[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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PLOI.  Note 6

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| To explain this, we must remember that the process of induction is a
| process of adding to our knowledge;  it differs therein from deduction --
| which merely explicates what we know -- and is on this very account called
| scientific inference.  Now deduction rests as we have seen upon the inverse
| proportionality of the extension and comprehension of every term;  and this
| principle makes it impossible apparently to proceed in the direction of
| ascent to universals.  But a little reflection will show that when our
| knowledge receives an addition this principle does not hold.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 463

Peirce's usage varies here, but for the sake of simplicity let's regard
extension as a set of instances and comprehension as a set of properties.
This will allow us to treat breadth as a measure of extension and depth
as a measure of comprehension, writing breadth(t) = |extension(t)| and
depth(t) = |comprehension(t)| for any term t.

In the classical model, where the measures of extension and comprehension
are inversely proportional, we can write an equation of the following sort:

   breadth * depth  =  constant

In Peirce's theory, this constant is identified as the measure of information,
which may remain constant in certain situations, but which may vary in others:

   breadth * depth  =  information

A rough plot may serve to illustrate certain aspects of the situation.
For convenience in graphing, let's indicate the measure of information
by means of the symbols [1, ..., 9, a, b, c, d, e, f, *] to signify the
range of integral values from 1 to 16.  If we can imagine interpolating
smooth curves through the symbols plotted for a given information value,
then that would give us some idea of the "constant information curves"
or the "isoformal curves" of this information topography.

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` Depth ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ^ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | f ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | e ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | d ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | b ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 9 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 8 * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 7 e ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 6 c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 5 a f ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 4 8 c * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 3 6 9 c f ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 2 4 6 8 a c e * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` | 1 2 3 4 5 6 7 8 9 a b c d e f * ` ` ` ` ` ` ` ` `
` ` ` ` ` o----------------------------------> Breadth` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Jon Awbrey

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