[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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

PLOI.  Note 9

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

I am beginning to have third and fourth thoughts about my
current reading of Peirce's intent with the first example --
it seems more likely that he had in mind an unarticulated
number of things that have colors other than red -- still,
the present scenario doesn't appear to be inconsistent in
its own right, and it does seem to give us something near
to the minimal model of Peirce's descriptions, so I think
that it may be worthwhile following through for any light
that it can shed on the topic area of information process.

It will help, in the long run, to tighten up some of the terminology
that we are using to discuss the abstract forms of graphs and their
various and sundry styles of relatively concrete representation.
As a side benefit, we will find this general paradigm of usage
readily adaptable to making the needed distinctions between
abstract graphs and their concrete "replicas", to mention
Peirce's own favorite word for the same distinction.

Consider the graphical matrix that we last looked on:

` ` ` ` ` ` ` ` ` ` ` ` A ` ` B ` ` C ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Color | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` Red ` | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This is the "incidence matrix" of a "concretely labeled graph".

Taking one small step of abstraction up from there would give us
the "incidence matrix" of an "abstractly labeled graph", like so:

` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 2 ` ` 3 ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` 4 | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` 5 | ` 1 ` ` 1 ` ` 1 ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` o-------------------o ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This is the incidence matrix of a "labeled bipartite graph", or
a "labeled bigraph", for short.  The adjective "bipartite" means
that its set of points (aka nodes or vertices) can be partitioned
into just two parts, such that all the lines (aka edges or blocks)
of the graph lie between the points of one part of the partition
and the points of the other part of the partition, with no lines
that lie between the points in any one part of the partition.

As it happens, the particular example that we are contemplating here
has all of the lines that it can have for the partition in question,
that is, a line between each point of one part and every point of
the other part, and so it's called a "complete labeled bigraph".
On account of the partition of points into a 2-set and a 3-set,
the abstract graph that is said to underlie this labeled graph
is standardly notated by graph theorists as "K_2,3", where the
letter "K" is evidently intended as a mnemonic for "complete".

Look's like a good place to take a break ...

Jon Awbrey

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