Re: ONT What Is Information That A Sign May Bear It?
Jon --
This is a nice exposition of the foundational situation and
questions that motivate your ongoing project of inquiry into
inquiry. I wonder if part of your project will ever come to
a consideration of one (as in " ... one finds oneself ... ")
as a sign emitter, as well as a sign receptor?
Doug McDavid
IBM Global Services
Member, IBM Academy of Technology
mcdavid@us.ibm.com
"Imagine all the people ... living life in peace."
Jon Awbrey <jawbrey@oakland.edu>@majordomo.ieee.org on 08/16/2002 09:45:36
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Subject: ONT What Is Information That A Sign May Bear It?
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What Is Information That A Sign May Bear It?
Here is a trio of questions that I try to keep in mind:
1. How is a sign empowered to contain information?
2. What is the practical context of communication?
3. Why do we care about these bits of information?
My way of addressing these questions is a bit like this:
We are mostly concerned with our own lives,
but then a world obtrudes on our existence,
and so we find ourselves forced to take up
an interest in the realities of its nature.
In pragmatic terms our initial pragma is a bit
like the verbal infinitive "to live", but then
it gets turned into the derivative substantial
forms of "nature", "reality", "the world", etc.
Against this backdrop one finds oneself cast as
a protagonist in a "scene of uncertainty" (SOU).
I picture this as a juncture where I have a set
of n options that fan out before me. It may be
a question of "What is true?", or "What to do?",
or "What to hope?", where the last is a codebit
for "What regulative principle has any chance?",
but the main uncertainty is that I am called on
to make a choice and often do not have any clue
what is fit to pick. (By the way, this picture
of the human practical fix is credited to Kant.)
Just to make up a discrete example let us suppose
that the cardinality of this choice is a finite n,
and just to make it fully concrete let us say n=5.
Here is the picture that I would have in mind for
such a situation:
o-------------------------------------------------o
| |
| ? ? ? ? ? |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| o o o o o |
| |
| ooooo |
| |
| @ n = 5 |
| |
o-------------------------------------------------o
Figure 1. Juncture of Degree 5
This pictures a juncture, represented by "@",
where there are n options for the outcome of
a conduct, and I do not have a clue which it
must be. In a sense the degree of this node,
in this case n = 5, measures the uncertainty
that I have at this point.
As best I can figure, this is the minimal sort of
setting in which a sign can make any sense at all.
A sign has significance for an agent, interpreter,
or observer because its actualization, its being
given or its being present, serves to reduce the
uncertainty of a decision that the agent has to
make, whether it concerns the actions that the
agent ought to take in order to achieve some
objective of interest, or whether it concerns
the predicates that the agent ought to treat
as being true of some object in the world.
The way that signs come into this setting,
to make the scene, as one used to say, is
something that I could picture as follows:
o-------------------------------------------------o
| |
| k_1 = 3 k_2 = 2 |
| o-----o-----o o-----o |
| "A" "B" |
| o----o----o o----o |
| |
| o---o---o o---o |
| |
| o--o--o o--o |
| |
| o-o-o o-o |
| |
| ooooo |
| |
| @ n = 5 |
| |
o-------------------------------------------------o
Figure 2. Partition of Degrees 3 and 2
This illustrates a situation of uncertainty
that has been augmented by a classification.
In the particular pattern of classification that is shown here,
the first three outcomes fall under the sign "A", and the next
two outcomes fall under the sign "B". If the outcomes make up
a set of "things that might be true about an object", then the
signs could be read as nomens (terms) or notions (concepts) of
a relevant empirical, ontological, taxonomical, or theoretical
scheme, that is, as predicates and predictions of the outcomes.
If the outcomes make up a set of "things that might be good to
do in order to achieve an objective", then the signs could be
read as bits of advice or other sorts of indicators that tell
us what to do in the situation, relative to our active goals.
Just to unpack some of the many things that may be getting
glossed over in this little word "sign", it encompasses all
of the "data of the senses" (DOTS) that we take as informing
us about inner and outer worlds, plus all of the concepts and
terms that we use to argue, to communicate, to inquire, or even
to speculate about our ontologies for beings and our policies for
action in the world.
This is the basic framework for talking about information and signs
in regard to communication, decision, and the uncertainties thereof.
Here is one of the places where it is tempting to try to
collapse the 3-adic sign relation into a 2-adic relation.
