[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 59

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Arisbe List, Cybernetics List,

Analyzing Contingent Truths (cont.)

Let's pause to refresh ourselves with a few morsels of lemmas bread.
One lemma that I can see just far enough ahead to see our imminent
need of is the principle that I canonize as the "Emptiness Rule".
It says that a bare lobe expression like "(... , ...)", with any
number of places for arguments but nothing but blanks as filler,
is logically tantamount to the proto-typical expression of its
type, namely, the constant expression "()" that ExG interprets
as denoting the logical value "false".  To depict the rule in
graphical form, we have this continuing sequence of equations:

o-----------------------------------------------------------o
| Emptiness Rule` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` o---o ` ` ` o-o-o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` O ` ` = ` ` O ` ` = ` ` O ` ` = ` `...` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` `( )` ` = ` ( , ) ` = `( , , )` = ` `...` ` ` ` ` ` |
o-----------------------------------------------------------o

Yet another rule that we'll need is the following:

o-----------------------------------------------------------o
| Indistinctness Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` a ` ` ` a a a ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` o---o ` ` ` o-o-o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` O ` ` = ` ` O ` ` = ` ` O ` ` = ` `...` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` `( )` ` = ` (a,a) ` = `(a,a,a)` = ` `...` ` ` ` ` ` |
o-----------------------------------------------------------o

This one is easy enough to derive from rules that are
already known, but I call it the "Indistinctness Rule"
just on behalf of ready reference and easy employment.

Finally, let me introduce a rule-of-thumb that is a bit
more suited to routine computation, and that will serve
to replace the indistinctness rule in many of the cases
where we actually have to call on it.  This is actually
just a special case of the evaluation rule listed above:

o-----------------------------------------------------------o
| Evaluation Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | `x_2` `... x_k` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o---o-...-o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` x_2 ... x_k ` ` ` ` |
| ` ` ` ` ` ` `O` ` ` ` ` ` ` = ` ` ` ` ` ` `O` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ((), x_2, ..., x_k) ` ` = ` ` ` ` x_2 ... x_k ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` `Setup` ` ` <---- | ----> ` ` `Spike` ` ` ` ` ` |
o-----------------------------------------------------------o

Jon Awbrey

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