[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 48

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Arisbe List, Cybernetics List,

The Case Analysis-Synthesis Theorem (cont.)

In order to think of tackling even the roughest sketch toward a proof
of this theorem, we need to add a number of axioms and axiom schemata.
Because I abandoned proof-theoretic purity somewhere in the middle of
grinding this calculus into computational form, I never got around to
finding the most elegant and minimal, or anything near a complete set
of axioms for the "cactus language", so what I list here are just the
slimmest rudiments of the hodge-podge of "rules of thumb" that I have
found over time to be necessary and useful in most working settings.
Some of these special "precepts" are probably provable from genuine
axioms, but I have yet to go looking for a more proper formulation.

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| Precept L_1.` Indifference` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `(a, (a)) ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` Split <---- | ----> Merge ` ` ` ` ` ` ` ` |
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| Precept L_2.` Equality. `The Following Are Equivalent:` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` b ` ` ` ` ` ` ` a ` b ` ` ` ` ` ` ` a ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` ` ` ` ` o---o ` ` ` ` ` ` ` o ` ` ` ` ` |
| ` ` ` a ` | ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` | ` b ` ` ` |
| ` ` ` o---o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` o---o ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `\ /` ` ` ` |
| ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `(a, (b)) ` ` = ` ` ((a , b)) ` ` = ` ` ((a), b)` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

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| Precept L_3.` Dispersion` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| For k > 1, the following equation holds:` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` y_1 ` `y_2` `...` ` y_k ` ` x y_1 `x y_2` `...` x y_k ` |
| ` `o------o-...-o------o` ` ` ` `o------o-...-o------o` ` |
| ` ` \ ` ` ` ` ` ` ` ` / ` ` ` ` ` \ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` |
| ` ` ` ` ` `x O` ` ` ` ` ` ` = ` ` ` ` ` ` `O` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` x (y_1, ..., y_k) ` ` ` = ` ` (x y_1, ..., x y_k) ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` Distill ` ` <---- | ----> ` ` Disperse` ` ` ` ` |
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To see why the Dispersion Rule holds, look at it this way:
If "x" is "o" (a terminal node), then the presence of "x"
makes no difference on either side of the equation, but
if "x" is "|" (a terminal edge), then both sides of
the equation reduce to "|".

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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