Changes

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 ''X'' collect the objects of thought that fall within a particular OM, and let ''X'' include the whole world of a sign relation plus everything needed to support and contain it.  That is, ''X'' collects all the types of things that go into a sign relation, ''O''&nbsp;&cup;&nbsp;''S''&nbsp;&cup;&nbsp;''I'' = ''W''&nbsp;&sube;&nbsp;''X'', plus whatever else in the way of distinct object qualities (OQ's) and object exemplars (OE's) 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 ''X'' simple enough to contemplate on a single pass, but still make it deep enough to cover the issues of interest at present, I limit ''X'' to having just three disjoint layers of things to worry about.  The middle layer ''X''₀ is the initial collection of objects and signs, ''X''₀ = ''W''.  The top layer ''Q'' is the relevant class of object qualities, ''Q'' = ''X''₀<math>\lessdot</math> = ''W''&nbsp;<math>\cdot</math>&nbsp;<math>\lessdot</math>.  The bottom layer ''E'' is a suitable set of object exemplars, ''E'' = ''X''₀<math>\gtrdot</math> = ''W''&nbsp;<math>\cdot</math>&nbsp;<math>\gtrdot</math>.

In order to keep this ''X'' simple enough to contemplate on a single pass, but still make it deep enough to cover the issues of interest at present, I limit ''X'' to having just three disjoint layers of things to worry about.  The middle layer ''X''₀ is the initial collection of objects and signs, ''X''₀ = ''W''.  The top layer ''Q'' is the relevant class of object qualities, ''Q'' = ''X''₀<math>\lessdot</math> = ''W''&nbsp;<math>\cdot</math>&nbsp;<math>\lessdot</math>.  The bottom layer ''E'' is a suitable set of object exemplars, ''E'' = ''X''₀<math>\gtrdot</math> = ''W''&nbsp;<math>\cdot</math>&nbsp;<math>\gtrdot</math>.