[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D8

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Theory of Control and Control of Theory

| You will hardly know who I am or what I mean,
| But I shall be good health to you nevertheless,
| And filter and fibre your blood.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]

In the boolean context, a function f : X -> B is tantamount to a "proposition"
about elements of X, and the elements of X constitute the "interpretations" of
that proposition.  The fiber f^(-1)(1) comprises the set of "models" of f, or
examples of elements in X satisfying the proposition f.  The fiber f^(-1)(0)
collects the complementary set of "anti-models", or the exceptions to the
proposition f that exist in X.  Of course, the space of functions (X -> B)
is isomorphic to the set of all subsets of X, called the "power set" of X
and often denoted as Pow(X) or 2^X.

The operation of replacing X by (X -> B) in a type schema corresponds
to a certain shift of attitude towards the space X, in which one passes
from a focus on the ostensibly individual elements of X to a concern with
the states of information and uncertainty that one possesses about objects
and situations in X.  The conceptual obstacles in the path of this transition
can be smoothed over by using singular functions (X ::> B) as stepping stones.
First of all, it's an easy step from an element x of type B^n to the equivalent
information of a singular proposition x : X ::> B, and then only a small jump of
generalization remains to reach the type of an arbitrary proposition f : X -> B,
perhaps understood to indicate a relaxed constraint on the singularity of points
or a neighborhood circumscribing the original x.  I have frequently discovered
this to be a useful transformation, communicating between the "objective" and
the "intentional" perspectives, in spite perhaps of the open objection that
this distinction is transient in the mean time and ultimately superficial.

It is hoped that this measure of flexibility, allowing us to stretch a point
into a proposition, can be useful in the examination of inquiry driven systems,
where the differences between empirical, intentional, and theoretical propositions
constitute the discrepancies and the distributions that drive experimental activity.
I can give this model of inquiry a cybernetic cast by realizing that theory change
and theory evolution, as well as the choice and the evaluation of experiments, are
actions that are taken by a system or its agent in response to the differences
that are detected between observational contents and theoretical coverage.

All of the above notwithstanding, there are several points that distinguish
these two tasks, namely, the "theory of control" and the "control of theory",
features that are often obscured by too much precipitation in the quickness
with which we understand their similarities.  In the control of uncertainty
through inquiry, some of the actuators that we need to be concerned with are
axiom changers and theory modifiers, operators with the power to compile and
to revise the theories that generate expectations and predictions, effectors
that form and edit our grammars for the languages of observational data, and
agencies that rework the proposed model to fit the actual sequences of events
and the realized relationships of values that are observed in the environment.
Moreover, when steps must be taken to carry out an experimental action, there
must be something about the particular shape of our uncertainty that guides us
in choosing what directions to explore, and this impression is more than likely
influenced by previous accumulations of experience.  Thus it must be anticipated
that much of what goes into scientific progress, or any sustainable effort toward
a goal of knowledge, is necessarily predicated on long term observation and modal
expectations, not only on the more local or short term prediction and correction.

Jon Awbrey

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