[Inquiry] Futures Of Logical Graphs

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

FOLG.  Note 1

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Cybernetics List, Peirce List,

I think I am finally ready to speculate on
the futures of logical graphs that will be
able to rise to the challenge of embodying
the fundamental logical insights of Peirce.

For the sake of those who may be unfamiliar with it,
let us first divert ourselves with an exposition of
a standard way that graphs of the order that Peirce
considered, those embedded in a continuous manifold
like a plane sheet of paper, without or without the
paper bridges that Peirce used to augment his genus,
can be represented as parse-strings in Asciiish and
sculpted into pointer-structures in computer memory.

A blank sheet of paper can be represented as a blank space in line,
but that way of doing it tends to be confusing unless the logical
expression under consideration is set off in a separate display.

For example, consider the axiom drawn in box form below:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` = ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

We can write this in line as "(()) = " or set it off as:

              (( ))  =

When we turn to representing the corresponding expressions in computer memory,
where they can be manipulated with utmost facility, we begin by transforming
the planar graphs into their topological duals.  The planar regions of the
original graph correspond to nodes (or points) of the dual graph, and the
boundaries between planar regions in the original graph correspond to
edges (or lines) between the nodes of the dual graph.

For example, overlaying the corresponding dual graphs
on the plane-embedded graphs shown above, we get the
following composite picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----------o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o-------o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` ` ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` o ` | | ` ` ` = ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | | ` | ` | | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | o---|---o | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` o ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o-----|-----o ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` O ` ` ` ` ` ` = ` ` ` ` O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Though it's not really there in the most abstract topology of the matter,
for all sorts of pragmatic reasons we find ourselves almost compelled to
single out the outermost region of the plane in a distinctive way and to
mark it as the "root node" of the corresponding dual graph, in the above
picture drawn as a slightly bigger node "O".

Extracting the dual graph from its composite matrix, we get this picture:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` O ` ` ` ` ` = ` ` ` ` ` O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

To be continued ...

Jon Awbrey

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