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{{DISPLAYTITLE:Dynamics And Logic}}
 +
<pre>
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Dynamics And Logic
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 1
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I am going to excerpt some of my previous explorations
 +
on differential logic and dynamic systems and bring them
 +
to bear on the sorts of discrete dynamical themes that we
 +
find of interest in the NKS Forum.  This adaptation draws on
 +
the "Cactus Rules", "Propositional Equation Reasoning Systems",
 +
and "Reductions Among Relations" threads, and will in time be
 +
applied to the "Differential Analytic Turing Automata" thread:
 +
 +
CR.    http://forum.wolframscience.com/showthread.php?threadid=256
 +
PERS.  http://forum.wolframscience.com/showthread.php?threadid=297
 +
RAR.  http://forum.wolframscience.com/showthread.php?threadid=400
 +
DATA.  http://forum.wolframscience.com/showthread.php?threadid=228
 +
 +
One of the first things that you can do, once you have
 +
a moderately functional calculus for boolean functions
 +
or propositional logic, whatever you choose to call it,
 +
is to start thinking about, and even start computing,
 +
the differentials of these functions or propositions.
 +
 +
Let us start with a proposition of the form "p and q",
 +
that is graphed as two labels attached to a root node:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      p q                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                    p and q                    |
 +
o-------------------------------------------------o
 +
 +
Written as a string, this is just the concatenation "p q".
 +
 +
The proposition pq may be taken as a boolean function f<p, q>
 +
having the abstract type f : B x B -> B, where B = {0, 1} is
 +
read in such a way that 0 means "false" and 1 means "true".
 +
 +
In this style of graphical representation,
 +
the value "true" looks like a blank label
 +
and the value "false" looks like an edge.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                      true                      |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                        |
 +
|                        |                        |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                      false                      |
 +
o-------------------------------------------------o
 +
 +
Back to the proposition pq.  Imagine yourself standing
 +
in a fixed cell of the corresponding venn diagram, say,
 +
the cell where the proposition pq is true, as pictured:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|          o-----------o  o-----------o          |
 +
|        /            \ /            \        |
 +
|        /              o              \        |
 +
|      /              /%\              \      |
 +
|      /              /%%%\              \      |
 +
|    o              o%%%%%o              o    |
 +
|    |              |%%%%%|              |    |
 +
|    |              |%%%%%|              |    |
 +
|    |      P      |%%%%%|      Q      |    |
 +
|    |              |%%%%%|              |    |
 +
|    |              |%%%%%|              |    |
 +
|    o              o%%%%%o              o    |
 +
|      \              \%%%/              /      |
 +
|      \              \%/              /      |
 +
|        \              o              /        |
 +
|        \            / \            /        |
 +
|          o-----------o  o-----------o          |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
Now ask yourself:  What is the value of the
 +
proposition pq at a distance of dp and dq
 +
from the cell pq where you are standing?
 +
 +
Don't think about it -- just compute:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  dp o  o dq                  |
 +
|                    / \ / \                    |
 +
|                  p o---@---o q                  |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
 +
To make future graphs easier to draw in Asciiland,
 +
I'll use devices like @=@=@ and o=o=o to identify
 +
several nodes into one, as in this next redrawing:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
 +
However you draw it, these expressions follow because the
 +
expression p + dp, where the plus sign indicates addition
 +
in B and thus corresponds to the exclusive-or in logic,
 +
parses to a graph of the following form:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    p    dp                    |
 +
|                      o---o                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                    (p, dp)                    |
 +
o-------------------------------------------------o
 +
 +
Next question:  What is the difference between the value of the
 +
proposition pq "over there", at a remove of dp dq, and the value
 +
of the proposition pq where you are, all expressed in the form of
 +
a general formula, of course?  Here is the appropriate formulation:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            p  dp q  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/        p q                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|              ((p, dp)(q, dq), p q)              |
 +
o-------------------------------------------------o
 +
 +
There is one thing that I ought to mention at this point:
 +
Computed over B, plus and minus are identical operations.
 +
This will make the relation between the differential and
 +
the integral parts of the appropriate calculus slightly
 +
stranger than usual, but we will get into that later.
 +
 +
Last question, for now:  What is the value of this expression
 +
from your current standpoint, that is, evaluated at the point
 +
where pq is true?  Well, substituting 1 for p and 1 for q in
 +
the graph amounts to erasing the labels "p" and "q", like so:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/                            |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|              (( , dp)( , dq),    )              |
 +
o-------------------------------------------------o
 +
 +
And this is equivalent to the following graph:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    dp  dq                    |
 +
|                      o  o                      |
 +
|                      \ /                      |
 +
|                        o                        |
 +
|                        |                        |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                  ((dp) (dq))                  |
 +
o-------------------------------------------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 2
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have just met with the fact that
 +
the differential of the "and" is
 +
the "or" of the differentials.
 +
 +
p and q --Diff--> dp or dq.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                    dp  dq      |
 +
|                                    o  o      |
 +
|                                      \ /        |
 +
|                                      o        |
 +
|        p q                            |        |
 +
|        @          --Diff-->          @        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|        p q        --Diff-->    ((dp) (dq))    |
 +
o-------------------------------------------------o
 +
 +
It will be necessary to develop a more refined analysis of
 +
this statement directly, but that is roughly the nub of it.
 +
 +
If the form of the above statement reminds you of DeMorgan's rule,
 +
it is no accident, as differentiation and negation turn out to be
 +
closely related operations.  Indeed, one can find discussions of
 +
logical difference calculus in the Boole-DeMorgan correspondence
 +
and Peirce also made use of differential operators in a logical
 +
context, but the exploration of these ideas has been hampered
 +
by a number of factors, not the least of which has been the
 +
lack of a syntax that was up to handling the complexity of
 +
the expressions that evolve.
 +
 +
Let us run through the initial example again, this time attempting
 +
to interpret the formulas that develop at each stage along the way.
 +
 +
We begin with a proposition, or a boolean function, f<p, q> = pq.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|          o-----------o  o-----------o          |
 +
|        /            \ /            \        |
 +
|        /              o              \        |
 +
|      /              /%\              \      |
 +
|      /              /%%%\              \      |
 +
|    o              o%%%%%o              o    |
 +
|    |              |%%%%%|              |    |
 +
|    |              |%%%%%|              |    |
 +
|    |      P      |% F %|      Q      |    |
 +
|    |              |%%%%%|              |    |
 +
|    |              |%%%%%|              |    |
 +
|    o              o%%%%%o              o    |
 +
|      \              \%%%/              /      |
 +
|      \              \%/              /      |
 +
|        \              o              /        |
 +
|        \            / \            /        |
 +
|          o-----------o  o-----------o          |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      p q                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| f =                  p q                      |
 +
o-------------------------------------------------o
 +
 +
A function like this has an abstract type and a concrete type.
 +
The abstract type is what we invoke when we write things like
 +
f : B x B -> B or f : B^2 -> B.  The concrete type takes into
 +
account the qualitative dimensions or the "units" of the case,
 +
which can be explained as follows.
 +
 +
  Let !P! be the set of two values {(p), p} = {not-p, p} ~=~ B
 +
 +
  Let !Q! be the set of two values {(q), q} = {not-q, q} ~=~ B
 +
 +
Then interpret the usual propositions about p, q
 +
as functions of concrete type f : !P! x !Q! -> B.
 +
 +
We are going to consider various "operators" on these functions.
 +
Here, an operator W is a function that takes one function f into
 +
another function Wf.
 +
 +
The first couple of operators that we need to consider are
 +
logical analogues of the pair that play a founding role
 +
in the classical "finite difference calculus", namely:
 +
 +
  The "difference" operator [capital Delta], written here as D.
 +
 +
  The "enlargement" operator [capital Epsilon], written here as E.
 +
 +
These days, E is more often called the "shift" operator.
 +
 +
In order to describe the universe in which these operators operate,
 +
it will be necessary to enlarge our original universe of discourse.
 +
 +
Starting out from the initial space X = !P! x !Q!, we
 +
construct its (first order) "differential extension":
 +
 +
  EX  =  X x dX  =  !P! x !Q! x d!P! x d!Q!
 +
 +
  where:
 +
 +
    X  =  !P! x !Q!
 +
 +
    dX  =  d!P! x d!Q!
 +
 +
  d!P!  =  {(dp), dp}
 +
 +
  d!Q!  =  {(dq), dq}
 +
 +
The interpretations of these new symbols can be diverse,
 +
but the easiest interpretation for now is just to say
 +
that "dp" means "change p" and "dq" means "change q".
 +
 +
Drawing a venn diagram for the differential extension EX = X x dX
 +
requires four logical dimensions, !P!, !Q!, d!P!, d!Q!, but it is
 +
possible to project a suggestion of what the differential features
 +
dp and dq are about on the 2-dimensional base space X = !P! x !Q!
 +
by drawing arrows that cross the boundaries of the basic circles
 +
in the venn diagram for X, reading an arrow as dp if it crosses
 +
the boundary between p and (p) in either direction and reading
 +
an arrow as dq if it crosses the boundary between q and (q)
 +
in either direction.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|          o-----------o  o-----------o          |
 +
|        /            \ /            \        |
 +
|        /      p      o      q      \        |
 +
|      /              /%\              \      |
 +
|      /              /%%%\              \      |
 +
|    o              o%%%%%o              o    |
 +
|    |              |%%%%%|              |    |
 +
|    |        dq    |%%%%%|    dp        |    |
 +
|    |    <---------|--o--|--------->    |    |
 +
|    |              |%%%%%|              |    |
 +
|    |              |%%%%%|              |    |
 +
|    o              o%%%%%o              o    |
 +
|      \              \%%%/              /      |
 +
|      \              \%/              /      |
 +
|        \              o              /        |
 +
|        \            / \            /        |
 +
|          o-----------o  o-----------o          |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
We can form propositions from these differential variables
 +
in the same way that we would any other logical variables,
 +
for example, taking the differential proposition (dp (dq))
 +
as saying that dp implies dq, in other words, that there
 +
is "no change in p without a change in q".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 3
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Given the proposition f<p, q> over the space X = !P! x !Q!,
 +
the (first order) "enlargement" of f is the proposition Ef
 +
over the differential extension EX that is defined by the
 +
following formula:
 +
 +
  Ef<p, q, dp, dq>
 +
 +
  =  f<p + dp, q + dq>
 +
 +
  =  f<(p, dp), (q, dq)>
 +
 +
In the example f<p, q> = pq, the enlargement Ef is given by:
 +
 +
  Ef<p, q, dp, dq>
 +
 +
  =  [p + dp][q + dq]
 +
 +
  =  (p, dp)(q, dq)
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef =            (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
 +
Given the proposition f<p, q> over X = !P! x !Q!, the
 +
(first order) "difference" of f is the proposition Df
 +
over EX that is defined by the formula Df = Ef - f, or,
 +
written out in full:
 +
 +
  Df<p, q, dp, dq>
 +
 +
  =  f<p + dp, q + dq> - f<p, q>
 +
 +
  =  (f<(p, dp), (q, dq)>, f<p, q>)
 +
 +
In the example f<p, q> = pq, the difference Df is given by:
 +
 +
  Df<p, q, dp, dq>
 +
 +
  =  [p + dp][q + dq] - pq
 +
 +
  =  ((p, dp)(q, dq), pq)
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            p  dp q  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/        p q                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df =          ((p, dp)(q, dq), pq)            |
 +
o-------------------------------------------------o
 +
 +
We did not yet go through the trouble to interpret this (first order)
 +
"difference of conjunction" fully, but were happy simply to evaluate
 +
it with respect to a single location in the universe of discourse,
 +
namely, at the point picked out by the singular proposition pq,
 +
in as much as if to say at the place where p = 1 and q = 1.
 +
This evaluation is written in the form Df|pq or Df|<1, 1>,
 +
and we arrived at the locally applicable law that states
 +
that f = pq = p and q => Df|pq = ((dp)(dq)) = dp or dq.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|          o-----------o  o-----------o          |
 +
|        /            \ /            \        |
 +
|        /      p      o      q      \        |
 +
|      /              /%\              \      |
 +
|      /              /%%%\              \      |
 +
|    o              o%%%%%o              o    |
 +
|    |              |%%%%%|              |    |
 +
|    |      dq (dp)  |%%%%%|  dp (dq)      |    |
 +
|    |  o<----------|--o--|---------->o  |    |
 +
|    |              |%%|%%|              |    |
 +
|    |              |%%|%%|              |    |
 +
|    o              o%%|%%o              o    |
 +
|      \              \%|%/              /      |
 +
|      \              \|/              /      |
 +
|        \              |              /        |
 +
|        \            /|\            /        |
 +
|          o-----------o | o-----------o          |
 +
|                        |                        |
 +
|                      dp|dq                      |
 +
|                        |                        |
 +
|                        v                        |
 +
|                        o                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    dp  dq                    |
 +
|                      o  o                      |
 +
|                      \ /                      |
 +
|                        o                        |
 +
|                        |                        |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|pq =          ((dp) (dq))                  |
 +
o-------------------------------------------------o
 +
 +
The picture illustrates the analysis of the inclusive
 +
disjunction ((dp)(dq)) into the exclusive disjunction:
 +
dp(dq) + (dp)dq + dp dq, a differential proposition that
 +
may be interpreted to say "change p or change q or both".
 +
And this can be recognized as just what you need to do if
 +
you happen to find yourself in the center cell and require
 +
a complete and detailed description of ways to escape it.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 4
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Last time we computed what will variously be called
 +
the "difference map", the "difference proposition",
 +
or the "local proposition" Df_x for the proposition
 +
f<p, q> = pq at the point x where p = 1 and q = 1.
 +
 +
In the universe X = !P! x !Q!, the four propositions
 +
pq, p(q), (p)q, (p)(q) that indicate the "cells",
 +
or the smallest regions of the venn diagram, are
 +
called "singular propositions".  These serve as
 +
an alternative notation for naming the points
 +
<1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.
 +
 +
Thus, we can write Df_x = Df|x = Df|<1, 1> = Df|pq,
 +
so long as we know the frame of reference in force.
 +
 +
Sticking with the example f<p, q> = pq, let us compute the
 +
value of the difference proposition Df at all of the points.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            p  dp q  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/        p q                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df =        ((p, dp)(q, dq), pq)                |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/                            |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|pq =          ((dp) (dq))                  |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o                            |
 +
|                dp |  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /        o                |
 +
|                \| |/          |                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|p(q) =          (dp) dq                      |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            o                                  |
 +
|            |  dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /        o                |
 +
|                \| |/          |                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|(p)q =            dp (dq)                    |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            o    o                            |
 +
|            |  dp |  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /      o  o              |
 +
|                \| |/        \ /                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|(p)(q) =          dp dq                      |
 +
o-------------------------------------------------o
 +
 +
The easy way to visualize the values of these graphical
 +
expressions is just to notice the following equivalents:
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  e                                              |
 +
|  o-o-o-...-o-o-o                                |
 +
|  \          /                                |
 +
|    \        /                                  |
 +
|    \      /                                  |
 +
|      \    /                          e        |
 +
|      \  /                          o        |
 +
|        \ /                            |        |
 +
|        @              =              @        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  (e, , ... , , )      =            (e)        |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                o                                |
 +
| e_1 e_2  e_k  |                                |
 +
|  o---o-...-o---o                                |
 +
|  \          /                                |
 +
|    \        /                                  |
 +
|    \      /                                  |
 +
|      \    /                                    |
 +
|      \  /                                    |
 +
|        \ /                      e_1 ... e_k    |
 +
|        @              =              @        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  (e_1, ..., e_k, ())  =        e_1 ... e_k    |
 +
o-------------------------------------------------o
 +
 +
Laying out the arrows on the augmented venn diagram,
 +
one gets a picture of a "differential vector field".
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                        |
 +
|                        |                        |
 +
|                      dp|dq                      |
 +
|                        |                        |
 +
|          o-----------o | o-----------o          |
 +
|        /            \|/            \        |
 +
|        /      p      |      q      \        |
 +
|      /              /|\              \      |
 +
|      /              /%|%\              \      |
 +
|    o              o%%|%%o              o    |
 +
|    |      (dp) dq  |%%v%%|  dp (dq)      |    |
 +
|    |  o-----------|->o<-|-----------o  |    |
 +
|    |              |%%%%%|              |    |
 +
|    |  o<----------|--o--|---------->o  |    |
 +
|    |      (dp) dq  |%%|%%|  dp (dq)      |    |
 +
|    o              o%%|%%o              o    |
 +
|      \              \%|%/              /      |
 +
|      \              \|/              /      |
 +
|        \              |              /        |
 +
|        \            /|\            /        |
 +
|          o-----------o | o-----------o          |
 +
|                        |                        |
 +
|                      dp|dq                      |
 +
|                        |                        |
 +
|                        v                        |
 +
|                        o                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
This just amounts to a depiction of the points,
 +
truth-value assignments, or interpretations in
 +
EX = !P! x !Q! x d!P! x d!Q! that are indicated
 +
by the difference map Df : EX -> B, namely, the
 +
