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Directory:Jon Awbrey/Papers/Futures Of Logical Graphs (view source)
Revision as of 18:08, 4 August 2017
, 18:08, 4 August 2017→Categories of structured individuals: use "Rooted Node Big" & "Rooted Edge Big"
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]], [[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]], [[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%" |
| width="5%" |
| 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>
|
|
|
|
| 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>
|
|
| <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>
|
|
|
|
| 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>