[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 2

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Cybernetics List, Peirce List,

It is easy to see the relationship between the parenthetical expressions
of Peirce's logical graphs, that somewhat clippedly picture the ordered
containments of their formal contents, and the associated dual graphs,
that constitute the species of rooted trees here to be described.

In the case of our last example, a moment's contemplation
of the following picture will lead us to see that we can
get the corresponding parenthesis string by starting at
the root of the tree, climbing up the left side of the
tree until we reach the top, then climbing back down
the right side of the tree until we return to the
root, all the while reading off the symbols, in
this particular case either "(" or ")", that
we happen to encounter in our travels.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ( | ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ( | ) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

This ritual is called "traversing" the tree, and the string
read off is often called the "traversal string" of the tree.
The reverse ritual, that passes from the string to the tree,
is called "parsing" the string, and the tree constructed is
often called the "parse graph" of the string.  I tend to be
a bit loose in this language, often using "parse string" to
mean the string that gets parsed into the associated graph.

This much preparation allows us to present the two most basic axioms of
logical graphs, shown in graph and string forms below, along with handy
names for referring to the different directions of applying the axioms.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Jon Awbrey

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