following six points or singular propositions:
 +
 +
  1.  p  q  dp  dq
 +
  2.  p  q  dp (dq)
 +
  3.  p  q (dp) dq
 +
  4.  p (q)(dp) dq
 +
  5.  (p) q  dp (dq)
 +
  6.  (p)(q) dp  dq
 +
 +
By inspection, it is fairly easy to understand Df
 +
as telling you what you have to do from each point
 +
of X in order to change the value borne by f<p, q>
 +
at the point in question, that is, in order to get
 +
to a point where the value of f<p, q> is different
 +
from what it is where you started.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 5
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have been studying the action of the difference operator D,
 +
also known as the "localization operator", on the proposition
 +
f : !P! x !Q! -> B that is commonly called the conjunction pq.
 +
We categorized Df as a (first order) differential proposition,
 +
a proposition of the type Df : !P! x !Q! x d!P! x d!Q! -> B.
 +
 +
Abstracting from the augmented venn diagram that shows how the
 +
models or the satisfying interpretations of Df distribute over
 +
the (first order) extended space EX = !P! x !Q! x d!P! x d!Q!,
 +
we can represent Df in the form of a digraph or directed graph,
 +
one whose points are labeled with the elements of X = !P! x !Q!
 +
and whose arcs are labeled with the elements of dX = d!P! x d!Q!.
 +
 +
o-------------------------------------------------o
 +
|  f =                  p q                      |
 +
o-------------------------------------------------o
 +
|                                                |
 +
| Df =              p  q  ((dp)(dq))              |
 +
|                                                |
 +
|          +      p (q)  (dp) dq                |
 +
|                                                |
 +
|          +      (p) q    dp (dq)              |
 +
|                                                |
 +
|          +      (p)(q)  dp  dq                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      p q                      |
 +
|  p (q) o<------------->o<------------->o (p) q  |
 +
|            (dp) dq    ^    dp (dq)            |
 +
|                        |                        |
 +
|                        |                        |
 +
|                    dp | dq                    |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        v                        |
 +
|                        o                        |
 +
|                    (p) (q)                    |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
Any proposition worth its salt has many equivalent ways to view it,
 +
any one of which may reveal some unsuspected aspect of its meaning.
 +
We will encounter more and more of these variant readings as we go.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 6
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The enlargement operator E, also known as the "shift operator",
 +
has many interesting and very useful properties in its own right,
 +
so let us not fail to observe a few of the more salient features
 +
that play out on the surface of our simple example, f<p, q> = pq.
 +
 +
To begin we need to formulate a suitably generic
 +
definition of the extended universe of discourse:
 +
 +
  Relative to an initial domain X = X_1 x ... x X_k,
 +
 +
  EX = X x dX = X_1 x ... x X_k x dX_1 x ... x dX_k.
 +
 +
For a proposition f : X_1 x ... x X_k -> B,
 +
the (first order) "enlargement" of f is the
 +
proposition Ef : EX -> B that is defined by:
 +
 +
  Ef<x_1, ..., x_k, dx_1, ..., dx_k>
 +
 +
  =  f<x_1 + dx_1, ..., x_k + dx_k>
 +
 +
  =  f<(x_1, dx_1), ..., (x_k, dx_k)>
 +
 +
It should be noted that the so-called "differential variables" dx_j
 +
are really just the same type of boolean variables as the other x_j.
 +
It is conventional to give the additional variables these inflected
 +
names, but whatever extra connotations we attach to these syntactic
 +
conveniences are wholly external to their purely algebraic meanings.
 +
 +
In the case of the conjunction f<p, q> = pq,
 +
the enlargement Ef is formulated as follows:
 +
 +
  Ef<p, q, dp, dq>
 +
 +
  =  [p + dp][q + dq]
 +
 +
  =  (p, dp)(q, dq)
 +
 +
Given that this expression uses nothing more than the "boolean ring"
 +
operations of addition (+) and multiplication (*), it is permissible
 +
to "multiply things out" in the usual manner to arrive at the result:
 +
 +
  Ef<p, q, dp, dq>
 +
 +
  =  p q  +  p dq  +  q dp  +  dp dq
 +
 +
To understand what this means in logical terms,
 +
for instance, as expressed in a boolean expansion
 +
or a "disjunctive normal form" (DNF), it is perhaps
 +
a little better to go back and analyze the expression
 +
the same way that we did for Df.  Thus, let us compute
 +
the value of the enlarged proposition Ef at each of the
 +
points in the initial domain of discourse X = !P! x !Q!.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef =            (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      dp    dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|pq =            (dp) (dq)                    |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                      |
 +
|                      dp |  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|p(q) =          (dp)  dq                    |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o                            |
 +
|                  |  dp    dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|(p)q =          dp  (dq)                    |
 +
o-------------------------------------------------o
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o    o                      |
 +
|                  |  dp |  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|(p)(q) =        dp  dq                    |
 +
o-------------------------------------------------o
 +
 +
Given the kind of data that arises from this form of analysis,
 +
we can now fold the disjoined ingredients back into a boolean
 +
expansion or a DNF that is equivalent to the proposition Ef.
 +
 +
  Ef = pq Ef_pq + p(q) Ef_p(q) + (p)q Ef_(p)q + (p)(q) Ef_(p)(q)
 +
 +
Here is a summary of the result, illustrated by means of
 +
a digraph picture, where the "no change" element (dp)(dq)
 +
is drawn as a loop at the point p q.
 +
 +
o-------------------------------------------------o
 +
|  f =                  p q                      |
 +
o-------------------------------------------------o
 +
|                                                |
 +
| Ef =              p  q  (dp)(dq)              |
 +
|                                                |
 +
|          +      p (q)  (dp) dq                |
 +
|                                                |
 +
|          +      (p) q    dp (dq)              |
 +
|                                                |
 +
|          +      (p)(q)  dp  dq                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    (dp) (dq)                    |
 +
|                    .--->---.                    |
 +
|                    \    /                    |
 +
|                      \p q/                      |
 +
|                      \ /                      |
 +
|  p (q) o-------------->o<--------------o (p) q  |
 +
|            (dp) dq    ^    dp (dq)            |
 +
|                        |                        |
 +
|                        |                        |
 +
|                    dp | dq                    |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        o                        |
 +
|                    (p) (q)                    |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
We may understand the enlarged proposition Ef
 +
as telling us all the different ways to reach
 +
a model of f from any point of the universe X.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 7
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
To broaden our experience with simple examples, let us
 +
now contemplate the sixteen functions of concrete type
 +
!P! x !Q! -> B and abstract type B x B -> B.  For ease
 +
of future reference, I will set here a few tables that
 +
specify the actions of E and D on the 16 functions and
 +
allow us to view the results in several different ways.
 +
 +
By way of initial orientation, Table 7 lists equivalent expressions
 +
for the sixteen functions in several different formalisms, indexing
 +
systems, or languages for the propositional calculus, also known as
 +
"zeroth order logic" (ZOL).
 +
 +
Table 7.  Propositional Forms on Two Variables
 +
o---------o---------o---------o----------o------------------o----------o
 +
| L_1    | L_2    | L_3    | L_4      | L_5              | L_6      |
 +
|        |        |        |          |                  |          |
 +
| Decimal | Binary  | Vector  | Cactus  | English          | Ordinary |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|        |      p : 1 1 0 0 |          |                  |          |
 +
|        |      q : 1 0 1 0 |          |                  |          |
 +
o---------o---------o---------o----------o------------------o----------o
 +
|        |        |        |          |                  |          |
 +
| f_0    | f_0000  | 0 0 0 0 |    ()    | false            |    0    |
 +
|        |        |        |          |                  |          |
 +
| f_1    | f_0001  | 0 0 0 1 |  (p)(q)  | neither p nor q  | ~p & ~q  |
 +
|        |        |        |          |                  |          |
 +
| f_2    | f_0010  | 0 0 1 0 |  (p) q  | q and not p      | ~p &  q  |
 +
|        |        |        |          |                  |          |
 +
| f_3    | f_0011  | 0 0 1 1 |  (p)    | not p            | ~p      |
 +
|        |        |        |          |                  |          |
 +
| f_4    | f_0100  | 0 1 0 0 |  p (q)  | p and not q      |  p & ~q  |
 +
|        |        |        |          |                  |          |
 +
| f_5    | f_0101  | 0 1 0 1 |    (q)  | not q            |      ~q  |
 +
|        |        |        |          |                  |          |
 +
| f_6    | f_0110  | 0 1 1 0 |  (p, q)  | p not equal to q |  p +  q  |
 +
|        |        |        |          |                  |          |
 +
| f_7    | f_0111  | 0 1 1 1 |  (p  q)  | not both p and q | ~p v ~q  |
 +
|        |        |        |          |                  |          |
 +
| f_8    | f_1000  | 1 0 0 0 |  p  q  | p and q          |  p &  q  |
 +
|        |        |        |          |                  |          |
 +
| f_9    | f_1001  | 1 0 0 1 | ((p, q)) | p equal to q    |  p =  q  |
 +
|        |        |        |          |                  |          |
 +
| f_10    | f_1010  | 1 0 1 0 |      q  | q                |      q  |
 +
|        |        |        |          |                  |          |
 +
| f_11    | f_1011  | 1 0 1 1 |  (p (q)) | not p without q  |  p => q  |
 +
|        |        |        |          |                  |          |
 +
| f_12    | f_1100  | 1 1 0 0 |  p      | p                |  p      |
 +
|        |        |        |          |                  |          |
 +
| f_13    | f_1101  | 1 1 0 1 | ((p) q)  | not q without p  |  p <= q  |
 +
|        |        |        |          |                  |          |
 +
| f_14    | f_1110  | 1 1 1 0 | ((p)(q)) | p or q          |  p v  q  |
 +
|        |        |        |          |                  |          |
 +
| f_15    | f_1111  | 1 1 1 1 |  (())  | true            |    1    |
 +
|        |        |        |          |                  |          |
 +
o---------o---------o---------o----------o------------------o----------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 8
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The next four Tables expand the expressions of Ef and Df
 +
in two different ways, for each of the sixteen functions.
 +
Notice that the functions are given in a different order,
 +
partitioned into seven natural classes by a group action.
 +
 +
Table 8-a.  Ef Expanded Over Ordinary Features
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  Ef | pq  | Ef | p(q)  | Ef | (p)q  | Ef | (p)(q)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (p)(q)  |  dp  dq  |  dp (dq)  |  (dp) dq  |  (dp)(dq)  |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (p) q    |  dp (dq)  |  dp  dq  |  (dp)(dq)  |  (dp) dq  |
 +
|      |            |            |            |            |            |
 +
| f_4  |    p (q)  |  (dp) dq  |  (dp)(dq)  |  dp  dq  |  dp (dq)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    p  q    |  (dp)(dq)  |  (dp) dq  |  dp (dq)  |  dp  dq  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (p)      |  dp      |  dp      |  (dp)      |  (dp)      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    p      |  (dp)      |  (dp)      |  dp      |  dp      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (p, q)  |  (dp, dq)  | ((dp, dq)) | ((dp, dq)) |  (dp, dq)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((p, q))  | ((dp, dq)) |  (dp, dq)  |  (dp, dq)  | ((dp, dq)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (q)  |      dq  |      (dq)  |      dq  |      (dq)  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      q    |      (dq)  |      dq  |      (dq)  |      dq  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (p  q)  | ((dp)(dq)) | ((dp) dq)  |  (dp (dq)) |  (dp  dq)  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (p (q))  | ((dp) dq)  | ((dp)(dq)) |  (dp  dq)  |  (dp (dq)) |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((p) q)  |  (dp (dq)) |  (dp  dq)  | ((dp)(dq)) | ((dp) dq)  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((p)(q))  |  (dp  dq)  |  (dp (dq)) | ((dp) dq)  | ((dp)(dq)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
Table 8-b.  Df Expanded Over Ordinary Features
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  Df | pq  | Df | p(q)  | Df | (p)q  | Df | (p)(q)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (p)(q)  |  dp  dq  |  dp (dq)  |  (dp) dq  | ((dp)(dq)) |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (p) q    |  dp (dq)  |  dp  dq  | ((dp)(dq)) |  (dp) dq  |
 +
|      |            |            |            |            |            |
 +
| f_4  |    p (q)  |  (dp) dq  | ((dp)(dq)) |  dp  dq  |  dp (dq)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    p  q    | ((dp)(dq)) |  (dp) dq  |  dp (dq)  |  dp  dq  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (p)      |  dp      |  dp      |  dp      |  dp      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    p      |  dp      |  dp      |  dp      |  dp      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (p, q)  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((p, q))  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (q)  |      dq  |      dq  |      dq  |      dq  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      q    |      dq  |      dq  |      dq  |      dq  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (p  q)  | ((dp)(dq)) |  (dp) dq  |  dp (dq)  |  dp  dq  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (p (q))  |  (dp) dq  | ((dp)(dq)) |  dp  dq  |  dp (dq)  |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((p) q)  |  dp (dq)  |  dp  dq  | ((dp)(dq)) |  (dp) dq  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((p)(q))  |  dp  dq  |  dp (dq)  |  (dp) dq  | ((dp)(dq)) |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 9
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Table 9-a.  Ef Expanded Over Differential Features
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      |  T_11 f  |  T_10 f  |  T_01 f  |  T_00 f  |
 +
|      |            |            |            |            |            |
 +
|      |            | Ef| dp dq  | Ef| dp(dq) | Ef| (dp)dq | Ef|(dp)(dq)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (p)(q)  |    p  q    |    p (q)  |  (p) q    |  (p)(q)  |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (p) q    |    p (q)  |    p  q    |  (p)(q)  |  (p) q    |
 +
|      |            |            |            |            |            |
 +
| f_4  |    p (q)  |  (p) q    |  (p)(q)  |    p  q    |    p (q)  |
 +
|      |            |            |            |            |            |
 +
| f_8  |    p  q    |  (p)(q)  |  (p) q    |    p (q)  |    p  q    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (p)      |    p      |    p      |  (p)      |  (p)      |
 +
|      |            |            |            |            |            |
 +
| f_12 |    p      |  (p)      |  (p)      |    p      |    p      |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (p, q)  |  (p, q)  |  ((p, q))  |  ((p, q))  |  (p, q)  |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((p, q))  |  ((p, q))  |  (p, q)  |  (p, q)  |  ((p, q))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (q)  |      q    |      (q)  |      q    |      (q)  |
 +
|      |            |            |            |            |            |
 +
| f_10 |      q    |      (q)  |      q    |      (q)  |      q    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (p  q)  |  ((p)(q))  |  ((p) q)  |  (p (q))  |  (p  q)  |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (p (q))  |  ((p) q)  |  ((p)(q))  |  (p  q)  |  (p (q))  |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((p) q)  |  (p (q))  |  (p  q)  |  ((p)(q))  |  ((p) q)  |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((p)(q))  |  (p  q)  |  (p (q))  |  ((p) q)  |  ((p)(q))  |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|                  |            |            |            |            |
 +
| Fiped Point Total |      4    |      4    |      4    |    16    |
 +
|                  |            |            |            |            |
 +
o-------------------o------------o------------o------------o------------o
 +
 +
Table 9-b.  Df Expanded Over Differential Features
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
|      |    f      | Df| dp dq  | Df| dp(dq) | Df| (dp)dq | Df|(dp)(dq)|
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_0  |    ()    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_1  |  (p)(q)  |  ((p, q))  |    (q)    |    (p)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_2  |  (p) q    |  (p, q)  |    q      |    (p)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_4  |    p (q)  |  (p, q)  |    (q)    |    p      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_8  |    p  q    |  ((p, q))  |    q      |    p      |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_3  |  (p)      |    (())    |    (())    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_12 |    p      |    (())    |    (())    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_6  |  (p, q)  |    ()    |    (())    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_9  |  ((p, q))  |    ()    |    (())    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_5  |      (q)  |    (())    |    ()    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_10 |      q    |    (())    |    ()    |    (())    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_7  |  (p  q)  |  ((p, q))  |    q      |    p      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_11 |  (p (q))  |  (p, q)  |    (q)    |    p      |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_13 |  ((p) q)  |  (p, q)  |    q      |    (p)    |    ()    |
 +
|      |            |            |            |            |            |
 +
| f_14 |  ((p)(q))  |  ((p, q))  |    (q)    |    (p)    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
|      |            |            |            |            |            |
 +
| f_15 |    (())    |    ()    |    ()    |    ()    |    ()    |
 +
|      |            |            |            |            |            |
 +
o------o------------o------------o------------o------------o------------o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 10
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
If you think that I linger in the realm of logical difference calculus
 +
out of sheer vacillation about getting down to the differential proper,
 +
it is probably out of a prior expectation that you derive from the art
 +
or the long-engrained practice of real analysis.  But the fact is that
 +
ordinary calculus only rushes on to the sundry orders of approximation
 +
because the strain of comprehending the full import of E and D at once
 +
whelm over its discrete and finite powers to grasp them.  But here, in
 +
the fully serene idylls of ZOL, we find ourselves fit with the compass
 +
of a wit that is all we'd ever need to explore their effects with care.
 +
 +
So let us do just that.
 +
 +
I will first rationalize the novel grouping of propositional forms
 +
in the last set of Tables, as that will extend a gentle invitation
 +
to the mathematical subject of "group theory", and demonstrate its
 +
relevance to differential logic in a strikingly apt and useful way.
 +
The data for that account is contained in Table 9-a, above or here:
 +
 +
DAL 9.  http://forum.wolframscience.com/showthread.php?postid=1301#post1301
 +
DAL 9.  http://stderr.org/pipermail/inquiry/2004-May/001408.html
 +
 +
The shift operator E can be understood as enacting
 +
a substitution operation on the proposition f that
 +
is given as its argument.  In our present focus on
 +
propositional forms that involve two variables, we
 +
have the following datatype and applied definition:
 +
 +
  E : (X -> B)  ->  (EX -> B)
 +
 +
  E :  f<p, q>  ->  Ef<p, q, dp, dq>
 +
 +
  Ef<p, q, dp, dq>
 +
 +
  =  f<p + dp, q + dq)
 +
 +
  =  f<(p, dp), (q, dq)>
 +
 +
Therefore, if we evaluate Ef at particular values of dp and dq,
 +
for example, dp = i and dq = j, where i, j are in B, we obtain:
 +
 +
  E_ij : (X -> B)  ->  (X -> B)
 +
 +
  E_ij :    f      ->  E_ij f
 +
 +
  E_ij f
 +
 +
  =  Ef | <dp = i, dq = j>
 +
 +
  =  f<p + i, q + j>
 +
 +
  =  f<(p, i), (q, j)>
 +
 +
The notation is a little bit awkward, but the data of the Table should
 +
make the sense clear.  The important thing to observe is that E_ij has
 +
the effect of transforming each proposition f : X -> B into some other
 +
proposition f' : X -> B.  As it happens, the action is one-to-one and
 +
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
 +
is an example of what is called a "transformation group" on the set
 +
of sixteen propositions.  Bowing to a longstanding linear and local
 +
tradition, I will therefore redub the four elements of this group
 +
as T_00, T_01, T_10, T_11, to bear in mind their transformative
 +
character, or nature, as the case may be.  Abstractly viewed,
 +
this group of order four has the following operation table:
 +
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|    *    %  T_00  |  T_01  |  T_10  |  T_11  |
