[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 23

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Cybernetics List, Peirce List,

Corresponding to the Entitative and Existential interpretations
of the primary arithmetic, there are two distinct mappings from
the sign domain !S!, containing the topological equivalents of
bare and rooted trees, onto the object domain !O!, containing
the two objects whose conventional, ordinary, or meta-language
names are "Falsity" and "Truth", respectively.

The next two Figures suggests how we might view the interpretation maps
as mappings from a cosi !S! to a cosi !O!.  Here I have placed names of
categories at the bottom, indices of individuals at the next level, and
extended upward from there whatever structures the individuals may have.

Here is the Figure for the Entitative interpretation:

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

Here is the Figure for the Existential interpretation:

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

Note that the structure of a tree begins at its root, marked by an "O".
The objects in !O! have no further structure to speak of, so there is
nothing much happening in the object domain !O! between the level of
individuals and the level of structures.  In the sign domain !S!,
the individuals are the parts of the partition into referential
equivalence classes, each part of which contains a countable
infinity of syntactic structures, rooted trees, or whatever
form one views their structures taking.  The sense of the
Figures is that the interpretation under consideration
maps the individual on the left (right) side of !S!
to the individual on the left (right) side of !O!.

Jon Awbrey

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