in ~ :\!\gtrdot</math>, all of the following notations are equivalent:

{| align="center" cellpadding="8" style="text-align:center; width:75%"

{| align="center" cellpadding="10" style="text-align:center; width:75%"

| <math>j : x \lessdot y,</math>

| <math>j : x \lessdot y</math>,

| <math>x \lessdot_j y,</math>

| <math>x \lessdot_j y</math>,

| <math>x \lessdot y : j,</math>

| <math>x \lessdot y : j</math>,

|-

|-

| <math>j : y \gtrdot x,</math>

| <math>j : y \gtrdot x</math>,

| <math>y \gtrdot_j x,</math>

| <math>y \gtrdot_j x</math>,

| <math>y \gtrdot x : j.</math>

| <math>y \gtrdot x : j</math>.

|}

|}