[Arisbe] Re: Differential Logic & Dynamic Systems

[Jon Awbrey]

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Dif Log Dyn Sys SIG:

I will now make yet another attempt to re-introduce this subject
to which I have alluded before under some of the following names:
"Analytic Differential Ontology" (ADO), when we imagine it to be
a question of being itself, or the "Differential Extension" (DE)
of logic, for now, just the propositional or the sentential type
of logic, specifically, for my part, as expressed in what I call
a "Reflective Extension" (RE) of Peirce's Logical Graphs, in sum,
named "DiffLog" and "RefLog", respectively.  I will apologize at
this point for all of the "rude mnemonic mechanics", but I think
that you can probably guess by now that this is utterly the only
way that I have found to keep track of all this stuff in my head.

The quickest way for me to open up this topic is just to jump in,
and so I will cite here my very first outline of it, that formed
an appendix to my Master's (in Psych) Thesis Substitute Document.
I am hoping that the very simplicity of this will make it useful
as an initial invitation, aided by the circumstance that I wrote
it in such a way as to form a gentle bridge between the ordinary
difference calculus, executed over the reals R or the integers Z,
and the logical difference calculus, working over the booleans B.
Accordingly, all of the definitions, equations, expressions, and
formulas in the following presentation can be read independently
of whether you interpret the ostensible names as denoting values
in R, or Z, or B.

| Notes On Approach & On Notation:
|
| This Chapter is titled "Linear Topics" because that is the heading
| under which the derivatives and the differentials of any functions
| usually come up in mathematics, that is, related to the finding of
| a "local and linear approximation" (LALA) to the more unrestrained,
| arbitrary brands of functions that one is given in a given context.
|
| I will have to use the "numb-artifice" of "#A#" for a bold letter "A".

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Chapter 3.  Linear Topics

The Differential Theory of Qualitative Equations

To denote lists of propositions and to detail their components,
we use notations like:

   #a# = <a, b, c>,   #p# = <p, q, r>,   #x# = <x, y, z>,

or, in more complicated situations:

   x = <x1, x2, x3>,  y = <y1, y2, y3>,  z = <z1, z2, z3>.

In a universe where some region is ruled by a proposition,
it is natural to ask whether we can change the value of that
proposition by changing the features of a current state.

Given a Venn diagram with a shaded region and starting from
any cell in that universe, what sequences of feature changes,
what traverses of cell walls, will take us from shaded to
unshaded areas, or the reverse?

In order to discuss questions of this type, it is useful
to define several "operators" on functions.  An operator
is nothing more than a function between sets that happen
to have functions as members.

A typical operator F takes us from thinking about a given function f
to thinking about another function g.  To express the fact that g can
be obtained by applying F to f, we write g = Ff.
 
The first operator, E, associates with a function f : A -> B,
another function Ef, where Ef : AxA -> B is defined as follows:

   Ef(x, y) = f(x + y).

E is called a "shift operator" because it takes us from
contemplating the value of f at a place x to considering
the value of f at a shift of y away.  In effect, it tells
us the absolute effect on f of changing its argument from x
by an amount that is equal to y.

The second operator, D, associates with a function f : A -> B
another function Df, where Df: A x A -> B is defined by

   Df(x, y) = Ef(x, y) - f(x),

or, equivalently,

   Df(x, y) = f(x + y) - f(x).

D is called a "difference operator" because it informs us of the
relative change in the value of f along the shift from x to x + y.
 
In practice, one of the variables x or y is usually
considered as being "less variable" than the other,
being fixed in the context of a concrete discussion.
Thus, we may we see any one of the following idioms:

1.  Df : AxA -> B,

    Df(c, x) = f(c + x) - f(c).

Here, c is held constant and Df(c, x) is regarded
mainly as a function of the second variable x,
giving the relative change in f at various
distances x from the center c.

2.  Df : AxA -> B,

    Df(x, h) = f(x + h) - f(x).

Here, h is either a constant (usually 1), in discrete contexts,
or a variably "small" amount (near to 0) over which a limit is
being taken, as in continuous contexts.  Df(x, h) is regarded
mainly as a function of the first variable x, in effect, giving
the differences in the value of f between x and a neighbor that
is a distance of h away, all the while that x itself ranges over
its various possible locations.

3.  Df : AxA -> B,

    Df(x, dx) = f(x + dx) - f(x).

This is yet another variant of the previous form,
with dx denoting small changes contemplated in x.
 
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That's the basic idea.  After I take a little break,
I will return to develop the logical side of things
a bit more fully, and also take up the elaboration
of some moderately simple applications of these
ideas to some relatively concrete examples.

Until Then,

Jon Awbrey

P.S.  I have not read this old treatise in more than ten years,
      so if there are mistakes I will ferret them out as we go.
J.A.

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