Changes

Returning to <math>\mathrm{En}</math> and <math>\mathrm{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S~\!</math> to the object domain <math>O,~\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Returning to <math>\mathrm{En}</math> and <math>\mathrm{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S~\!</math> to the object domain <math>O,~\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.

Taking <math>\mathrm{En}~\!</math> and <math>\mathrm{Ex}~\!</math> as arrows of the form <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}~\!</math> and <math>O = \{ \mathrm{falsity}, \mathrm{truth} \},~\!</math> it is possible to factor each arrow across the domain <math>S_0~\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]],&nbsp;[[Image:Rooted Edge.jpg|12px]]<math>\}.~\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\mathrm{En}_\text{syn}, \mathrm{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : S_0 \to O.</math>

Taking <math>\mathrm{En}~\!</math> and <math>\mathrm{Ex}~\!</math> as arrows of the form <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}~\!</math> and <math>O = \{ \mathrm{falsity}, \mathrm{truth} \},~\!</math> it is possible to factor each arrow across the domain <math>S_0~\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[File:Rooted Node Big.jpg|16px]],&nbsp;[[File:Rooted Edge Big.jpg|12px]]<math>\}.~\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\mathrm{En}_\text{syn}, \mathrm{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : S_0 \to O.</math>

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:

| width="5%" | &nbsp;

| width="5%" | &nbsp;

| width="5%" | <math>\mathrm{En}_\text{sem} :</math>

| width="5%" | <math>\mathrm{En}_\text{sem} :</math>

| width="5%" align="center" | [[Image:Rooted Node.jpg|16px]]

| width="5%" align="center" | [[Image:Rooted Node Big.jpg|16px]]

| width="5%" | <math>\mapsto</math>

| width="5%" | <math>\mapsto</math>

| <math>\mathrm{false},</math>

| <math>\mathrm{false},</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge Big.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\mathrm{true}.</math>

| <math>\mathrm{true}.</math>

| &nbsp;

| &nbsp;

| <math>\mathrm{Ex}_\text{sem} :</math>

| <math>\mathrm{Ex}_\text{sem} :</math>

| align="center" | [[Image:Rooted Node.jpg|16px]]

| align="center" | [[Image:Rooted Node Big.jpg|16px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\mathrm{true},</math>

| <math>\mathrm{true},</math>

| &nbsp;

| &nbsp;

| &nbsp;

| &nbsp;

| align="center" | [[Image:Rooted Edge.jpg|12px]]

| align="center" | [[Image:Rooted Edge Big.jpg|12px]]

| <math>\mapsto</math>

| <math>\mapsto</math>

| <math>\mathrm{false}.</math>

| <math>\mathrm{false}.</math>