 +
|          %          |          |          |          |
 +
o==========o==========o==========o==========o==========o
 +
|          %          |          |          |          |
 +
|  T_00  %  T_00  |  T_01  |  T_10  |  T_11  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_01  %  T_01  |  T_00  |  T_11  |  T_10  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_10  %  T_10  |  T_11  |  T_00  |  T_01  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
|          %          |          |          |          |
 +
|  T_11  %  T_11  |  T_10  |  T_01  |  T_00  |
 +
|          %          |          |          |          |
 +
o----------o----------o----------o----------o----------o
 +
 +
It happens that there are just two possible groups of 4 elements.
 +
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
 +
The other is Klein's four-group V_4 (German "Vier"), which it is.
 +
 +
More concretely viewed, the group as a whole pushes the set
 +
of sixteen propositions around in such a way that they fall
 +
into seven natural classes, called "orbits".  One says that
 +
the orbits are preserved by the action of the group.  There
 +
is an "Orbit Lemma" of immense utility to "those who count"
 +
which, depending on your upbringing, you may associate with
 +
the names of Burnside, Cauchy, Frobenius, or some subset or
 +
superset of these three, vouching that the number of orbits
 +
is equal to the mean number of fixed points, in other words,
 +
the total number of points (in our case, propositions) that
 +
are left unmoved by the separate operations, divided by the
 +
order of the group.  In this instance, T_00 operates as the
 +
group identity, fixing all 16 propositions, while the other
 +
three group elements fix 4 propositions each, and so we get:
 +
Number of orbits  =  (4 + 4 + 4 + 16) / 4  =  7. -- Amazing!
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 11
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have been contemplating functions of the type f : X -> B,
 +
studying the action of the operators E and D on this family.
 +
These functions, that we may identify for our present aims
 +
with propositions, inasmuch as they capture their abstract
 +
forms, are logical analogues of "scalar potential fields".
 +
These are the sorts of fields that are so picturesquely
 +
presented in elementary calculus and physics textbooks
 +
by images of snow-covered hills and parties of skiers
 +
who trek down their slopes like least action heroes.
 +
The analogous scene in propositional logic presents
 +
us with forms more reminiscent of plateaunic idylls,
 +
being all plains at one of two levels, the mesas of
 +
verity and falsity, as it were, with nary a niche
 +
to inhabit between them, restricting our options
 +
for a sporting gradient of downhill dynamics to
 +
just one of two, standing still on level ground
 +
or falling off a bluff.
 +
 +
We are still working well within the logical analogue of the
 +
classical finite difference calculus, taking in the novelties
 +
that the logical transmutation of familiar elements is able to
 +
bring to light.  Soon we will take up several different notions
 +
of approximation relationships that may be seen to organize the
 +
space of propositions, and these will allow us to define several
 +
different forms of differential analysis applying to propositions.
 +
In time we will find reason to consider more general types of maps,
 +
having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n
 +
and abstract types B^k -> B^n.  We will think of these mappings as
 +
transforming universes of discourse into themselves or into others,
 +
in short, as "transformations of discourse".
 +
 +
Before we continue with this intinerary, however, I would like
 +
to highlight another sort of "differential aspect" that concerns
 +
the "boundary operator" or the "marked connective" that serves as
 +
one of a pair of basic connectives in the cactus language for ZOL.
 +
 +
Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
 +
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
 +
cactus syntax.  Taken as an assertion in what C.S. Peirce called the
 +
"existential interpretation", the so-called boundary form "(p, q, r)"
 +
asserts that one and only one of the propositions p, q, r is false.
 +
It is instructive to consider this assertion in relation to the
 +
conjunction "p q r" of the same propositions.  A venn diagram
 +
for the boundary form (p, q, r) is shown in Figure 11.
 +
 +
o-----------------------------------------------------------o
 +
|                                                          |
 +
|                                                          |
 +
|                      o-------------o                      |
 +
|                    /              \                    |
 +
|                    /                \                    |
 +
|                  /                  \                  |
 +
|                  /                    \                  |
 +
|                /                      \                |
 +
|                o                        o                |
 +
|                |                        |                |
 +
|                |            P            |                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|                |                        |                |
 +
|            o--o----------o  o----------o--o            |
 +
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
 +
|          /      \%%%%%%%%%%o%%%%%%%%%%/      \          |
 +
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
 +
|        /          \%%%%%%/  \%%%%%%/          \        |
 +
|        /            \%%%%/    \%%%%/            \        |
 +
|      o              o--o-------o--o              o      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      |        Q        |%%%%%%%|        R        |      |
 +
|      |                |%%%%%%%|                |      |
 +
|      o                o%%%%%%%o                o      |
 +
|        \                \%%%%%/                /        |
 +
|        \                \%%%/                /        |
 +
|          \                \%/                /          |
 +
|          \                o                /          |
 +
|            \              / \              /            |
 +
|            o-------------o  o-------------o            |
 +
|                                                          |
 +
|                                                          |
 +
o-----------------------------------------------------------o
 +
Figure 11.  Boundary Form (p, q, r)
 +
 +
In relation to the center cell indicated by the conjunction pqr
 +
the region indicated by (p, q, r) is comprised of the "adjacent"
 +
or the "bordering" cells.  Thus they are the cells that are just
 +
across the boundary of the center cell, as if reached by way of
 +
Leibniz's "minimal changes" from the point of origin, here, pqr.
 +
 +
More generally speaking, in a k-dimensional universe of discourse
 +
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
 +
the same form of boundary relationship is manifested for any cell
 +
of origin that one might choose to indicate, say, by means of the
 +
conjunction of positive and negative basis features "u_1 ... u_k",
 +
where u_j = x_j or u_j = (x_j), for j = 1 to k.  The proposition
 +
(u_1, ..., u_k) indicates the disjunctive region consisting of
 +
the cells that are "just next door" to the cell u_1 ... u_k.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 12
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Consider what effects that might conceivably have
 +
| practical bearings you conceive the objects of your
 +
| conception to have.  Then, your conception of those
 +
| effects is the whole of your conception of the object.
 +
|
 +
| C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438
 +
 +
One other subject that it would be opportune to mention at this point,
 +
while we have an object example of a mathematical group fresh in mind,
 +
is the relationship between the pragmatic maxim and what are commonly
 +
known in mathematics as "representation principles".  As it turns out,
 +
with regard to its formal characteristics, the pragmatic maxim unites
 +
the aspects of a representation principle with the attributes of what
 +
would ordinarily be known as a "closure principle".  We will consider
 +
the form of closure that is invoked by the pragmatic maxim on another
 +
occasion, focusing here and now on the topic of group representations.
 +
 +
Let us return to the example of the so-called "four-group" V_4.
 +
We encountered this group in one of its concrete representations,
 +
namely, as a "transformation group" that acts on a set of objects,
 +
in this particular case a set of sixteen functions or propositions.
 +
Forgetting about the set of objects that the group transforms among
 +
themselves, we may take the abstract view of the group's operational
 +
structure, say, in the form of the group operation table copied here:
 +
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  *  %  e  |  f  |  g  |  h  |
 +
|      %      |      |      |      |
 +
o=======o=======o=======o=======o=======o
 +
|      %      |      |      |      |
 +
|  e  %  e  |  f  |  g  |  h  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  f  %  f  |  e  |  h  |  g  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  g  %  g  |  h  |  e  |  f  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  h  %  h  |  g  |  f  |  e  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
 +
This table is abstractly the same as, or isomorphic to, the versions
 +
with the E_ij operators and the T_ij transformations that we took up
 +
earlier.  That is to say, the story is the same, only the names have
 +
been changed.  An abstract group can have a variety of significantly
 +
and superficially different representations.  But even after we have
 +
long forgotten the details of any particular representation there is
 +
a type of concrete representations, called "regular representations",
 +
that are always readily available, as they can be generated from the
 +
mere data of the abstract operation table itself.
 +
 +
For example, select a group element from the top margin of the Table,
 +
and "consider its effects" on each of the group elements as they are
 +
listed along the left margin.  We may record these effects as Peirce
 +
usually did, as a logical "aggregate" of elementary dyadic relatives,
 +
that is to say, a disjunction or a logical sum whose terms represent
 +
the ordered pairs of <input : output> transactions that are produced
 +
by each group element in turn.  This yields what is usually known as
 +
one of the "regular representations" of the group, specifically, the
 +
"first", the "post-", or the "right" regular representation.  It has
 +
long been conventional to organize the terms in the form of a matrix:
 +
 +
Reading "+" as a logical disjunction:
 +
 +
  G  =  e  +  f  +  g  + h,
 +
 +
And so, by expanding effects, we get:
 +
 +
  G =
 +
 +
  e:e  +  f:f  +  g:g  +  h:h  +
 +
 +
  e:f  +  f:e  +  g:h  +  h:g  +
 +
 +
  e:g  +  f:h  +  g:e  +  h:f  +
 +
 +
  e:h  +  f:g  +  g:f  +  h:e
 +
 +
More on the pragmatic maxim as a representation principle later.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 13
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The above-mentioned fact about the regular representations
 +
of a group is universally known as "Cayley's Theorem".  It
 +
is usually stated in the form:  "Every group is isomorphic
 +
to a subgroup of Aut(X), where X is a suitably chosen set
 +
and Aut(X) is the group of its automorphisms".  There is
 +
in Peirce's early papers a considerable generalization
 +
of the concept of regular representations to a broad
 +
class of relational algebraic systems.  The crux of
 +
the whole idea can be summed up as follows:
 +
 +
  Contemplate the effects of the symbol
 +
  whose meaning you wish to investigate
 +
  as they play out on all the stages of
 +
  conduct on which you have the ability
 +
  to imagine that symbol playing a role.
 +
 +
This idea of definition by way of context transforming operators
 +
is basically the same as Jeremy Bentham's notion of "paraphrasis",
 +
a "method of accounting for fictions by explaining various purported
 +
terms away" (Quine, in Van Heijenoort, 'From Frege to Gödel', p. 216).
 +
Today we'd call these constructions "term models".  This, again, is
 +
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
 +
of lambda calculus, and I reckon you all know where that leads.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 14
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The next few excursions in this series will provide
 +
a scenic tour of various ideas in group theory that
 +
will turn out to be of constant guidance in several
 +
of the settings that are associated with our topic.
 +
 +
Let me return to Peirce's early papers on the algebra of relatives
 +
to pick up the conventions that he used there, and then rewrite my
 +
account of regular representations in a way that conforms to those.
 +
 +
Peirce expresses the action of an "elementary dual relative" like so:
 +
 +
| [Let] A:B be taken to denote
 +
| the elementary relative which
 +
| multiplied into B gives A.
 +
|
 +
| Peirce, 'Collected Papers', CP 3.123.
 +
 +
Peirce is well aware that it is not at all necessary to arrange the
 +
elementary relatives of a relation into arrays, matrices, or tables,
 +
but when he does so he tends to prefer organizing 2-adic relations
 +
in the following manner:
 +
 +
  a:b  +  a:b  +  a:c  +
 +
 +
  b:a  +  b:b  +  b:c  +
 +
 +
  c:a  +  c:b  +  c:c
 +
 +
For example, given the set X = {a, b, c}, suppose that
 +
we have the 2-adic relative term m = "marker for" and
 +
the associated 2-adic relation M c X x X, the general
 +
pattern of whose common structure is represented by
 +
the following matrix:
 +
 +
  M  =
 +
 +
  M_aa a:a  +  M_ab a:b  +  M_ac a:c  +
 +
 +
  M_ba b:a  +  M_bb b:b  +  M_bc b:c  +
 +
 +
  M_ca c:a  +  M_cb c:b  +  M_cc c:c
 +
 +
It has long been customary to omit the implicit plus signs
 +
in these matrical displays, but I have restored them here
 +
simply as a way of separating terms in this blancophage
 +
web format.
 +
 +
For at least a little while, I will make explicit
 +
the distinction between a "relative term" like m
 +
and a "relation" like M c X x X, but it is best
 +
to think of both of these entities as involving
 +
different applications of the same information,
 +
and so we could just as easily write this form:
 +
 +
  m  =
 +
 +
  m_aa a:a  +  m_ab a:b  +  m_ac a:c  +
 +
 +
  m_ba b:a  +  m_bb b:b  +  m_bc b:c  +
 +
 +
  m_ca c:a  +  m_cb c:b  +  m_cc c:c
 +
 +
By way of making up a concrete example,
 +
let us say that M is given as follows:
 +
 +
  a is a marker for a
 +
 +
  a is a marker for b
 +
 +
  b is a marker for b
 +
 +
  b is a marker for c
 +
 +
  c is a marker for c
 +
 +
  c is a marker for a
 +
 +
In sum, we have this matrix:
 +
 +
  M  =
 +
 +
  1 a:a  +  1 a:b  +  0 a:c  +
 +
 +
  0 b:a  +  1 b:b  +  1 b:c  +
 +
 +
  1 c:a  +  0 c:b  +  1 c:c
 +
 +
I think that will serve to fix notation
 +
and set up the remainder of the account.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 15
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In Peirce's time, and even in some circles of mathematics today,
 +
the information indicated by the elementary relatives (i:j), as
 +
i, j range over the universe of discourse, would be referred to
 +
as the "umbral elements" of the algebraic operation represented
 +
by the matrix, though I seem to recall that Peirce preferred to
 +
call these terms the "ingredients".  When this ordered basis is
 +
understood well enough, one will tend to drop any mention of it
 +
from the matrix itself, leaving us nothing but these bare bones:
 +
 +
  M  =
 +
 +
  1  1  0
 +
 +
  0  1  1
 +
 +
  1  0  1
 +
 +
However the specification may come to be written, this
 +
is all just convenient schematics for stipulating that:
 +
 +
  M  =  a:a  +  b:b  +  c:c  +  a:b  +  b:c  +  c:a
 +
 +
Recognizing !1! = a:a + b:b + c:c as the identity transformation,
 +
the 2-adic relative term m = "marker for" can be represented as an
 +
element !1! + a:b + b:c + c:a of the so-called "group ring", all of
 +
which makes this element just a special sort of linear transformation.
 +
 +
Up to this point, we are still reading the elementary relatives
 +
of the form i:j in the way that Peirce customarily read them in
 +
logical contexts:  i is the relate, j is the correlate, and in
 +
our current example we reading i:j, or more exactly, m_ij = 1,
 +
to say that i is a marker for j.  This is the mode of reading
 +
that we call "multiplying on the left".
 +
 +
In the algebraic, permutational, or transformational contexts of
 +
application, however, Peirce converts to the alternative mode of
 +
reading, although still calling i the relate and j the correlate,
 +
the elementary relative i:j now means that i gets changed into j.
 +
In this scheme of reading, the transformation a:b + b:c + c:a is
 +
a permutation of the aggregate $1$ = a + b + c, or what we would
 +
now call the set {a, b, c}, in particular, it is the permutation
 +
that is otherwise notated as:
 +
 +
  ( a b c )
 +
  <      >
 +
  ( b c a )
 +
 +
This is consistent with the convention that Peirce uses in
 +
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 16
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We've been exploring the applications of a certain technique
 +
for clarifying abstruse concepts, a rough-cut version of the
 +
pragmatic maxim that I've been accustomed to refer to as the
 +
"operationalization" of ideas.  The basic idea is to replace
 +
the question of "What it is", which modest people comprehend
 +
is far beyond their powers to answer any time soon, with the
 +
question of "What it does", which most people know at least
 +
a modicum about.
 +
 +
In the case of regular representations of groups we found
 +
a non-plussing surplus of answers to sort our way through.
 +
So let us track back one more time to see if we can learn
 +
any lessons that might carry over to more realistic cases.
 +
 +
Here is is the operation table of V_4 once again:
 +
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  *  %  e  |  f  |  g  |  h  |
 +
|      %      |      |      |      |
 +
o=======o=======o=======o=======o=======o
 +
|      %      |      |      |      |
 +
|  e  %  e  |  f  |  g  |  h  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  f  %  f  |  e  |  h  |  g  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  g  %  g  |  h  |  e  |  f  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
|      %      |      |      |      |
 +
|  h  %  h  |  g  |  f  |  e  |
 +
|      %      |      |      |      |
 +
o-------o-------o-------o-------o-------o
 +
 +
A group operation table is really just a device for recording
 +
a certain 3-adic relation, specifically, the set of 3-tuples
 +
of the form <x, y, z> that satisfy the equation x * y = z,
 +
where the sign '*' that indicates the group operation is
 +
frequently omitted in contexts where it is understood.
 +
 +
In the case of V_4 = (G, *), where G is the "underlying set"
 +
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
 +
whose triples are listed below:
 +
 +
  e:e:e
 +
  e:f:f
 +
  e:g:g
 +
  e:h:h
 +
 +
  f:e:f
 +
  f:f:e
 +
  f:g:h
 +
  f:h:g
 +
 +
  g:e:g
 +
  g:f:h
 +
  g:g:e
 +
  g:h:f
 +
 +
  h:e:h
 +
  h:f:g
 +
  h:g:f
 +
  h:h:e
 +
 +
It is part of the definition of a group that the 3-adic
 +
relation L c G^3 is actually a function L : G x G -> G.
 +
It is from this functional perspective that we can see
 +
an easy way to derive the two regular representations.
 +
 +
Since we have a function of the type L : G x G -> G,
 +
we can define a couple of substitution operators:
 +
 +
1.  Sub(x, <_, y>) puts any specified x into
 +
    the empty slot of the rheme <_, y>, with
 +
    the effect of producing the saturated
 +
    rheme <x, y> that evaluates to xy.
 +
 +
2.  Sub(x, <y, _>) puts any specified x into
 +
    the empty slot of the rheme <y, _>, with
 +
    the effect of producing the saturated
 +
    rheme <y, x> that evaluates to yx.
 +
 +
In (1), we consider the effects of each x in its
 +
practical bearing on contexts of the form <_, y>,
 +
as y ranges over G, and the effects are such that
 +
x takes <_, y> into xy, for y in G, all of which
 +
is summarily notated as x = {<y : xy> : y in G}.
 +
The pairs <y : xy> can be found by picking an x
 +
from the left margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run across the top margin.  This aspect of
 +
pragmatic definition we recognize as the regular
 +
ante-representation:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
  f  =  e:f  +  f:e  +  g:h  +  h:g
 +
 +
  g  =  e:g  +  f:h  +  g:e  +  h:f
 +
 +
  h  =  e:h  +  f:g  +  g:f  +  h:e
 +
 +
In (2), we consider the effects of each x in its
 +
practical bearing on contexts of the form <y, _>,
 +
as y ranges over G, and the effects are such that
 +
x takes <y, _> into yx, for y in G, all of which
 +
is summarily notated as x = {<y : yx> : y in G}.
 +
The pairs <y : yx> can be found by picking an x
 +
from the top margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run down the left margin.  This aspect of
 +
pragmatic definition we recognize as the regular
 +
post-representation:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h
 +
 +
  f  =  e:f  +  f:e  +  g:h  +  h:g
 +
 +
  g  =  e:g  +  f:h  +  g:e  +  h:f
 +
 +
  h  =  e:h  +  f:g  +  g:f  +  h:e
 +
 +
If the ante-rep looks the same as the post-rep,
 +
now that I'm writing them in the same dialect,
 +
that is because V_4 is abelian (commutative),
 +
and so the two representations have the very
 +
same effects on each point of their bearing.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 17
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
So long as we're in the neighborhood, we might as well take in
 +
some more of the sights, for instance, the smallest example of
 +
a non-abelian (non-commutative) group.  This is a group of six
 +
elements, say, G = {e, f, g, h, i, j}, with no relation to any
 +
other employment of these six symbols being implied, of course,
 +
and it can be most easily represented as the permutation group
 +
on a set of three letters, say, X = {a, b, c}, usually notated
 +
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
 +
Here are the permutation (= substitution) operations in Sym(X):
 +
 +
Table 17-a.  Permutations or Substitutions in Sym_{a, b, c}
