Changes

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 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> the domain that consists of a single rooted node plus a single rooted edge. As a strategic tactic, 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 its 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: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>

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: