[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 4

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Cybernetics List, Peirce List,

We have already seen various forms of the axiom that is formulated
in string form as "(( )) =  ".  For the sake of comparison, let's
record the planar and dual forms of the axiom that is formulated
in string form as "( )( ) = ( )".

First the planar form:

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

Next the planar and dual forms superimposed:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` o-------o ` ` ` o-------o ` ` ` ` ` ` ` o-------o ` ` `
` ` ` | ` ` ` | ` ` ` | ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` ` `
` ` ` | ` o ` | ` ` ` | ` o ` | ` ` ` = ` ` ` | ` o ` | ` ` `
` ` ` | ` `\` | ` ` ` | `/` ` | ` ` ` ` ` ` ` | ` | ` | ` ` `
` ` ` o-----\-o ` ` ` o-/-----o ` ` ` ` ` ` ` o---|---o ` ` `
` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` = ` ` ` ` ` O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Finally the dual form by itself:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` `
` ` ` ` ` `\` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` = ` ` ` ` ` O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

We have at this point enough material to begin thinking about
the forms of analogy, iconicity, metaphor, morphism, whatever
you want to call it, that are pertinent to the use of logical
graphs in their various logical interpretations, for instance,
those that Peirce described as "entitative" and "existential".

By way of providing a conceptual-technical framework for
organizing that discussion, let me introduce the concept
of a "category of structured individuals" (COSI).  There
may be some cause for some readers to rankle at the very
idea of a "structured individual", for taking the notion
of an individual in its strictest etymology would render
it absurd that an atom could have parts, but all we mean
by "individual" in this context is an individual by dint
of some conversational convention currently in play, not
an individual on account of its intrinsic indivisibility.
Incidentally, though, it will also be convenient to take
in the case of a class or collection of individuals with
no pertinent inner structure as a trivial case of a COSI.

To be continued ...

Jon Awbrey

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