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |
 +
|        |        |        |        |        |        |
 +
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
 +
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
 +
|        |        |        |        |        |        |
 +
|  a b c  |  c a b  |  b c a  |  a c b  |  c b a  |  b a c  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
 +
Here is the operation table for S_3, given in abstract fashion:
 +
 +
Table 17-b.  Symmetric Group S_3
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                        |
 +
|                    e / \ e                    |
 +
|                      /  \                      |
 +
|                    /  e  \                    |
 +
|                  f / \  / \ f                  |
 +
|                  /  \ /  \                  |
 +
|                  /  f  \  f  \                  |
 +
|              g / \  / \  / \ g              |
 +
|                /  \ /  \ /  \                |
 +
|              /  g  \  g  \  g  \              |
 +
|            h / \  / \  / \  / \ h            |
 +
|            /  \ /  \ /  \ /  \            |
 +
|            /  h  \  e  \  e  \  h  \            |
 +
|        i / \  / \  / \  / \  / \ i        |
 +
|          /  \ /  \ /  \ /  \ /  \          |
 +
|        /  i  \  i  \  f  \  j  \  i  \        |
 +
|      j / \  / \  / \  / \  / \  / \ j      |
 +
|      /  \ /  \ /  \ /  \ /  \ /  \      |
 +
|      o  j  \  j  \  j  \  i  \  h  \  j  o      |
 +
|      \  / \  / \  / \  / \  / \  /      |
 +
|        \ /  \ /  \ /  \ /  \ /  \ /        |
 +
|        \  h  \  h  \  e  \  j  \  i  /        |
 +
|          \  / \  / \  / \  / \  /          |
 +
|          \ /  \ /  \ /  \ /  \ /          |
 +
|            \  i  \  g  \  f  \  h  /            |
 +
|            \  / \  / \  / \  /            |
 +
|              \ /  \ /  \ /  \ /              |
 +
|              \  f  \  e  \  g  /              |
 +
|                \  / \  / \  /                |
 +
|                \ /  \ /  \ /                |
 +
|                  \  g  \  f  /                  |
 +
|                  \  / \  /                  |
 +
|                    \ /  \ /                    |
 +
|                    \  e  /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        o                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
 +
I think that the NKS reader can guess how we might apply
 +
this group to the space of propositions of type B^3 -> B.
 +
 +
By the way, we will meet with the symmetric group S_3 again
 +
when we return to take up the study of Peirce's early paper
 +
"On a Class of Multiple Algebras" (CP 3.324-327), and also
 +
his late unpublished work "The Simplest Mathematics" (1902)
 +
(CP 4.227-323), with particular reference to the section
 +
that treats of "Trichotomic Mathematics" (CP 4.307-323).
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 18
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
By way of collecting a short-term pay-off for all the work that we
 +
did on the regular representations of the Klein 4-group V_4, let us
 +
write out as quickly as possible in "relative form" a minimal budget
 +
of representations for the symmetric group on three letters, Sym(3).
 +
After doing the usual bit of compare and contrast among the various
 +
representations, we will have enough concrete material beneath our
 +
abstract belts to tackle a few of the presently obscured details
 +
of Peirce's early "Algebra + Logic" papers.
 +
 +
Writing the permutations or substitutions of Sym {a, b, c}
 +
in relative form generates what is generally thought of as
 +
a "natural representation" of S_3.
 +
 +
  e  =  a:a + b:b + c:c
 +
 +
  f  =  a:c + b:a + c:b
 +
 +
  g  =  a:b + b:c + c:a
 +
 +
  h  =  a:a + b:c + c:b
 +
 +
  i  =  a:c + b:b + c:a
 +
 +
  j  =  a:b + b:a + c:c
 +
 +
I have without stopping to think about it written out this natural
 +
representation of S_3 in the style that comes most naturally to me,
 +
to wit, the "right" way, whereby an ordered pair configured as x:y
 +
constitutes the turning of x into y.  It is possible that the next
 +
time we check in with CSP that we will have to adjust our sense of
 +
direction, but that will be an easy enough bridge to cross when we
 +
come to it.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 19
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
To construct the regular representations of S_3,
 +
we pick up from the data of its operation table,
 +
DAL 17, Table 17-b, at either one of these sites:
 +
 +
http://stderr.org/pipermail/inquiry/2004-May/001419.html
 +
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
 +
 +
Just by way of staying clear about what we are doing,
 +
let's return to the recipe that we worked out before:
 +
 +
It is part of the definition of a group that the 3-adic
 +
relation L c G^3 is actually a function L : G x G -> G.
 +
It is from this functional perspective that we can see
 +
an easy way to derive the two regular representations.
 +
 +
Since we have a function of the type L : G x G -> G,
 +
we can define a couple of substitution operators:
 +
 +
1.  Sub(x, <_, y>) puts any specified x into
 +
    the empty slot of the rheme <_, y>, with
 +
    the effect of producing the saturated
 +
    rheme <x, y> that evaluates to xy.
 +
 +
2.  Sub(x, <y, _>) puts any specified x into
 +
    the empty slot of the rheme <y, _>, with
 +
    the effect of producing the saturated
 +
    rheme <y, x> that evaluates to yx.
 +
 +
In (1), we consider the effects of each x in its
 +
practical bearing on contexts of the form <_, y>,
 +
as y ranges over G, and the effects are such that
 +
x takes <_, y> into xy, for y in G, all of which
 +
is summarily notated as x = {<y : xy> : y in G}.
 +
The pairs <y : xy> can be found by picking an x
 +
from the left margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run along the right margin.  This produces
 +
the regular ante-representation of S_3, like so:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j
 +
 +
  f  =  e:f  +  f:g  +  g:e  +  h:j  +  i:h  +  j:i
 +
 +
  g  =  e:g  +  f:e  +  g:f  +  h:i  +  i:j  +  j:h
 +
 +
  h  =  e:h  +  f:i  +  g:j  +  h:e  +  i:f  +  j:g
 +
 +
  i  =  e:i  +  f:j  +  g:h  +  h:g  +  i:e  +  j:f
 +
 +
  j  =  e:j  +  f:h  +  g:i  +  h:f  +  i:g  +  j:e
 +
 +
In (2), we consider the effects of each x in its
 +
practical bearing on contexts of the form <y, _>,
 +
as y ranges over G, and the effects are such that
 +
x takes <y, _> into yx, for y in G, all of which
 +
is summarily notated as x = {<y : yx> : y in G}.
 +
The pairs <y : yx> can be found by picking an x
 +
on the right margin of the group operation table
 +
and considering its effects on each y in turn as
 +
these run along the left margin.  This generates
 +
the regular post-representation of S_3, like so:
 +
 +
  e  =  e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j
 +
 +
  f  =  e:f  +  f:g  +  g:e  +  h:i  +  i:j  +  j:h
 +
 +
  g  =  e:g  +  f:e  +  g:f  +  h:j  +  i:h  +  j:i
 +
 +
  h  =  e:h  +  f:j  +  g:i  +  h:e  +  i:g  +  j:f
 +
 +
  i  =  e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g
 +
 +
  j  =  e:j  +  f:i  +  g:h  +  h:g  +  i:f  +  j:e
 +
 +
If the ante-rep looks different from the post-rep,
 +
it is just as it should be, as S_3 is non-abelian
 +
(non-commutative), and so the two representations
 +
differ in the details of their practical effects,
 +
though, of course, being representations of the
 +
same abstract group, they must be isomorphic.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 20
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
You may be wondering what happened to the announced subject
 +
of "Dynamics And Logic".  What occurred was a bit like this:
 +
 +
We happened to make the observation that the shift operators {E_ij}
 +
form a transformation group that acts on the set of propositions of
 +
the form f : B x B -> B.  Group theory is a very attractive subject,
 +
but it did not draw us so far from our intended course as one might
 +
initially think.  For one thing, groups, especially the groups that
 +
are named after the Norwegian mathematician Marius Sophus Lie, turn
 +
out to be of critical importance in solving differential equations.
 +
For another thing, group operations provide us with an ample supply
 +
of triadic relations that have been extremely well-studied over the
 +
years, and thus they give us no small measure of useful guidance in
 +
the study of sign relations, another brand of 3-adic relations that
 +
have significance for logical studies, and in our acquaintance with
 +
which we have scarcely begun to break the ice.  Finally, I couldn't
 +
resist taking up the links between group representations, amounting
 +
to the very archetypes of logical models, and the pragmatic maxim.
 +
 +
Biographical Data for Marius Sophus Lie (1842-1899):
 +
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 21
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
We have seen a couple of groups, V_4 and S_3, represented in
 +
several different ways, and we have seen each of these types
 +
of representation presented in several different fashions.
 +
Let us look at one other stylistic variant for presenting
 +
a group representation that is often used, the so-called
 +
"matrix representation" of a group.
 +
 +
Returning to the example of Sym(3), we first encountered
 +
this group in concrete form as a set of permutations or
 +
substitutions acting on a set of letters X = {a, b, c}.
 +
This set of permutations was displayed in Table 17-a,
 +
copies of which can be found here:
 +
 +
http://stderr.org/pipermail/inquiry/2004-May/001419.html
 +
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
 +
 +
These permutations were then converted to "relative form":
 +
 +
  e  =  a:a + b:b + c:c
 +
 +
  f  =  a:c + b:a + c:b
 +
 +
  g  =  a:b + b:c + c:a
 +
 +
  h  =  a:a + b:c + c:b
 +
 +
  i  =  a:c + b:b + c:a
 +
 +
  j  =  a:b + b:a + c:c
 +
 +
From this relational representation of Sym {a, b, c} ~=~ S_3,
 +
one easily derives a "linear representation", regarding each
 +
permutation as a linear transformation that maps the elements
 +
of a suitable vector space into each other, and representing
 +
each of these linear transformations by means of a matrix,
 +
resulting in the following set of matrices for the group:
 +
 +
Table 21.  Matrix Representations of the Permutations in S_3
 +
o---------o---------o---------o---------o---------o---------o
 +
|        |        |        |        |        |        |
 +
|    e    |    f    |    g    |    h    |    i    |    j    |
 +
|        |        |        |        |        |        |
 +
o=========o=========o=========o=========o=========o=========o
 +
|        |        |        |        |        |        |
 +
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
 +
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
 +
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
 +
|        |        |        |        |        |        |
 +
o---------o---------o---------o---------o---------o---------o
 +
 +
The key to the mysteries of these matrices is revealed by
 +
observing that their coefficient entries are arrayed and
 +
overlayed on a place mat that's marked like so:
 +
 +
o-----o-----o-----o
 +
| a:a | a:b | a:c |
 +
o-----o-----o-----o
 +
| b:a | b:b | b:c |
 +
o-----o-----o-----o
 +
| c:a | c:b | c:c |
 +
o-----o-----o-----o
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 22
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
It would be good to summarize, in rough but intuitive terms,
 +
the outlook on differential logic that we have reached so far.
 +
 +
We've been considering a class of operators on universes
 +
of discourse, each of which takes us from considering one
 +
universe of discourse, X%, to considering a larger universe
 +
of discourse, EX%.
 +
 +
Each of these operators, in broad terms having the form
 +
W : X% -> EX%, acts on each proposition f : X -> B of the
 +
source universe X% to produce a proposition Wf : EX -> B
 +
of the target universe EX%.
 +
 +
The two main operators that we have worked with up to this
 +
point are the enlargement or shift operator E : X% -> EX%
 +
and the difference operator D : X% -> EX%.
 +
 +
E and D take a proposition in X%, that is, a proposition f : X -> B
 +
that is said to be "about" the subject matter of X, and produce the
 +
extended propositions Ef, Df : EX -> B, which may be interpreted as
 +
being about specified collections of changes that might occur in X.
 +
 +
Here we have need of visual representations,
 +
some array of concrete pictures to anchor our
 +
more earthy intuitions and to help us keep our
 +
wits about us before we try to climb any higher
 +
into the ever more rarefied air of abstractions.
 +
 +
One good picture comes to us by way of the "field" concept.
 +
Given a space X, a "field" of a specified type Y over X is
 +
formed by assigning to each point of X an object of type Y.
 +
If that sounds like the same thing as a function from X to
 +
the space of things of type Y -- it is -- but it does seem
 +
helpful to vary the mental images and to take advantage of
 +
the figures of speech that spring to mind under the emblem
 +
of this field idea.
 +
 +
In the field picture, a proposition f : X -> B becomes
 +
a "scalar" field, that is, a field of values in B, or
 +
a "field of model indications" (FOMI).
 +
 +
Let us take a moment to view an old proposition
 +
in this new light, for example, the conjunction
 +
pq : X -> B that is depicted in Figure 22-a.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        o-------------o  o-------------o        |
 +
|      /              \ /              \      |
 +
|      /                o                \      |
 +
|    /                /%\                \    |
 +
|    /                /%%%\                \    |
 +
|  o                o%%%%%o                o  |
 +
|  |                |%%%%%|                |  |
 +
|  |        P        |%%%%%|        Q        |  |
 +
|  |                |%%%%%|                |  |
 +
|  o                o%%%%%o                o  |
 +
|    \                \%%%/                /    |
 +
|    \                \%/                /    |
 +
|      \                o                /      |
 +
|      \              / \              /      |
 +
|        o-------------o  o-------------o        |
 +
|                                                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  f =                  p q                      |
 +
o-------------------------------------------------o
 +
Figure 22-a.  Conjunction pq : X -> B
 +
 +
Each of the operators E, D : X% -> EX% takes us from considering
 +
propositions f : X -> B, here viewed as "scalar fields" over X,
 +
to considering the corresponding "differential fields" over X,
 +
analogous to what are usually called "vector fields" over X.
 +
 +
The structure of these differential fields can be described this way.
 +
To each point of X there is attached an object of the following type:
 +
a proposition about changes in X, that is, a proposition g : dX -> B.
 +
In this frame, if X% is the universe that is generated by the set of
 +
coordinate propositions {p, q}, then dX% is the differential universe
 +
that is generated by the set of differential propositions {dp, dq}.
 +
These differential propositions may be interpreted as indicating
 +
"change in p" and "change in q", respectively.
 +
 +
A differential operator W, of the first order sort that we have
 +
been considering, takes a proposition f : X -> B and gives back
 +
a differential proposition Wf: EX -> B.
 +
 +
In the field view, we see the proposition f : X -> B as a scalar field
 +
and we see the differential proposition Wf: EX -> B as a vector field,
 +
specifically, a field of propositions about contemplated changes in X.
 +
 +
The field of changes produced by E on pq is shown in Figure 22-b.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        o-------------o  o-------------o        |
 +
|      /              \ /              \      |
 +
|      /        P        o        Q        \      |
 +
|    /                /%\                \    |
 +
|    /                /%%%\                \    |
 +
|  o                o.->-.o                o  |
 +
|  |    p(q)(dp)dq  |%\%/%|  (p)q dp(dq)    |  |
 +
|  | o---------------|->o<-|---------------o |  |
 +
|  |                |%%^%%|                |  |
 +
|  o                o%%|%%o                o  |
 +
|    \                \%|%/                /    |
 +
|    \                \|/                /    |
 +
|      \                o                /      |
 +
|      \              /|\              /      |
 +
|        o-------------o | o-------------o        |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        o                        |
 +
|                  (p)(q) dp dq                  |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  f =                  p q                      |
 +
o-------------------------------------------------o
 +
|                                                |
 +
| Ef =              p  q  (dp)(dq)              |
 +
|                                                |
 +
|          +      p (q)  (dp) dq                |
 +
|                                                |
 +
|          +      (p) q    dp (dq)              |
 +
|                                                |
 +
|          +      (p)(q)  dp  dq                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 22-b.  Enlargement E[pq] : EX -> B
 +
 +
The differential field E[pq] specifies the changes
 +
that need to be made from each point of X in order
 +
to reach one of the models of the proposition pq,
 +
that is, in order to satisfy the proposition pq.
 +
 +
The field of changes produced by D on pq is shown in Figure 22-c.
 +
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|        o-------------o  o-------------o        |
 +
|      /              \ /              \      |
 +
|      /        P        o        Q        \      |
 +
|    /                /%\                \    |
 +
|    /                /%%%\                \    |
 +
|  o                o%%%%%o                o  |
 +
|  |      (dp)dq    |%%%%%|    dp(dq)      |  |
 +
|  | o<--------------|->o<-|-------------->o |  |
 +
|  |                |%%^%%|                |  |
 +
|  o                o%%|%%o                o  |
 +
|    \                \%|%/                /    |
 +
|    \                \|/                /    |
 +
|      \                o                /      |
 +
|      \              /|\              /      |
 +
|        o-------------o | o-------------o        |
 +
|                        |                        |
 +
|                        |                        |
 +
|                        v                        |
 +
|                        o                        |
 +
|                      dp dq                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  f =                  p q                      |
 +
o-------------------------------------------------o
 +
|                                                |
 +
| Df =              p  q  ((dp)(dq))              |
 +
|                                                |
 +
|          +      p (q)  (dp) dq                |
 +
|                                                |
 +
|          +      (p) q    dp (dq)              |
 +
|                                                |
 +
|          +      (p)(q)  dp  dq                |
 +
|                                                |
 +
o-------------------------------------------------o
 +
Figure 22-c.  Difference D[pq] : EX -> B
 +
 +
The differential field D[pq] specifies the changes
 +
that need to be made from each point of X in order
 +
to feel a change in the felt value of the field pq.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 23
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
I want to continue developing the basic tools of differential logic,
 +
which arose out of many years of thinking about the connections
 +
between dynamics and logic -- those there are and those there
 +
ought to be -- but I also wanted to give some hint of the
 +
applications that have motivated this work all along.
 +
One of these applications is to cybernetic systems,
 +
whether we see these systems as agents or cultures,
 +
individuals or species, organisms or organizations.
 +
 +
A cybernetic system has goals and actions for reaching them.
 +
It has a state space X, giving us all of the states that the
 +
system can be in, plus it has a goal space G c X, the set of
 +
states that the system "likes" to be in, in other words, the
 +
distinguished subset of possible states where the system is
 +
regarded as living, surviving, or thriving, depending on the
 +
type of goal that one has in mind for the system in question.
 +
As for actions, there is to begin with the full set !T! of all
 +
possible actions, each of which is a transformation of the form
 +
T : X -> X, but a given cybernetic system will most likely have
 +
but a subset of these actions available to it at any given time.
 +
And even if we begin by thinking of actions in very general and
 +
very global terms, as arbitrarily complex transformations acting
 +
on the whole state space X, we quickly find a need to analyze and
 +
approximate them in terms of simple transformations acting locally.
 +
The preferred measure of "simplicity" will of course vary from one
 +
paradigm of research to another.
 +
 +
A generic enough picture at this stage of the game, and one that will
 +
remind us of these fundamental features of the cybernetic system even
 +
as things get far more complex, is afforded by Figure 23.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o                                    |
 +
|          /                    \                                  |
 +
|          /                      \                                  |
 +
|        /                        \                                |
 +
|        /                          \                                |
 +
|      /                            \                              |
 +
|      /                              \                              |
 +
|    /                                \                            |
 +
|    o                                  o                            |
 +
|    |                                  |                            |
 +
|    |                                  |                            |
 +
|    |                                  |                            |
