[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 21

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Cybernetics List, Peirce List,

Cf: FOLG 5.  http://stderr.org/pipermail/inquiry/2005-October/003113.html
In: FOLG.    http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104

Toward the outset of this excursion, I mentioned the distinction between a
"pointwise-restricted iconic map" or a "pointedly rigid iconic map" (PRIM)
and a "system-wide iconic map" (SWIM).  The time has come to make use of
this mention.

We are once again concerned with "categories of structured items" (COSI's)
and the categories of mappings between them, indeed, the two ideas are all
but inseparable, there being many good reasons to consider the very notion
of structure to be most clearly defined in terms of the brands of "arrows",
maps, or morphisms between items that are admitted to the category in view.

At the level of the "primary arithmetic" (PAR), we have a set-up like this:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
Categories` ` ` ` `!O!` ` ` ` ` ` ` ` ` ` ` ` `!S!` ` ` ` ` `
` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` `
` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` `
` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` `
` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` `
` ` ` ` ` ` ` `/` ` ` ` `\` ` Denotes ` ` `/` ` ` ` `\` ` ` `
Individuals `{F}` ` ` ` `{T}` <------ ` {...} ` ` ` {...} ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `/|\` ` ` ` `/|\` ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` / | \ ` ` ` / | \ ` `
` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` `/` | `\` ` `/` | `\` `
Structures` ` F ` ` ` ` ` T ` ` ` ` ` o ` o ` o ` o ` o ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` o ` ` ` ` o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | `\`/` ` ` ` ` | `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` o ` o o ` o o `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` | ` | `\`/` | `
` ` ` ` ` ` ` F ` ` ` ` ` T ` ` ` ` ` O ` O ` O ` O ` O ` O `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The object domain !O! is the boolean domain !B! = {Falsity, Truth},
the semiotic domain !S! is any of the spaces isomorphic to the set
of rooted trees, matched-up parentheses, or unlabeled alpha graphs,
and we treat a couple of "denotation maps" D_en, D_ex : !S! -> !O!.

Either one of the denotation maps induces the same partition of !S!
into REC's, a partition whose structure can be suggested by writing
the following set of sets of strings:

{{" ", "(( ))", "(( )( ))", ...}, {"( )", "( )( )", "((( )))", ...}

These are of course the parenthesis strings that correspond to the
rooted trees that are shown in the lower right corner of the Figure.

Jon Awbrey

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