[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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PLOI.  Note 2

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In my current state of information I'm aware of Peirce
having investigated at least two different measures of
information, one that we may call "multiplicative" and
the other that we may call "exponential".  Though I'll
take up with the multiplicative measure first, it will
be one of my interests here to understand the possible
relation between these two measures of information and
also their relation to current concepts of information.

Somewhere in his manuscripts that I saw many years ago, Peirce illustrates
multiplicative measures in discrete situations by drawing bipartite graphs,
called "bigraphs", and counting the lines between the points of the graphs.
In the same vein, he illustrates exponential measures by counting how many
functional bigraphs can be drawn between two sets of points of given sizes.

The conical picture that I drew before illustrates the multiplicative case,
if we are given that we have j properties in the comprehension of object x
and k instances in the extension of object x, then the information measure
associated with object x is given by the product j*k, a product which also
counts the number of edges in the complete bigraph between the two domains
of properties and instances.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o.......o ` ` Properties` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` Object` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o.......o ` ` Instances ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

This "area measure" of information, given by Peirce's formula,
"Information = Comprehension x Extension", is the background
against which the change of information associated with all
further constraints can be reckoned.

That's a very rough sketch of just one aspect
of Peirce's theory, but we can refine it as
we go.  The next order of business, though,
is to balance these abstract speculations
with a few well-posed concrete examples.

Jon Awbrey

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