 +
|    |                G                |                            |
 +
|    |                                  |                            |
 +
|    |                                  |                            |
 +
|    |                                  |                            |
 +
|    o                                  o                            |
 +
|    \                                /                            |
 +
|      \                              /                              |
 +
|      \                          T /                              |
 +
|        \            o<------------/-------------o                  |
 +
|        \                        /                                |
 +
|          \                      /                                  |
 +
|          \                    /                                  |
 +
|            o-------------------o                                    |
 +
|                                                                    |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 23.  Elements of a Cybernetic System
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 24
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Now that we've introduced the field picture for thinking about
 +
propositions and their analytic series, a very pleasing way of
 +
picturing the relationship among a proposition f : X -> B, its
 +
enlargement or shift map Ef : EX -> B, and its difference map
 +
Df : EX -> B can now be drawn.
 +
 +
To illustrate this possibility, let's return to the differential
 +
analysis of the conjunctive proposition f<p, q> = pq, giving the
 +
development a slightly different twist at the appropriate point.
 +
 +
Figure 24-1 shows the proposition pq once again, which we now view
 +
as a scalar field, in effect, a potential "plateau" of elevation 1
 +
over the shaded region, with an elevation of 0 everywhere else.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /                      o                      \          |
 +
|        /                      /%\                      \        |
 +
|        /                      /%%%\                      \        |
 +
|      /                      /%%%%%\                      \      |
 +
|      /                      /%%%%%%%\                      \      |
 +
|    /                      /%%%%%%%%%\                      \    |
 +
|    o                      o%%%%%%%%%%%o                      o    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    |          P            |%%%%%%%%%%%|            Q          |    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    |                      |%%%%%%%%%%%|                      |    |
 +
|    o                      o%%%%%%%%%%%o                      o    |
 +
|    \                      \%%%%%%%%%/                      /    |
 +
|      \                      \%%%%%%%/                      /      |
 +
|      \                      \%%%%%/                      /      |
 +
|        \                      \%%%/                      /        |
 +
|        \                      \%/                      /        |
 +
|          \                      o                      /          |
 +
|          \                    / \                    /          |
 +
|            o-------------------o  o-------------------o            |
 +
|                                                                    |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 24-1.  Proposition pq : X -> B
 +
 +
Given any proposition f : X -> B, the "tacit extension" of f to EX
 +
is notated !e!f : EX -> B and defined by the equation !e!f = f, so
 +
it's really just the same proposition living in a bigger universe.
 +
 +
Tacit extensions formalize the intuitive idea that a new function
 +
is related to an old function in such a way that it obeys the same
 +
constraints on the old variables, with a "don't care" condition on
 +
the new variables.
 +
 +
Figure 24-2 illustrates the "tacit extension" of the proposition
 +
or scalar field f = pq : X -> B to give the extended proposition
 +
or differential field that we notate as !e!f = !e![pq] : EX -> B.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /  P                    o                    Q  \          |
 +
|        /                      / \                      \        |
 +
|        /                      /  \                      \        |
 +
|      /                      /    \                      \      |
 +
|      /                      /      \                      \      |
 +
|    /                      /        \                      \    |
 +
|    o                      o (dp) (dq) o                      o    |
 +
|    |                      |  o-->--o  |                      |    |
 +
|    |                      |  \  /  |                      |    |
 +
|    |            (dp) dq  |    \ /    |  dp (dq)            |    |
 +
|    |          o<-----------------o----------------->o          |    |
 +
|    |                      |    |    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    o                      o    |    o                      o    |
 +
|    \                      \    |    /                      /    |
 +
|      \                      \  |  /                      /      |
 +
|      \                      \  |  /                      /      |
 +
|        \                      \ | /                      /        |
 +
|        \                      \|/                      /        |
 +
|          \                      |                      /          |
 +
|          \                    /|\                    /          |
 +
|            o-------------------o | o-------------------o            |
 +
|                                  |                                  |
 +
|                              dp | dq                              |
 +
|                                  |                                  |
 +
|                                  v                                  |
 +
|                                  o                                  |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 24-2.  Tacit Extension !e![pq] : EX -> B
 +
 +
Thus we have a pictorial way of visualizing the following data:
 +
 +
  !e![pq]
 +
 +
    =
 +
 +
    p q . dp dq
 +
 +
    +
 +
 +
    p q . dp (dq)
 +
 +
    +
 +
 +
    p q . (dp) dq
 +
 +
    +
 +
 +
    p q . (dp)(dq)
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 25
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Staying with the example pq : X -> B, Figure 25-1 shows
 +
the enlargement or shift map E[pq] : EX -> B in the same
 +
style of differential field picture that we drew for the
 +
tacit extension !e![pq] : EX -> B.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /  P                    o                    Q  \          |
 +
|        /                      / \                      \        |
 +
|        /                      /  \                      \        |
 +
|      /                      /    \                      \      |
 +
|      /                      /      \                      \      |
 +
|    /                      /        \                      \    |
 +
|    o                      o (dp) (dq) o                      o    |
 +
|    |                      |  o-->--o  |                      |    |
 +
|    |                      |  \  /  |                      |    |
 +
|    |            (dp) dq  |    \ /    |  dp (dq)            |    |
 +
|    |          o----------------->o<-----------------o          |    |
 +
|    |                      |    ^    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    o                      o    |    o                      o    |
 +
|    \                      \    |    /                      /    |
 +
|      \                      \  |  /                      /      |
 +
|      \                      \  |  /                      /      |
 +
|        \                      \ | /                      /        |
 +
|        \                      \|/                      /        |
 +
|          \                      |                      /          |
 +
|          \                    /|\                    /          |
 +
|            o-------------------o | o-------------------o            |
 +
|                                  |                                  |
 +
|                              dp | dq                              |
 +
|                                  |                                  |
 +
|                                  |                                  |
 +
|                                  o                                  |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 25-1.  Enlargement E[pq] : EX -> B
 +
 +
A very important conceptual transition has just occurred here,
 +
almost tacitly, as it were.  Generally speaking, having a set
 +
of mathematical objects of compatible types, in this case the
 +
two differential fields !e!f and Ef, both of the type EX -> B,
 +
is very useful, because it allows us to consider these fields
 +
as integral mathematical objects that can be operated on and
 +
combined in the ways that we usually associate with algebras.
 +
 +
In this case one notices that the tacit extension !e!f and the
 +
enlargement Ef are in a certain sense dual to each other, with
 +
!e!f indicating all of the arrows out of the region where f is
 +
true, and with Ef indicating all of the arrows into the region
 +
where f is true.  The only arc that they have in common is the
 +
no-change loop (dp)(dq) at pq.  If we add the two sets of arcs
 +
mod 2, then the common loop drops out, leaving the 6 arrows of
 +
D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /  P                    o                    Q  \          |
 +
|        /                      / \                      \        |
 +
|        /                      /  \                      \        |
 +
|      /                      /    \                      \      |
 +
|      /                      /      \                      \      |
 +
|    /                      /        \                      \    |
 +
|    o                      o          o                      o    |
 +
|    |                      |          |                      |    |
 +
|    |                      |          |                      |    |
 +
|    |            (dp) dq  |          |  dp (dq)            |    |
 +
|    |          o<---------------->o<---------------->o          |    |
 +
|    |                      |    ^    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    |                      |    |    |                      |    |
 +
|    o                      o    |    o                      o    |
 +
|    \                      \    |    /                      /    |
 +
|      \                      \  |  /                      /      |
 +
|      \                      \  |  /                      /      |
 +
|        \                      \ | /                      /        |
 +
|        \                      \|/                      /        |
 +
|          \                      |                      /          |
 +
|          \                    /|\                    /          |
 +
|            o-------------------o | o-------------------o            |
 +
|                                  |                                  |
 +
|                              dp | dq                              |
 +
|                                  |                                  |
 +
|                                  v                                  |
 +
|                                  o                                  |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 25-2.  Difference Map D[pq] : EX -> B
 +
 +
The differential features of D[pq] may be collected cell by cell of
 +
the underlying universe X% = [p, q] to give the following expansion:
 +
 +
  D[pq]
 +
 +
  =
 +
 +
  p q . ((dp)(dq))
 +
 +
  +
 +
 +
  p (q) . (dp) dq
 +
 +
  +
 +
 +
  (p) q . dp (dq)
 +
 +
  +
 +
 +
  (p)(q) . dp dq
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Note 26
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
If we follow the classical line that singles out linear functions
 +
as ideals of simplicity, then we may complete the analytic series
 +
of the proposition f = pq : X -> B in the following way.
 +
 +
Figure 26-1 shows the differential proposition df = d[pq] : EX -> B
 +
that we get by extracting the cell-wise linear approximation to the
 +
difference map Df = D[pq] : EX -> B.  This is the logical analogue
 +
of what would ordinarily be called 'the' differential of pq, but
 +
since I've been attaching the adjective "differential" to just
 +
about everything in sight, the distinction tends to be lost.
 +
For the time being, I'll resort to using the alternative
 +
name "tangent map" for df.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /  P                    o                    Q  \          |
 +
|        /                      / \                      \        |
 +
|        /                      /  \                      \        |
 +
|      /                      /    \                      \      |
 +
|      /                      /  o  \                      \      |
 +
|    /                      /  ^ ^  \                      \    |
 +
|    o                      o  /  \  o                      o    |
 +
|    |                      |  /    \  |                      |    |
 +
|    |                      | /      \ |                      |    |
 +
|    |                      |/        \|                      |    |
 +
|    |                  (dp)/ dq    dp \(dq)                  |    |
 +
|    |                      /|          |\                      |    |
 +
|    |                    / |          | \                    |    |
 +
|    |                    /  |          |  \                    |    |
 +
|    o                  /  o          o  \                  o    |
 +
|    \                v    \  dp dq  /    v                /    |
 +
|      \              o<--------------------->o              /      |
 +
|      \                      \    /                      /      |
 +
|        \                      \  /                      /        |
 +
|        \                      \ /                      /        |
 +
|          \                      o                      /          |
 +
|          \                    / \                    /          |
 +
|            o-------------------o  o-------------------o            |
 +
|                                                                    |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 26-1.  Differential or Tangent d[pq] : EX -> B
 +
 +
Just to be clear about what's being indicated here,
 +
it's a visual way of specifying the following data:
 +
 +
  d[pq]
 +
 +
  =
 +
 +
  p q . (dp, dq)
 +
 +
  +
 +
 +
  p (q) . dq
 +
 +
  +
 +
 +
  (p) q . dp
 +
 +
  +
 +
 +
  (p)(q) . 0
 +
 +
To understand the extended interpretations, that is,
 +
the conjunctions of basic and differential features
 +
that are being indicated here, it may help to note
 +
the following equivalences:
 +
 +
  (dp, dq)  =  dp + dq  =  dp(dq) + (dp)dq
 +
 +
      dp      =  dp dq  +  dp(dq)
 +
 +
      dq      =  dp dq  +  (dp)dq
 +
 +
Capping the series that analyzes the proposition pq
 +
in terms of succeeding orders of linear propositions,
 +
Figure 26-2 shows the remainder map r[pq] : EX -> B,
 +
that happens to be linear in pairs of variables.
 +
 +
o---------------------------------------------------------------------o
 +
|                                                                    |
 +
|  X                                                                |
 +
|            o-------------------o  o-------------------o            |
 +
|          /                    \ /                    \          |
 +
|          /  P                    o                    Q  \          |
 +
|        /                      / \                      \        |
 +
|        /                      /  \                      \        |
 +
|      /                      /    \                      \      |
 +
|      /                      /      \                      \      |
 +
|    /                      /        \                      \    |
 +
|    o                      o          o                      o    |
 +
|    |                      |          |                      |    |
 +
|    |                      |          |                      |    |
 +
|    |                      |  dp dq  |                      |    |
 +
|    |            o<------------------------------->o            |    |
 +
|    |                      |          |                      |    |
 +
|    |                      |          |                      |    |
 +
|    |                      |    o    |                      |    |
 +
|    o                      o    ^    o                      o    |
 +
|    \                      \    |    /                      /    |
 +
|      \                      \  |  /                      /      |
 +
|      \                      \  |  /                      /      |
 +
|        \                      \ | /                      /        |
 +
|        \                      \|/                      /        |
 +
|          \                    dp | dq                    /          |
 +
|          \                    /|\                    /          |
 +
|            o-------------------o | o-------------------o            |
 +
|                                  |                                  |
 +
|                                  |                                  |
 +
|                                  |                                  |
 +
|                                  v                                  |
 +
|                                  o                                  |
 +
|                                                                    |
 +
o---------------------------------------------------------------------o
 +
Figure 26-2.  Remainder r[pq] : EX -> B
 +
 +
Reading the arrows off the map produces the following data:
 +
 +
    r[pq]
 +
 +
    =
 +
 +
    p q . dp dq
 +
 +
    +
 +
 +
    p (q) . dp dq
 +
 +
    +
 +
 +
    (p) q . dp dq
 +
 +
    +
 +
 +
    (p)(q) . dp dq
 +
 +
In short, r[pq] is a constant field,
 +
having the value dp dq at each cell.
 +
 +
A more detailed presentation of Differential Logic can be found here:
 +
 +
DLOG D.  http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
 +
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
 +
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
DAL.  Dynamics And Logic
 +
 +
Inquiry List
 +
 +
00.  http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400
 +
00.  http://stderr.org/pipermail/inquiry/2004-July/thread.html#1685
 +
01.  http://stderr.org/pipermail/inquiry/2004-May/001400.html
 +
02.  http://stderr.org/pipermail/inquiry/2004-May/001401.html
 +
03.  http://stderr.org/pipermail/inquiry/2004-May/001402.html
 +
04.  http://stderr.org/pipermail/inquiry/2004-May/001403.html
 +
05.  http://stderr.org/pipermail/inquiry/2004-May/001404.html
 +
06.  http://stderr.org/pipermail/inquiry/2004-May/001405.html
 +
07.  http://stderr.org/pipermail/inquiry/2004-May/001406.html
 +
08.  http://stderr.org/pipermail/inquiry/2004-May/001407.html
 +
09.  http://stderr.org/pipermail/inquiry/2004-May/001408.html
 +
10.  http://stderr.org/pipermail/inquiry/2004-May/001410.html
 +
11.  http://stderr.org/pipermail/inquiry/2004-May/001411.html
 +
12.  http://stderr.org/pipermail/inquiry/2004-May/001412.html
 +
13.  http://stderr.org/pipermail/inquiry/2004-May/001413.html
 +
14.  http://stderr.org/pipermail/inquiry/2004-May/001415.html
 +
15.  http://stderr.org/pipermail/inquiry/2004-May/001416.html
 +
16.  http://stderr.org/pipermail/inquiry/2004-May/001418.html
 +
17.  http://stderr.org/pipermail/inquiry/2004-May/001419.html
 +
18.  http://stderr.org/pipermail/inquiry/2004-May/001420.html
 +
19.  http://stderr.org/pipermail/inquiry/2004-May/001421.html
 +
20.  http://stderr.org/pipermail/inquiry/2004-May/001422.html
 +
21.  http://stderr.org/pipermail/inquiry/2004-May/001423.html
 +
22.  http://stderr.org/pipermail/inquiry/2004-May/001424.html
 +
23.  http://stderr.org/pipermail/inquiry/2004-July/001685.html
 +
24.  http://stderr.org/pipermail/inquiry/2004-July/001686.html
 +
25.  http://stderr.org/pipermail/inquiry/2004-July/001687.html
 +
26.  http://stderr.org/pipermail/inquiry/2004-July/001688.html
 +
 +
NKS Forum
 +
 +
00.  http://forum.wolframscience.com/showthread.php?threadid=420
 +
01.  http://forum.wolframscience.com/showthread.php?postid=1282#post1282
 +
02.  http://forum.wolframscience.com/showthread.php?postid=1285#post1285
 +
03.  http://forum.wolframscience.com/showthread.php?postid=1289#post1289
 +
04.  http://forum.wolframscience.com/showthread.php?postid=1292#post1292
 +
05.  http://forum.wolframscience.com/showthread.php?postid=1293#post1293
 +
06.  http://forum.wolframscience.com/showthread.php?postid=1294#post1294
 +
07.  http://forum.wolframscience.com/showthread.php?postid=1296#post1296
 +
08.  http://forum.wolframscience.com/showthread.php?postid=1299#post1299
 +
09.  http://forum.wolframscience.com/showthread.php?postid=1301#post1301
 +
10.  http://forum.wolframscience.com/showthread.php?postid=1304#post1304
 +
11.  http://forum.wolframscience.com/showthread.php?postid=1307#post1307
 +
12.  http://forum.wolframscience.com/showthread.php?postid=1309#post1309
 +
13.  http://forum.wolframscience.com/showthread.php?postid=1311#post1311
 +
14.  http://forum.wolframscience.com/showthread.php?postid=1314#post1314
 +
15.  http://forum.wolframscience.com/showthread.php?postid=1315#post1315
 +
16.  http://forum.wolframscience.com/showthread.php?postid=1318#post1318
 +
17.  http://forum.wolframscience.com/showthread.php?postid=1321#post1321
 +
18.  http://forum.wolframscience.com/showthread.php?postid=1323#post1323
 +
19.  http://forum.wolframscience.com/showthread.php?postid=1326#post1326
 +
20.  http://forum.wolframscience.com/showthread.php?postid=1327#post1327
 +
21.  http://forum.wolframscience.com/showthread.php?postid=1330#post1330
 +
22.  http://forum.wolframscience.com/showthread.php?postid=1331#post1331
 +
23.  http://forum.wolframscience.com/showthread.php?postid=1598#post1598
 +
24.  http://forum.wolframscience.com/showthread.php?postid=1601#post1601
 +
25.  http://forum.wolframscience.com/showthread.php?postid=1602#post1602
 +
26.  http://forum.wolframscience.com/showthread.php?postid=1603#post1603
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
</pre>

