Changes

Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision.  In support of this aim, the concept of an objective genre and the particular example provided by <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math> have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.

Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision.  In support of this aim, the concept of an objective genre and the particular example provided by <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math> have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.

The notion of an ''objective motive'' or ''objective motif'' (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account.  For example, pursuing the pattern of <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}),</math> a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving "a local habitation and a name" to the bunch.  What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live ''hierarchy of thought'', one that is currently known to and actively pursued by a designated interpreter of those thoughts.

The notion of an ''objective motive'' or ''objective motif'' (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account.  For example, pursuing the pattern of <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math>, a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving "a local habitation and a name" to the bunch.  What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live ''hierarchy of thought'', one that is currently known to and actively pursued by a designated interpreter of those thoughts.

The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind.  As always, it is best to regard OG's and OM's as ''filters'' and ''reticles'', as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.

The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind.  As always, it is best to regard OGs and OMs as ''filters'' and ''reticles'', as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.

With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to ''factor the facets'' or ''decompose the components'' of sign relations in a more systematic fashion.  Given a homogeneous sign relation <math>H\!</math> of iconic or indexical type, the dyadic projections <math>H_{OS}\!</math> and <math>H_{OI}\!</math> can be analyzed as compound relations over the basis supplied by the <math>G_j\!</math> in <math>G.\!</math>  As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.

With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to ''factor the facets'' or ''decompose the components'' of sign relations in a more systematic fashion.  Given a homogeneous sign relation <math>H\!</math> of iconic or indexical type, the dyadic projections <math>H_{OS}\!</math> and <math>H_{OI}\!</math> can be analyzed as compound relations over the basis supplied by the <math>G_j\!</math> in <math>G\!</math>. As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.

To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type <math>\langle \lessdot, \gtrdot \rangle</math> and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.

To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type <math>\langle \lessdot, \gtrdot \rangle</math> and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.

Let <math>X\!</math> collect the objects of thought that fall within a particular OM, and let <math>X\!</math> include the whole world of a sign relation plus everything needed to support and contain it.  That is, <math>X\!</math> collects all the types of things that go into a sign relation, <math>O \cup S \cup I = W \subseteq X,</math> plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.

Let <math>X\!</math> collect the objects of thought that fall within a particular OM, and let <math>X\!</math> include the whole world of a sign relation plus everything needed to support and contain it.  That is, <math>X\!</math> collects all the types of things that go into a sign relation, <math>O \cup S \cup I = W \subseteq X</math>, plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.

In order to keep this <math>X\!</math> simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit <math>X\!</math> to having just three disjoint layers of things to worry about:

In order to keep this <math>X\!</math> simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit <math>X\!</math> to having just three disjoint layers of things to worry about:

| <math>h : x \lessdot m</math>

| <math>h : x \lessdot m</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

| <math>h\ \operatorname{regards}\ x\ \operatorname{as~an~instance~of}\ m.</math>

| <math>h ~\operatorname{regards}~ x ~\operatorname{as~an~instance~of}~ m.</math>

|-

|-

| <math>h : m \gtrdot y</math>

| <math>h : m \gtrdot y</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

| <math>h\ \operatorname{regards}\ m\ \operatorname{as~a~property~of}\ y.</math>

| <math>h ~\operatorname{regards}~ m ~\operatorname{as~a~property~of}~ y.</math>

|-

|-

| <math>h : x \gtrdot n</math>

| <math>h : x \gtrdot n</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

| <math>h\ \operatorname{regards}\ x\ \operatorname{as~a~property~of}\ n.</math>

| <math>h ~\operatorname{regards}~ x ~\operatorname{as~a~property~of}~ n.</math>

|-

|-

| <math>h : n \lessdot y</math>

| <math>h : n \lessdot y</math>

| <math>\Leftrightarrow</math>

| <math>\Leftrightarrow</math>

| <math>h\ \operatorname{regards}\ n\ \operatorname{as~an~instance~of}\ y.</math>

| <math>h ~\operatorname{regards}~ n ~\operatorname{as~an~instance~of}~ y.</math>

|}

|}

These statements can be read to say:

These statements can be read to say:

:* <math>j\!</math> thinks <math>x\!</math> an icon of <math>y\!</math> if and only if there is an <math>m\!</math> such that <math>j\!</math> thinks <math>x\!</math> an instance of <math>m\!</math> and <math>j\!</math> thinks <math>m\!</math> a property of <math>y.\!</math>.

:* <math>j\!</math> thinks <math>x\!</math> an icon of <math>y\!</math> if and only if there is an <math>m\!</math> such that <math>j\!</math> thinks <math>x\!</math> an instance of <math>m\!</math> and <math>j\!</math> thinks <math>m\!</math> a property of <math>y\!</math>.

:* <math>k\!</math> thinks <math>x\!</math> an index of <math>y\!</math> if and only if there is an <math>n\!</math> such that <math>k\!</math> thinks <math>x\!</math> a property of <math>n\!</math> and <math>k\!</math> thinks <math>n\!</math> an instance of <math>y.\!</math>.

:* <math>k\!</math> thinks <math>x\!</math> an index of <math>y\!</math> if and only if there is an <math>n\!</math> such that <math>k\!</math> thinks <math>x\!</math> a property of <math>n\!</math> and <math>k\!</math> thinks <math>n\!</math> an instance of <math>y\!</math>.

Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb ''thinks'' with the phrase ''interprets &hellip; as'', or even the circumlocution ''acts in every formally relevant way as if'', changing what must be changed elsewhere.  For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.

Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb ''thinks'' with the phrase ''interprets &hellip; as'', or even the circumlocution ''acts in every formally relevant way as if'', changing what must be changed elsewhere.  For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.

In the discussion of the dialogue between <math>A\!</math> and <math>B\!</math> it was allowed that the same signs <math>^{\backprime\backprime} A ^{\prime\prime}</math> and <math>^{\backprime\backprime} B ^{\prime\prime}</math> could reference the different categories of things they name with a deliberate duality and a systematic ambiguity.  Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves.  Used formally within the focal dialogue, they denote the objects of two particular sign relations.  In just this way, or an elaboration of it, the signs <math>^{\backprime\backprime} j ^{\prime\prime}</math> and <math>^{\backprime\backprime} k ^{\prime\prime}</math> can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.

In the discussion of the dialogue between <math>\text{A}</math> and <math>\text{B}</math> it was allowed that the same signs <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math> could reference the different categories of things they name with a deliberate duality and a systematic ambiguity.  Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves.  Used formally within the focal dialogue, they denote the objects of two particular sign relations.  In just this way, or an elaboration of it, the signs <math>{}^{\backprime\backprime} j {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} k {}^{\prime\prime}</math> can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.

=====1.3.4.16. The Integration of Frameworks=====

=====1.3.4.16. The Integration of Frameworks=====