[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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PLOI.  Note 19

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By way of a recap, the current state of our inquiry into
Peirce's ideas about information can be stated as follows.

Peirce has offered us a couple of formulas for information:

   Information = Comprehension * Extension

   Information = Superfluous Comprehension

By way of explaining the latter idea he says the following:

| The information of a term is the measure of its superfluous comprehension.
| That is to say that the proper office of the comprehension is to determine
| the extension of the term.  For instance, you and I are men because we
| possess those attributes -- having two legs, being rational, etc. --
| which make up the comprehension of 'man'.  Every addition to the
| comprehension of a term lessens its extension up to a certain
| point, after that further additions increase the information
| instead.
|
| C.S. Peirce, 'Chronological Edition', CE 1, 467
| Cf: ICE 2.  http://stderr.org/pipermail/inquiry/2004-November/001914.html
| In: ICE.    http://stderr.org/pipermail/inquiry/2004-November/thread.html#1913

These ideas sound plausible enough at first -- at least they did to me --
but in trying to work out the kinds of details that it would take to
make a bona fide measure out of this notion of information, we run
into a bunch of questions that we have to answer before going on.

By way of illustrating these concepts in a concrete case,
Peirce invents a story about a blind man learning about
color terms, and describes the learner's initial state
of information as defined by the following data set:

   Comprehension("Red")  =  {'Color'}

   Extension("Red")      =  {'A', 'B', 'C'}

On this data we would calculate the measure
of information in the term "Red" as follows:

   |Information("Red")|  =  |{'Color'}| * |{'A', 'B', 'C'}|  =  1 * 3  =  3

Some of the questions that arise at this point are these:

   1.  What is the reason for excluding 'Red' from the comprehension of "Red"?

   2.  What is the criterion for superfluous comprehension?

   3.  What is the extension of "Color" in this example?

   4.  What is the measure of an indefinite extension?

Questions like these require us to think more carefully
about the context in which measures are usually defined.
I don't know if Peirce in 1865-1866 was aware of all of
these issues, but I know that he eventually became very
sophisticated about the conditions of defining measures,
at least in the case of probability measures, which are
roughly the extensional half of the information problem.

Jon Awbrey

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