[Inquiry] Re: Sign Relations -- Commentary

[Jon Awbrey]

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SR.  Commentary Note 27

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Re: SR-COM 25.  http://stderr.org/pipermail/inquiry/2005-September/003061.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-September/thread.html#3028

I've been meaning to get back to the gang of four 3-adic relations that
we were looking at in connection with the question of iconic minimalism,
that is, our inquiry into the simplest possible sign relations that can
be devised that might conceivably qualify as containing iconic signs.

One of the things that makes this type of investigation interesting,
over and above the focal question of icons, is that it illustrates
some of the themes of recursive inquiry, in particular, the ways
that an inquiry into one class of sign relations, that we might
call the "object" class, relies on our ability to use as ways,
means, and tools another class of sign relations, the "base"
or "resource" class.  For example, in the present case, we
are using the sign relations of propositional expressions
as a "base camp" for the ascent on iconic sign relations.

We can see this theme more clearly now if we look at the relationship
between the following portraits of 3-adic relations and their legends,
the propositional expressions that are inscribed beneath their frames:

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| x y z | ` ` | x y z | ` ` | x y z | ` ` | x y z |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| 0 0 0 | ` ` | 0 0 0 | ` ` | 0 0 1 | ` ` | 0 0 0 |
| 0 1 1 | ` ` | 0 1 0 | ` ` | 0 1 0 | ` ` | 0 1 1 |
| 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 0 | ` ` | 1 0 1 |
| 1 1 0 | ` ` | 1 1 1 | ` ` | 1 1 1 | ` ` | 1 1 0 |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
z = (x) y ` ` z `=` x y ` ` z=((x,y)) ` ` z = (x,y)

Indeed, acting on Peirce's discovery that graphical calculi are extremely
useful for the logical grasp of many subjects, largely by virtue of their
x-ray facility for revealing the functional articulations of propositions,
it might be useful to examine the graphical forms that these propositions
take, at least, as they allow of rendering into the present medium by way
of the "parse graphs" that are topologically dual to a moderate extension
of Peirce's own "alpha graphs".

To make a long story short, here is my first try
at graphing the above collection of propositions.

o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` P ` | ` ` | ` R ` | ` ` | ` S ` | ` ` | ` U ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
| ` ` ` | ` ` | ` ` ` | ` ` | x ` y | ` ` | ` ` ` |
| ` ` ` | ` ` | ` ` ` | ` ` | o---o | ` ` | ` ` ` |
| ` ` ` | ` ` | ` ` ` | ` ` | `\`/` | ` ` | x ` y |
| ` o x | ` ` | ` ` ` | ` ` | ` o ` | ` ` | o---o |
| ` | ` | ` ` | `x`y` | ` ` | ` | ` | ` ` | `\`/` |
| z=O y | ` ` | z=O ` | ` ` | z=O ` | ` ` | z=O ` |
o-------o ` ` o-------o ` ` o-------o ` ` o-------o
z = (x) y ` ` z `=` x y ` ` z=((x,y)) ` ` z = (x,y)

These graphs are "mixed mode", inasmuch as they use
the non-graphical symbol of equality "=", and thus
they will need to be converted into a non-hybrid
syntax before they become fully satisfactory as
logical graphs.

Jon Awbrey

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