Revision as of 02:16, 12 June 2009

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Dynamics And Logic

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 1

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

I am going to excerpt some of my previous explorations
on differential logic and dynamic systems and bring them
to bear on the sorts of discrete dynamical themes that we
find of interest in the NKS Forum.  This adaptation draws on
the "Cactus Rules", "Propositional Equation Reasoning Systems",
and "Reductions Among Relations" threads, and will in time be
applied to the "Differential Analytic Turing Automata" thread:

CR.    http://forum.wolframscience.com/showthread.php?threadid=256
PERS.  http://forum.wolframscience.com/showthread.php?threadid=297
RAR.   http://forum.wolframscience.com/showthread.php?threadid=400
DATA.  http://forum.wolframscience.com/showthread.php?threadid=228

One of the first things that you can do, once you have
a moderately functional calculus for boolean functions
or propositional logic, whatever you choose to call it,
is to start thinking about, and even start computing,
the differentials of these functions or propositions.

Let us start with a proposition of the form "p and q",
that is graphed as two labels attached to a root node:

o-------------------------------------------------o
|                                                 |
|                       p q                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                     p and q                     |
o-------------------------------------------------o

Written as a string, this is just the concatenation "p q".

The proposition pq may be taken as a boolean function f<p, q>
having the abstract type f : B x B -> B, where B = {0, 1} is
read in such a way that 0 means "false" and 1 means "true".

In this style of graphical representation,
the value "true" looks like a blank label
and the value "false" looks like an edge.

o-------------------------------------------------o
|                                                 |
|                                                 |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      true                       |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                      false                      |
o-------------------------------------------------o

Back to the proposition pq.  Imagine yourself standing
in a fixed cell of the corresponding venn diagram, say,
the cell where the proposition pq is true, as pictured:

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     |       P       |%%%%%|       Q       |     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     o               o%%%%%o               o     |
|      \               \%%%/               /      |
|       \               \%/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o

Now ask yourself:  What is the value of the
proposition pq at a distance of dp and dq
from the cell pq where you are standing?

Don't think about it -- just compute:

o-------------------------------------------------o
|                                                 |
|                   dp o   o dq                   |
|                     / \ / \                     |
|                  p o---@---o q                  |
|                                                 |
o-------------------------------------------------o
|                 (p, dp) (q, dq)                 |
o-------------------------------------------------o

To make future graphs easier to draw in Asciiland,
I'll use devices like @=@=@ and o=o=o to identify
several nodes into one, as in this next redrawing:

o-------------------------------------------------o
|                                                 |
|                   p  dp q  dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
|                 (p, dp) (q, dq)                 |
o-------------------------------------------------o

However you draw it, these expressions follow because the
expression p + dp, where the plus sign indicates addition
in B and thus corresponds to the exclusive-or in logic,
parses to a graph of the following form:

o-------------------------------------------------o
|                                                 |
|                     p    dp                     |
|                      o---o                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                     (p, dp)                     |
o-------------------------------------------------o

Next question:  What is the difference between the value of the
proposition pq "over there", at a remove of dp dq, and the value
of the proposition pq where you are, all expressed in the form of
a general formula, of course?  Here is the appropriate formulation:

o-------------------------------------------------o
|                                                 |
|             p  dp q  dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         p q                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|              ((p, dp)(q, dq), p q)              |
o-------------------------------------------------o

There is one thing that I ought to mention at this point:
Computed over B, plus and minus are identical operations.
This will make the relation between the differential and
the integral parts of the appropriate calculus slightly
stranger than usual, but we will get into that later.

