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Returning to <math>\operatorname{En}</math> and <math>\operatorname{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>\operatorname{En}</math> and <math>\operatorname{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>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{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:Cactus Node Big Fat.jpg|16px]],&nbsp;[[Image:Cactus Spike Big Fat.jpg|12px]]<math>\}.\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : S_0 \to O.</math>

Taking <math>\operatorname{En}</math> and <math>\operatorname{Ex}</math> as arrows of the form <math>\operatorname{En}, \operatorname{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}\!</math> and <math>O = \{ \operatorname{falsity}, \operatorname{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>\operatorname{En}_\text{syn}, \operatorname{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\operatorname{En}_\text{sem}, \operatorname{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:

{| align="center" cellpadding="10" width="90%"

{| align="center" cellpadding="10" width="90%"

| <math>\operatorname{En}_\text{sem} :</math> &nbsp; [[Image:Cactus Node Big Fat.jpg|16px]] &nbsp; <math>\mapsto \operatorname{false},</math> &nbsp; [[Image:Cactus Spike Big Fat.jpg|12px]] &nbsp; <math>\mapsto \operatorname{true}.</math>

| <math>\operatorname{En}_\text{sem} :</math> &nbsp; [[Image:Rooted Node.jpg|16px]] &nbsp; <math>\mapsto \operatorname{false},</math> &nbsp; [[Image:Rooted Edge.jpg|12px]] &nbsp; <math>\mapsto \operatorname{true}.</math>

|-

|-

| <math>\operatorname{Ex}_\text{sem} :</math> &nbsp; [[Image:Cactus Node Big Fat.jpg|16px]] &nbsp; <math>\mapsto \operatorname{true},</math> &nbsp; [[Image:Cactus Spike Big Fat.jpg|12px]] &nbsp; <math>\mapsto \operatorname{false}.</math>

| <math>\operatorname{Ex}_\text{sem} :</math> &nbsp; [[Image:Rooted Node.jpg|16px]] &nbsp; <math>\mapsto \operatorname{true},</math> &nbsp; [[Image:Rooted Edge.jpg|12px]] &nbsp; <math>\mapsto \operatorname{false}.</math>

|}

|}

| <math>S_0\!</math>

| <math>S_0\!</math>

| <math>=\!</math>

| <math>=\!</math>

| <math>\{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]], [[Image:Cactus Spike Big Fat.jpg|12px]]<math>\}\!</math>

| <math>\{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]], [[Image:Rooted Edge.jpg|12px]]<math>\}\!</math>

|}

|}

Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Cactus Node Big Fat.jpg|16px]] or else to a rooted edge [[Image:Cactus Spike Big Fat.jpg|12px]]&nbsp;.

Simple intuition, or a simple inductive proof, assures us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Rooted Node.jpg|16px]] or else to a rooted edge [[Image:Rooted Edge.jpg|12px]]&nbsp;.

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows: