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Next time we'll apply this general scheme to the ''En'' and ''Ex'' interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Next time we'll apply this general scheme to the <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> interpretations of logical graphs, and see how it helps us to sort out the varieties of iconic mapping that are involved in that setting.

Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain '''S''', containing the topological equivalents of bare and rooted trees, onto the object domain '''O''', containing the two objects whose conventional, ordinary, or meta-language names are ''Falsity'' and ''Truth'', respectively.

Corresponding to the Entitative and Existential interpretations of the primary arithmetic, there are two distinct mappings from the sign domain <math>S,\!</math> containing the topological equivalents of bare and rooted trees, onto the object domain <math>O,\!</math> containing the two objects whose conventional, ordinary, or meta-language names are ''falsity'' and ''truth'', respectively.

The next two Figures suggest how one might view the interpretation maps as mappings from a COSI '''S''' to a COSI '''O'''.  Here I have placed names of categories at the bottom, indices of individuals at the next level, and extended upward from there whatever structures the individuals may have.

The next two Figures suggest how one might view the interpretation maps as mappings from a COSI <math>S\!</math> to a COSI <math>O.\!</math> Here I have placed names of categories at the bottom, indices of individuals at the next level, and extended upward from there whatever structures the individuals may have.

Here is the Figure for the Entitative interpretation:

Here is the Figure for the Entitative interpretation:

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Note that the structure of a tree begins at its root, marked by an "O".  The objects in '''O''' have no further structure to speak of, so there is nothing much happening in the object domain '''O''' between the level of individuals and the level of structures.  In the sign domain '''S''', the individuals are the parts of the partition into referential equivalence classes, each part of which contains a countable infinity of syntactic structures, rooted trees, or whatever form one views their structures taking.  The sense of the Figures is that the interpretation under consideration maps the individual on the left (right) side of '''S''' to the individual on the left (right) side of '''O'''.

Note that the structure of a tree begins at its root, marked by an "O".  The objects in <math>O\!</math> have no further structure to speak of, so there is nothing much happening in the object domain <math>O\!</math> between the level of individuals and the level of structures.  In the sign domain <math>S\!</math> the individuals are the parts of the partition into referential equivalence classes, each part of which contains a countable infinity of syntactic structures, rooted trees, or whatever form one views their structures taking.  The sense of the Figures is that the interpretation under consideration maps the individual on the left side of <math>S\!</math> to the individual on the left side of <math>O\!</math> and maps the individual on the right side of <math>S\!</math> to the individual on the right side of <math>O.\!</math>

An iconic mapping, that gets formalized in mathematical terms as a ''morphism'', is said to be a ''structure-preserving map''.  This does not mean that all of the structure of the source domain is preserved in the map ''images'' of the target domain, but only ''some'' of the structure, that is, specific types of relation that are defined among the elements of the source and target, respectively.

An iconic mapping, that gets formalized in mathematical terms as a ''morphism'', is said to be a ''structure-preserving map''.  This does not mean that all of the structure of the source domain is preserved in the map ''images'' of the target domain, but only ''some'' of the structure, that is, specific types of relation that are defined among the elements of the source and target, respectively.

For example, let's start with the archetype of all morphisms, namely, a ''linear function'' or a ''linear mapping'' ''f'' : ''X'' &rarr; ''Y''.

For example, let's start with the archetype of all morphisms, namely, a ''linear function'' or a ''linear mapping'' <math>f : X \to Y.</math>

To say that the function ''f'' is ''linear'' is to say that we have already got in mind a couple of relations on ''X'' and ''Y'' that have forms roughly analogous to "addition tables", so let's signify their operation by means of the symbols "#" for "addition in ''X''" and "+" for "addition in ''Y''".

To say that the function ''f'' is ''linear'' is to say that we have already got in mind a couple of relations on ''X'' and ''Y'' that have forms roughly analogous to "addition tables", so let's signify their operation by means of the symbols "#" for "addition in ''X''" and "+" for "addition in ''Y''".