Re: Information = Comprehension x Extension -- Discussion

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ICE.  Discussion Note 19

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One of the themes that appears to be arising from
this complex of issues is the relationship between
global thought and local action, not exactly to coin
a new phrase, as Goethe long time passingly aphorized
the prevalent but hopefully not the prevailing problem
in his succinct formula, "Action limits, thought lames".

One of the best ways to understand the relationship between
global abstraction and local application is by examining the
way that represented information constrains the structures of
various order systems, such as pre-orders, partial orders, and
lattices.  At this point we run into a problem of exposition in
the fact that "order" is too general, even ambiguous when taken
out of context, while "lattice" is too specific in the strictest
use of the term, so we find ourselves using the word "lattice"
in both a strict and a loose sense, depending on the context.

In mathematics, the action of an informational constraint on a pre-existing
structure is described as a "quotient" operation.  One frequently thinks of
this process as using a set of "generator" elements to over-generate a more
abundant variety of structure than one really needs and then using various
sets of "relator" formulas to "divide out" the extraneous portions of the
structure or variety in question.

When we turn to the problems of logic, semiotics, and scientific inference
that Peirce was trying to tackle with his information theory, our attempt
to apply the quotient method is complicated by the fact that we have two
sorts of quotients to think about.  There is the "syntax to semantics"
quotient by which the diversity of syntactic signs is partitioned into
"logical equivalence classes" (LEC's) by whatever laws of logic are
applicable by default.  Then there is the "semantics to semantics"
quotient by which the object domain in one state of information
turns into the object domain in another state of information.

That indicates some of the work of understanding that
we have yet to do, when next we turn to Peirce's text.

But first, to breakfast, and another cup of coffee.

Jon Awbrey

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