Last question, for now:  What is the value of this expression
from your current standpoint, that is, evaluated at the point
where pq is true?  Well, substituting 1 for p and 1 for q in
the graph amounts to erasing the labels "p" and "q", like so:

o-------------------------------------------------o
|                                                 |
|                dp    dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|              (( , dp)( , dq),    )              |
o-------------------------------------------------o

And this is equivalent to the following graph:

o-------------------------------------------------o
|                                                 |
|                     dp   dq                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                   ((dp) (dq))                   |
o-------------------------------------------------o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 2

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

We have just met with the fact that
the differential of the "and" is
the "or" of the differentials.

p and q --Diff--> dp or dq.

o-------------------------------------------------o
|                                                 |
|                                    dp   dq      |
|                                     o   o       |
|                                      \ /        |
|                                       o         |
|        p q                            |         |
|         @          --Diff-->          @         |
|                                                 |
o-------------------------------------------------o
|        p q         --Diff-->     ((dp) (dq))    |
o-------------------------------------------------o

It will be necessary to develop a more refined analysis of
this statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of DeMorgan's rule,
it is no accident, as differentiation and negation turn out to be
closely related operations.  Indeed, one can find discussions of
logical difference calculus in the Boole-DeMorgan correspondence
and Peirce also made use of differential operators in a logical
context, but the exploration of these ideas has been hampered
by a number of factors, not the least of which has been the
lack of a syntax that was up to handling the complexity of
the expressions that evolve.

Let us run through the initial example again, this time attempting
to interpret the formulas that develop at each stage along the way.

We begin with a proposition, or a boolean function, f<p, q> = pq.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /               o               \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     |       P       |% F %|       Q       |     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     o               o%%%%%o               o     |
|      \               \%%%/               /      |
|       \               \%/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       p q                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| f =                   p q                       |
o-------------------------------------------------o

A function like this has an abstract type and a concrete type.
The abstract type is what we invoke when we write things like
f : B x B -> B or f : B^2 -> B.  The concrete type takes into
account the qualitative dimensions or the "units" of the case,
which can be explained as follows.

   Let !P! be the set of two values {(p), p} = {not-p, p} ~=~ B

   Let !Q! be the set of two values {(q), q} = {not-q, q} ~=~ B

Then interpret the usual propositions about p, q
as functions of concrete type f : !P! x !Q! -> B.

We are going to consider various "operators" on these functions.
Here, an operator W is a function that takes one function f into
another function Wf.

The first couple of operators that we need to consider are
logical analogues of the pair that play a founding role
in the classical "finite difference calculus", namely:

   The "difference" operator [capital Delta], written here as D.

   The "enlargement" operator [capital Epsilon], written here as E.

These days, E is more often called the "shift" operator.

In order to describe the universe in which these operators operate,
it will be necessary to enlarge our original universe of discourse.

Starting out from the initial space X = !P! x !Q!, we
construct its (first order) "differential extension":

   EX  =  X x dX  =  !P! x !Q! x d!P! x d!Q!

   where:

     X   =   !P! x !Q!

    dX   =  d!P! x d!Q!

   d!P!  =  {(dp), dp}

   d!Q!  =  {(dq), dq}

The interpretations of these new symbols can be diverse,
but the easiest interpretation for now is just to say
that "dp" means "change p" and "dq" means "change q".

Drawing a venn diagram for the differential extension EX = X x dX
requires four logical dimensions, !P!, !Q!, d!P!, d!Q!, but it is
possible to project a suggestion of what the differential features
dp and dq are about on the 2-dimensional base space X = !P! x !Q!
by drawing arrows that cross the boundaries of the basic circles
in the venn diagram for X, reading an arrow as dp if it crosses
the boundary between p and (p) in either direction and reading
an arrow as dq if it crosses the boundary between q and (q)
in either direction.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       p       o       q       \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |         dq    |%%%%%|    dp         |     |
|     |     <---------|--o--|--------->     |     |
|     |               |%%%%%|               |     |
|     |               |%%%%%|               |     |
|     o               o%%%%%o               o     |
|      \               \%%%/               /      |
|       \               \%/               /       |
|        \               o               /        |
|         \             / \             /         |
|          o-----------o   o-----------o          |
|                                                 |
|                                                 |
o-------------------------------------------------o

We can form propositions from these differential variables
in the same way that we would any other logical variables,
for example, taking the differential proposition (dp (dq))
as saying that dp implies dq, in other words, that there
is "no change in p without a change in q".

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 3

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Given the proposition f<p, q> over the space X = !P! x !Q!,
the (first order) "enlargement" of f is the proposition Ef
over the differential extension EX that is defined by the
following formula:

   Ef<p, q, dp, dq>

   =  f<p + dp, q + dq>

   =  f<(p, dp), (q, dq)>

In the example f<p, q> = pq, the enlargement Ef is given by:

   Ef<p, q, dp, dq>

   =  [p + dp][q + dq]

   =  (p, dp)(q, dq)

o-------------------------------------------------o
|                                                 |
|                   p  dp q  dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (p, dp) (q, dq)                 |
o-------------------------------------------------o

Given the proposition f<p, q> over X = !P! x !Q!, the
(first order) "difference" of f is the proposition Df
over EX that is defined by the formula Df = Ef - f, or,
written out in full:

   Df<p, q, dp, dq>

   =  f<p + dp, q + dq> - f<p, q>

   =  (f<(p, dp), (q, dq)>, f<p, q>)

In the example f<p, q> = pq, the difference Df is given by:

   Df<p, q, dp, dq>

   =  [p + dp][q + dq] - pq

   =  ((p, dp)(q, dq), pq)

o-------------------------------------------------o
|                                                 |
|             p  dp q  dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         p q                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =           ((p, dp)(q, dq), pq)             |
o-------------------------------------------------o

We did not yet go through the trouble to interpret this (first order)
"difference of conjunction" fully, but were happy simply to evaluate
it with respect to a single location in the universe of discourse,
namely, at the point picked out by the singular proposition pq,
in as much as if to say at the place where p = 1 and q = 1.
This evaluation is written in the form Df|pq or Df|<1, 1>,
and we arrived at the locally applicable law that states
that f = pq = p and q => Df|pq = ((dp)(dq)) = dp or dq.

o-------------------------------------------------o
|                                                 |
|                                                 |
|          o-----------o   o-----------o          |
|         /             \ /             \         |
|        /       p       o       q       \        |
|       /               /%\               \       |
|      /               /%%%\               \      |
|     o               o%%%%%o               o     |
|     |               |%%%%%|               |     |
|     |      dq (dp)  |%%%%%|  dp (dq)      |     |
|     |   o<----------|--o--|---------->o   |     |
|     |               |%%|%%|               |     |
|     |               |%%|%%|               |     |
|     o               o%%|%%o               o     |
|      \               \%|%/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dp|dq                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                     dp   dq                     |
|                      o   o                      |
|                       \ /                       |
|                        o                        |
|                        |                        |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|pq =           ((dp) (dq))                   |
o-------------------------------------------------o

The picture illustrates the analysis of the inclusive
disjunction ((dp)(dq)) into the exclusive disjunction:
dp(dq) + (dp)dq + dp dq, a differential proposition that
may be interpreted to say "change p or change q or both".
And this can be recognized as just what you need to do if
you happen to find yourself in the center cell and require
a complete and detailed description of ways to escape it.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 4

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Last time we computed what will variously be called
the "difference map", the "difference proposition",
or the "local proposition" Df_x for the proposition
f<p, q> = pq at the point x where p = 1 and q = 1.

In the universe X = !P! x !Q!, the four propositions
pq, p(q), (p)q, (p)(q) that indicate the "cells",
or the smallest regions of the venn diagram, are
called "singular propositions".  These serve as
an alternative notation for naming the points
<1, 1>, <1, 0>, <0, 1>, <0, 0>, respectively.

Thus, we can write Df_x = Df|x = Df|<1, 1> = Df|pq,
so long as we know the frame of reference in force.

Sticking with the example f<p, q> = pq, let us compute the
value of the difference proposition Df at all of the points.

o-------------------------------------------------o
|                                                 |
|             p  dp q  dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/         p q                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df =        ((p, dp)(q, dq), pq)                |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                dp    dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /                           |
|                \| |/                            |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|pq =           ((dp) (dq))                   |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o                             |
|                dp |  dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|p(q) =          (dp) dq                      |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|             o                                   |
|             |  dp    dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /         o                 |
|                \| |/          |                 |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(p)q =            dp (dq)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|             o     o                             |
|             |  dp |  dq                         |
|             o---o o---o                         |
|              \  | |  /                          |
|               \ | | /       o   o               |
|                \| |/         \ /                |
|                 o=o-----------o                 |
|                  \           /                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o
| Df|(p)(q) =          dp dq                      |
o-------------------------------------------------o

The easy way to visualize the values of these graphical
expressions is just to notice the following equivalents:

o-------------------------------------------------o
|                                                 |
|  e                                              |
|  o-o-o-...-o-o-o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                          e         |
|       \   /                           o         |
|        \ /                            |         |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (e, , ... , , )       =             (e)        |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                o                                |
| e_1 e_2   e_k  |                                |
|  o---o-...-o---o                                |
|   \           /                                 |
|    \         /                                  |
|     \       /                                   |
|      \     /                                    |
|       \   /                                     |
|        \ /                       e_1 ... e_k    |
|         @              =              @         |
|                                                 |
o-------------------------------------------------o
|  (e_1, ..., e_k, ())   =         e_1 ... e_k    |
o-------------------------------------------------o

Laying out the arrows on the augmented venn diagram,
one gets a picture of a "differential vector field".

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                        |                        |
|                      dp|dq                      |
|                        |                        |
|          o-----------o | o-----------o          |
|         /             \|/             \         |
|        /       p       |       q       \        |
|       /               /|\               \       |
|      /               /%|%\               \      |
|     o               o%%|%%o               o     |
|     |      (dp) dq  |%%v%%|  dp (dq)      |     |
|     |   o-----------|->o<-|-----------o   |     |
|     |               |%%%%%|               |     |
|     |   o<----------|--o--|---------->o   |     |
|     |      (dp) dq  |%%|%%|  dp (dq)      |     |
|     o               o%%|%%o               o     |
|      \               \%|%/               /      |
|       \               \|/               /       |
|        \               |               /        |
|         \             /|\             /         |
|          o-----------o | o-----------o          |
|                        |                        |
|                      dp|dq                      |
|                        |                        |
|                        v                        |
|                        o                        |
|                                                 |
o-------------------------------------------------o

This just amounts to a depiction of the points,
truth-value assignments, or interpretations in
EX = !P! x !Q! x d!P! x d!Q! that are indicated
by the difference map Df : EX -> B, namely, the
following six points or singular propositions:

   1.   p  q  dp  dq
   2.   p  q  dp (dq)
   3.   p  q (dp) dq
   4.   p (q)(dp) dq
   5.  (p) q  dp (dq)
   6.  (p)(q) dp  dq

By inspection, it is fairly easy to understand Df
as telling you what you have to do from each point
of X in order to change the value borne by f<p, q>
at the point in question, that is, in order to get
to a point where the value of f<p, q> is different
from what it is where you started.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 5

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

We have been studying the action of the difference operator D,
also known as the "localization operator", on the proposition
f : !P! x !Q! -> B that is commonly called the conjunction pq.
We categorized Df as a (first order) differential proposition,
a proposition of the type Df : !P! x !Q! x d!P! x d!Q! -> B.

Abstracting from the augmented venn diagram that shows how the
models or the satisfying interpretations of Df distribute over
the (first order) extended space EX = !P! x !Q! x d!P! x d!Q!,
we can represent Df in the form of a digraph or directed graph,
one whose points are labeled with the elements of X = !P! x !Q!
and whose arcs are labeled with the elements of dX = d!P! x d!Q!.

o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Df =              p  q  ((dp)(dq))              |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                       p q                       |
|  p (q) o<------------->o<------------->o (p) q  |
|             (dp) dq    ^    dp (dq)             |
|                        |                        |
|                        |                        |
|                     dp | dq                     |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                     (p) (q)                     |
|                                                 |
o-------------------------------------------------o

Any proposition worth its salt has many equivalent ways to view it,
any one of which may reveal some unsuspected aspect of its meaning.
We will encounter more and more of these variant readings as we go.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 6

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The enlargement operator E, also known as the "shift operator",
has many interesting and very useful properties in its own right,
so let us not fail to observe a few of the more salient features
that play out on the surface of our simple example, f<p, q> = pq.

To begin we need to formulate a suitably generic
definition of the extended universe of discourse:

   Relative to an initial domain X = X_1 x ... x X_k,

   EX = X x dX = X_1 x ... x X_k x dX_1 x ... x dX_k.

For a proposition f : X_1 x ... x X_k -> B,
the (first order) "enlargement" of f is the
proposition Ef : EX -> B that is defined by:

   Ef<x_1, ..., x_k, dx_1, ..., dx_k>

   =  f<x_1 + dx_1, ..., x_k + dx_k>

   =  f<(x_1, dx_1), ..., (x_k, dx_k)>

It should be noted that the so-called "differential variables" dx_j
are really just the same type of boolean variables as the other x_j.
It is conventional to give the additional variables these inflected
names, but whatever extra connotations we attach to these syntactic
conveniences are wholly external to their purely algebraic meanings.

In the case of the conjunction f<p, q> = pq,
the enlargement Ef is formulated as follows:

   Ef<p, q, dp, dq>

   =  [p + dp][q + dq]

   =  (p, dp)(q, dq)

Given that this expression uses nothing more than the "boolean ring"
operations of addition (+) and multiplication (*), it is permissible
to "multiply things out" in the usual manner to arrive at the result:

   Ef<p, q, dp, dq>

   =  p q  +  p dq  +  q dp  +  dp dq

To understand what this means in logical terms,
for instance, as expressed in a boolean expansion
or a "disjunctive normal form" (DNF), it is perhaps
a little better to go back and analyze the expression
the same way that we did for Df.  Thus, let us compute
the value of the enlarged proposition Ef at each of the
points in the initial domain of discourse X = !P! x !Q!.

o-------------------------------------------------o
|                                                 |
|                   p  dp q  dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef =            (p, dp) (q, dq)                 |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                      dp    dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|pq =            (dp) (dq)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                         o                       |
|                      dp |  dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|p(q) =          (dp)  dq                     |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o                             |
|                   |  dp    dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(p)q =           dp  (dq)                    |
o-------------------------------------------------o

o-------------------------------------------------o
|                                                 |
|                   o     o                       |
|                   |  dp |  dq                   |
|                   o---o o---o                   |
|                    \  | |  /                    |
|                     \ | | /                     |
|                      \| |/                      |
|                       @=@                       |
|                                                 |
o-------------------------------------------------o
| Ef|(p)(q) =         dp   dq                     |
o-------------------------------------------------o

Given the kind of data that arises from this form of analysis,
we can now fold the disjoined ingredients back into a boolean
expansion or a DNF that is equivalent to the proposition Ef.

   Ef = pq Ef_pq + p(q) Ef_p(q) + (p)q Ef_(p)q + (p)(q) Ef_(p)(q)

Here is a summary of the result, illustrated by means of
a digraph picture, where the "no change" element (dp)(dq)
is drawn as a loop at the point p q.

o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Ef =              p  q   (dp)(dq)               |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
|                                                 |
|                    (dp) (dq)                    |
|                    .--->---.                    |
|                     \     /                     |
|                      \p q/                      |
|                       \ /                       |
|  p (q) o-------------->o<--------------o (p) q  |
|             (dp) dq    ^    dp (dq)             |
|                        |                        |
|                        |                        |
|                     dp | dq                     |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                     (p) (q)                     |
|                                                 |
o-------------------------------------------------o

We may understand the enlarged proposition Ef
as telling us all the different ways to reach
a model of f from any point of the universe X.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 7

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

To broaden our experience with simple examples, let us
now contemplate the sixteen functions of concrete type
!P! x !Q! -> B and abstract type B x B -> B.  For ease
of future reference, I will set here a few tables that
specify the actions of E and D on the 16 functions and
allow us to view the results in several different ways.

By way of initial orientation, Table 7 lists equivalent expressions
for the sixteen functions in several different formalisms, indexing
systems, or languages for the propositional calculus, also known as
"zeroth order logic" (ZOL).

