[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 45

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Arisbe List, Cybernetics List,

Let us examine the formal operation table for the next in our series
of reflective operations to see if we can elicit the general pattern.

o-------o-------o-------o-----------o
| ` a ` | ` b ` | ` c ` | (a, b, c) |
o-------o-------o-------o-----------o
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| blank | blank | blank | ` cross ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| blank | blank | cross | ` blank ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| blank | cross | blank | ` blank ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| blank | cross | cross | ` cross ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| cross | blank | blank | ` blank ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| cross | blank | cross | ` cross ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| cross | cross | blank | ` cross ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| cross | cross | cross | ` cross ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
o-------o-------o-------o-----------o

Or, thinking in terms of the graphic equivalents,
writing "o" for a blank node and "|" for an edge:

o-------o-------o-------o-----------o
| ` a ` | ` b ` | ` c ` | (a, b, c) |
o-------o-------o-------o-----------o
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` o ` | ` o ` | ` o ` | ` ` | ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` o ` | ` o ` | ` | ` | ` ` o ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` o ` | ` | ` | ` o ` | ` ` o ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` o ` | ` | ` | ` | ` | ` ` | ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` | ` | ` o ` | ` o ` | ` ` o ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` | ` | ` o ` | ` | ` | ` ` | ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` | ` | ` | ` | ` o ` | ` ` | ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
| ` | ` | ` | ` | ` | ` | ` ` | ` ` |
| ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` |
o-------o-------o-------o-----------o

Evidently, the rule is that "(a, b, c)" denotes the value denoted by "o"
if and only if exactly one of the variables has the value denoted by "|",
otherwise it denotes the value denoted by "|".  Examination of the whole
sequence of reflective negations will show that this is the general rule.

In the Entitative interpretation, where o = false and | = true,
(x_1, ..., x_k) interprets as "not just one of the x_j is true".

In the Existential interpretation, where o = true and | = false,
(x_1, ..., x_k) interprets as "just one of the x_j is not true".

Jon Awbrey

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