For if these DOTS are so closely identified with objects
that we can scarcely reckon how they might be discrepant,
then it will appear to us that one role of beings can be
eliminated from our picture of the world. In this event,
the only things that we are required to inform ourselves
about via the inspection of these DOTS are yet more DOTS,
whether past, or present, or prospective, just more DOTS.
This is the special form to which we frequently find the
idea of an information channel being reduced, namely, as
a "source" that has nothing better to tell us about than
its own conceivable conducts or its own potential issues.
As a matter of fact, at least in this discrete type of case, it would
be possible to use the degree of the node as a measure of uncertainty,
but it would operate as a multiplicative measure rather than the sort
of additive measure that we would normally prefer. To illustrate how
this would work out, let us consider an easier example, one where the
degree of the choice point is 4.
o-------------------------------------------------o
| |
| ? ? ? ? |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| oo oo |
| |
| @ n = 4 |
| |
o-------------------------------------------------o
Figure 3. Juncture of Degree 4
Suppose that we contemplate making another decision after
the present issue has been decided, one that has a degree
of 2 in every case. The compound situation looks like so:
o-------------------------------------------------o
| |
| o o o o o o o o |
| \ / \ / \ / \ / |
| o o o o n_2 = 2 |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| o o o o |
| |
| oo oo |
| |
| @ n_1 = 4 |
| |
o-------------------------------------------------o
Figure 4. Compound Junctures of Degrees 4 and 2
This depicts the fact that the compound uncertainty, 8,
is the product of the two component uncertainties, 4 x 2.
To convert this to an additive measure, we simply take the
logarithms to a convenient base, say 2, and thus we arrive
at the not too astounding fact that the uncertainty of the
first choice is 2 bits, the uncertainty of the next choice
is 1 bit, and the compound uncertainty is 3 = 2 + 1 bits.
In many ways, the provision of information, a process that
reduces uncertainty, is the inverse process to the kind of
uncertainty augmentation that occurs in compound decisions.
By way of illustrating this relationship, let us return to
our initial example.
A set of signs enters on a setup like this as a system
of middle terms, which I'm apt to regard as a "medium":
o-------------------------------------------------o
| |
| k_1 = 3 k_2 = 2 |
| o-----o-----o o-----o |
| "A" "B" |
| o----o----o o----o |
| |
| o---o---o o---o |
| |
| o--o--o o--o |
| |
| o-o-o o-o |
| |
| ooooo |
| |
| @ n = 5 |
| |
o-------------------------------------------------o
Figure 2. Partition of Degrees 3 and 2
The "language" or "medium" here is the set of signs {"A", "B"}.
On the assumption that the initial outcomes are equally likely,
we associate a frequency distribution <k_1, k_2> = <3, 2>, and
a probability distribution <p_1, p_2> = <3/5, 2/5> = <0.6, 0.4>
with this language, and thus define a communication "channel".
The most important thing here is really just to get a handle on
the "conditions for the possibility of signs making sense", but
once we have this much of a setup we find that we can begin to
construct some rough and ready bits of information-theoretic
furniture, like measures of uncertainty, channel capacity,
and the amount of information that can be associated with
the reception or the recognition of a single sign. Still,
before I get into all of this, I want to emphasize that,
even when these measures are too ad hoc and insufficient
to be of much use per se, the significance of the setup
that it takes to support them is not at all diminished.
Consider the augmented situation of uncertainty that was depicted above.
What happens if we receive one or the other of the two signs "A" or "B"?
A. If we receive "A" our uncertainty is reduced from log 5 to log 3.
B. If we receive "B" our uncertainty is reduced from log 5 to log 2.
It is from these characteristics that the "information capacity"
of a communication channel can be defined, specifically, as the
"average uncertainty reduction on receiving a sign" (AURORAS).
In the present example we have:
| Channel Capacity
|
| = (1/n)(k_1·(log n - log k_1) + k_2·(log n - log k_2))
|
| = (k_1/n)(log n - log k_1) + (k_2/n)(log n - log k_2)
|
| = (-k_1/n)(log k_1 - log n) + (-k_2/n)(log k_2 - log n)
|
| = (-k_1/n)(log k_1/n) + (-k_2/n)(log k_2/n)
|
| = - (p_1 log p_1 + p_2 log p_2)
|
| = - (0.6 log 0.6 + 0.4 log 0.4)
|
| = 0.971
In other words, the capacity of the channel is slightly under 1 bit.
This makes intuitive sense, as 3 versus 2 is a near-even split of 5,
and the measure of channel capacity or the "entropy" is supposed to
attain its maximum of 1 bit whenever a two-way partition is 50-50.
Jon Awbrey
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