Table 7.  Propositional Forms on Two Variables
o---------o---------o---------o----------o------------------o----------o
| L_1     | L_2     | L_3     | L_4      | L_5              | L_6      |
|         |         |         |          |                  |          |
| Decimal | Binary  | Vector  | Cactus   | English          | Ordinary |
o---------o---------o---------o----------o------------------o----------o
|         |       p : 1 1 0 0 |          |                  |          |
|         |       q : 1 0 1 0 |          |                  |          |
o---------o---------o---------o----------o------------------o----------o
|         |         |         |          |                  |          |
| f_0     | f_0000  | 0 0 0 0 |    ()    | false            |    0     |
|         |         |         |          |                  |          |
| f_1     | f_0001  | 0 0 0 1 |  (p)(q)  | neither p nor q  | ~p & ~q  |
|         |         |         |          |                  |          |
| f_2     | f_0010  | 0 0 1 0 |  (p) q   | q and not p      | ~p &  q  |
|         |         |         |          |                  |          |
| f_3     | f_0011  | 0 0 1 1 |  (p)     | not p            | ~p       |
|         |         |         |          |                  |          |
| f_4     | f_0100  | 0 1 0 0 |   p (q)  | p and not q      |  p & ~q  |
|         |         |         |          |                  |          |
| f_5     | f_0101  | 0 1 0 1 |     (q)  | not q            |      ~q  |
|         |         |         |          |                  |          |
| f_6     | f_0110  | 0 1 1 0 |  (p, q)  | p not equal to q |  p +  q  |
|         |         |         |          |                  |          |
| f_7     | f_0111  | 0 1 1 1 |  (p  q)  | not both p and q | ~p v ~q  |
|         |         |         |          |                  |          |
| f_8     | f_1000  | 1 0 0 0 |   p  q   | p and q          |  p &  q  |
|         |         |         |          |                  |          |
| f_9     | f_1001  | 1 0 0 1 | ((p, q)) | p equal to q     |  p =  q  |
|         |         |         |          |                  |          |
| f_10    | f_1010  | 1 0 1 0 |      q   | q                |       q  |
|         |         |         |          |                  |          |
| f_11    | f_1011  | 1 0 1 1 |  (p (q)) | not p without q  |  p => q  |
|         |         |         |          |                  |          |
| f_12    | f_1100  | 1 1 0 0 |   p      | p                |  p       |
|         |         |         |          |                  |          |
| f_13    | f_1101  | 1 1 0 1 | ((p) q)  | not q without p  |  p <= q  |
|         |         |         |          |                  |          |
| f_14    | f_1110  | 1 1 1 0 | ((p)(q)) | p or q           |  p v  q  |
|         |         |         |          |                  |          |
| f_15    | f_1111  | 1 1 1 1 |   (())   | true             |    1     |
|         |         |         |          |                  |          |
o---------o---------o---------o----------o------------------o----------o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 8

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The next four Tables expand the expressions of Ef and Df
in two different ways, for each of the sixteen functions.
Notice that the functions are given in a different order,
partitioned into seven natural classes by a group action.

Table 8-a.  Ef Expanded Over Ordinary Features
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Ef | pq   | Ef | p(q)  | Ef | (p)q  | Ef | (p)(q)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (p)(q)   |   dp  dq   |   dp (dq)  |  (dp) dq   |  (dp)(dq)  |
|      |            |            |            |            |            |
| f_2  |   (p) q    |   dp (dq)  |   dp  dq   |  (dp)(dq)  |  (dp) dq   |
|      |            |            |            |            |            |
| f_4  |    p (q)   |  (dp) dq   |  (dp)(dq)  |   dp  dq   |   dp (dq)  |
|      |            |            |            |            |            |
| f_8  |    p  q    |  (dp)(dq)  |  (dp) dq   |   dp (dq)  |   dp  dq   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (p)      |   dp       |   dp       |  (dp)      |  (dp)      |
|      |            |            |            |            |            |
| f_12 |    p       |  (dp)      |  (dp)      |   dp       |   dp       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (p, q)   |  (dp, dq)  | ((dp, dq)) | ((dp, dq)) |  (dp, dq)  |
|      |            |            |            |            |            |
| f_9  |  ((p, q))  | ((dp, dq)) |  (dp, dq)  |  (dp, dq)  | ((dp, dq)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (q)   |       dq   |      (dq)  |       dq   |      (dq)  |
|      |            |            |            |            |            |
| f_10 |       q    |      (dq)  |       dq   |      (dq)  |       dq   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (p  q)   | ((dp)(dq)) | ((dp) dq)  |  (dp (dq)) |  (dp  dq)  |
|      |            |            |            |            |            |
| f_11 |   (p (q))  | ((dp) dq)  | ((dp)(dq)) |  (dp  dq)  |  (dp (dq)) |
|      |            |            |            |            |            |
| f_13 |  ((p) q)   |  (dp (dq)) |  (dp  dq)  | ((dp)(dq)) | ((dp) dq)  |
|      |            |            |            |            |            |
| f_14 |  ((p)(q))  |  (dp  dq)  |  (dp (dq)) | ((dp) dq)  | ((dp)(dq)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

Table 8-b.  Df Expanded Over Ordinary Features
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |  Df | pq   | Df | p(q)  | Df | (p)q  | Df | (p)(q)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (p)(q)   |   dp  dq   |   dp (dq)  |  (dp) dq   | ((dp)(dq)) |
|      |            |            |            |            |            |
| f_2  |   (p) q    |   dp (dq)  |   dp  dq   | ((dp)(dq)) |  (dp) dq   |
|      |            |            |            |            |            |
| f_4  |    p (q)   |  (dp) dq   | ((dp)(dq)) |   dp  dq   |   dp (dq)  |
|      |            |            |            |            |            |
| f_8  |    p  q    | ((dp)(dq)) |  (dp) dq   |   dp (dq)  |   dp  dq   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (p)      |   dp       |   dp       |   dp       |   dp       |
|      |            |            |            |            |            |
| f_12 |    p       |   dp       |   dp       |   dp       |   dp       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (p, q)   |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |
|      |            |            |            |            |            |
| f_9  |  ((p, q))  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |  (dp, dq)  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (q)   |       dq   |       dq   |       dq   |       dq   |
|      |            |            |            |            |            |
| f_10 |       q    |       dq   |       dq   |       dq   |       dq   |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (p  q)   | ((dp)(dq)) |  (dp) dq   |   dp (dq)  |   dp  dq   |
|      |            |            |            |            |            |
| f_11 |   (p (q))  |  (dp) dq   | ((dp)(dq)) |   dp  dq   |   dp (dq)  |
|      |            |            |            |            |            |
| f_13 |  ((p) q)   |   dp (dq)  |   dp  dq   | ((dp)(dq)) |  (dp) dq   |
|      |            |            |            |            |            |
| f_14 |  ((p)(q))  |   dp  dq   |   dp (dq)  |  (dp) dq   | ((dp)(dq)) |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 9

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Table 9-a.  Ef Expanded Over Differential Features
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      |   T_11 f   |   T_10 f   |   T_01 f   |   T_00 f   |
|      |            |            |            |            |            |
|      |            | Ef| dp dq  | Ef| dp(dq) | Ef| (dp)dq | Ef|(dp)(dq)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (p)(q)   |    p  q    |    p (q)   |   (p) q    |   (p)(q)   |
|      |            |            |            |            |            |
| f_2  |   (p) q    |    p (q)   |    p  q    |   (p)(q)   |   (p) q    |
|      |            |            |            |            |            |
| f_4  |    p (q)   |   (p) q    |   (p)(q)   |    p  q    |    p (q)   |
|      |            |            |            |            |            |
| f_8  |    p  q    |   (p)(q)   |   (p) q    |    p (q)   |    p  q    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (p)      |    p       |    p       |   (p)      |   (p)      |
|      |            |            |            |            |            |
| f_12 |    p       |   (p)      |   (p)      |    p       |    p       |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (p, q)   |   (p, q)   |  ((p, q))  |  ((p, q))  |   (p, q)   |
|      |            |            |            |            |            |
| f_9  |  ((p, q))  |  ((p, q))  |   (p, q)   |   (p, q)   |  ((p, q))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (q)   |       q    |      (q)   |       q    |      (q)   |
|      |            |            |            |            |            |
| f_10 |       q    |      (q)   |       q    |      (q)   |       q    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (p  q)   |  ((p)(q))  |  ((p) q)   |   (p (q))  |   (p  q)   |
|      |            |            |            |            |            |
| f_11 |   (p (q))  |  ((p) q)   |  ((p)(q))  |   (p  q)   |   (p (q))  |
|      |            |            |            |            |            |
| f_13 |  ((p) q)   |   (p (q))  |   (p  q)   |  ((p)(q))  |  ((p) q)   |
|      |            |            |            |            |            |
| f_14 |  ((p)(q))  |   (p  q)   |   (p (q))  |  ((p) q)   |  ((p)(q))  |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |    (())    |    (())    |    (())    |    (())    |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|                   |            |            |            |            |
| Fiped Point Total |      4     |      4     |      4     |     16     |
|                   |            |            |            |            |
o-------------------o------------o------------o------------o------------o

Table 9-b.  Df Expanded Over Differential Features
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
|      |     f      | Df| dp dq  | Df| dp(dq) | Df| (dp)dq | Df|(dp)(dq)|
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_0  |     ()     |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_1  |   (p)(q)   |  ((p, q))  |    (q)     |    (p)     |     ()     |
|      |            |            |            |            |            |
| f_2  |   (p) q    |   (p, q)   |     q      |    (p)     |     ()     |
|      |            |            |            |            |            |
| f_4  |    p (q)   |   (p, q)   |    (q)     |     p      |     ()     |
|      |            |            |            |            |            |
| f_8  |    p  q    |  ((p, q))  |     q      |     p      |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_3  |   (p)      |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
| f_12 |    p       |    (())    |    (())    |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_6  |   (p, q)   |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
| f_9  |  ((p, q))  |     ()     |    (())    |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_5  |      (q)   |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
| f_10 |       q    |    (())    |     ()     |    (())    |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_7  |   (p  q)   |  ((p, q))  |     q      |     p      |     ()     |
|      |            |            |            |            |            |
| f_11 |   (p (q))  |   (p, q)   |    (q)     |     p      |     ()     |
|      |            |            |            |            |            |
| f_13 |  ((p) q)   |   (p, q)   |     q      |    (p)     |     ()     |
|      |            |            |            |            |            |
| f_14 |  ((p)(q))  |  ((p, q))  |    (q)     |    (p)     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o
|      |            |            |            |            |            |
| f_15 |    (())    |     ()     |     ()     |     ()     |     ()     |
|      |            |            |            |            |            |
o------o------------o------------o------------o------------o------------o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 10

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

If you think that I linger in the realm of logical difference calculus
out of sheer vacillation about getting down to the differential proper,
it is probably out of a prior expectation that you derive from the art
or the long-engrained practice of real analysis.  But the fact is that
ordinary calculus only rushes on to the sundry orders of approximation
because the strain of comprehending the full import of E and D at once
whelm over its discrete and finite powers to grasp them.  But here, in
the fully serene idylls of ZOL, we find ourselves fit with the compass
of a wit that is all we'd ever need to explore their effects with care.

So let us do just that.

I will first rationalize the novel grouping of propositional forms
in the last set of Tables, as that will extend a gentle invitation
to the mathematical subject of "group theory", and demonstrate its
relevance to differential logic in a strikingly apt and useful way.
The data for that account is contained in Table 9-a, above or here:

DAL 9.  http://forum.wolframscience.com/showthread.php?postid=1301#post1301
DAL 9.  http://stderr.org/pipermail/inquiry/2004-May/001408.html

The shift operator E can be understood as enacting
a substitution operation on the proposition f that
is given as its argument.  In our present focus on
propositional forms that involve two variables, we
have the following datatype and applied definition:

   E : (X -> B)  ->  (EX -> B)

   E :  f<p, q>  ->   Ef<p, q, dp, dq>

   Ef<p, q, dp, dq>

   =  f<p + dp, q + dq)

   =  f<(p, dp), (q, dq)>

Therefore, if we evaluate Ef at particular values of dp and dq,
for example, dp = i and dq = j, where i, j are in B, we obtain:

   E_ij : (X -> B)  ->  (X -> B)

   E_ij :    f      ->   E_ij f

   E_ij f

   =  Ef | <dp = i, dq = j>

   =  f<p + i, q + j>

   =  f<(p, i), (q, j)>

The notation is a little bit awkward, but the data of the Table should
make the sense clear.  The important thing to observe is that E_ij has
the effect of transforming each proposition f : X -> B into some other
proposition f' : X -> B.  As it happens, the action is one-to-one and
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
is an example of what is called a "transformation group" on the set
of sixteen propositions.  Bowing to a longstanding linear and local
tradition, I will therefore redub the four elements of this group
as T_00, T_01, T_10, T_11, to bear in mind their transformative
character, or nature, as the case may be.  Abstractly viewed,
this group of order four has the following operation table:

o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|    *     %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o==========o==========o==========o==========o==========o
|          %          |          |          |          |
|   T_00   %   T_00   |   T_01   |   T_10   |   T_11   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_01   %   T_01   |   T_00   |   T_11   |   T_10   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_10   %   T_10   |   T_11   |   T_00   |   T_01   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o
|          %          |          |          |          |
|   T_11   %   T_11   |   T_10   |   T_01   |   T_00   |
|          %          |          |          |          |
o----------o----------o----------o----------o----------o

It happens that there are just two possible groups of 4 elements.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
The other is Klein's four-group V_4 (German "Vier"), which it is.

More concretely viewed, the group as a whole pushes the set
of sixteen propositions around in such a way that they fall
into seven natural classes, called "orbits".  One says that
the orbits are preserved by the action of the group.  There
is an "Orbit Lemma" of immense utility to "those who count"
which, depending on your upbringing, you may associate with
the names of Burnside, Cauchy, Frobenius, or some subset or
superset of these three, vouching that the number of orbits
is equal to the mean number of fixed points, in other words,
the total number of points (in our case, propositions) that
are left unmoved by the separate operations, divided by the
order of the group.  In this instance, T_00 operates as the
group identity, fixing all 16 propositions, while the other
three group elements fix 4 propositions each, and so we get:
Number of orbits  =  (4 + 4 + 4 + 16) / 4  =  7. -- Amazing!

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 11

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

We have been contemplating functions of the type f : X -> B,
studying the action of the operators E and D on this family.
These functions, that we may identify for our present aims
with propositions, inasmuch as they capture their abstract
forms, are logical analogues of "scalar potential fields".
These are the sorts of fields that are so picturesquely
presented in elementary calculus and physics textbooks
by images of snow-covered hills and parties of skiers
who trek down their slopes like least action heroes.
The analogous scene in propositional logic presents
us with forms more reminiscent of plateaunic idylls,
being all plains at one of two levels, the mesas of
verity and falsity, as it were, with nary a niche
to inhabit between them, restricting our options
for a sporting gradient of downhill dynamics to
just one of two, standing still on level ground
or falling off a bluff.

We are still working well within the logical analogue of the
classical finite difference calculus, taking in the novelties
that the logical transmutation of familiar elements is able to
bring to light.  Soon we will take up several different notions
of approximation relationships that may be seen to organize the
space of propositions, and these will allow us to define several
different forms of differential analysis applying to propositions.
In time we will find reason to consider more general types of maps,
having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n
and abstract types B^k -> B^n.   We will think of these mappings as
transforming universes of discourse into themselves or into others,
in short, as "transformations of discourse".

Before we continue with this intinerary, however, I would like
to highlight another sort of "differential aspect" that concerns
the "boundary operator" or the "marked connective" that serves as
one of a pair of basic connectives in the cactus language for ZOL.

Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
cactus syntax.  Taken as an assertion in what C.S. Peirce called the
"existential interpretation", the so-called boundary form "(p, q, r)"
asserts that one and only one of the propositions p, q, r is false.
It is instructive to consider this assertion in relation to the
conjunction "p q r" of the same propositions.  A venn diagram
for the boundary form (p, q, r) is shown in Figure 11.

o-----------------------------------------------------------o
|                                                           |
|                                                           |
|                      o-------------o                      |
|                     /               \                     |
|                    /                 \                    |
|                   /                   \                   |
|                  /                     \                  |
|                 /                       \                 |
|                o                         o                |
|                |                         |                |
|                |            P            |                |
|                |                         |                |
|                |                         |                |
|                |                         |                |
|             o--o----------o   o----------o--o             |
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
|           /      \%%%%%%%%%%o%%%%%%%%%%/      \           |
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
|         /          \%%%%%%/   \%%%%%%/          \         |
|        /            \%%%%/     \%%%%/            \        |
|       o              o--o-------o--o              o       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |                 |%%%%%%%|                 |       |
|       |        Q        |%%%%%%%|        R        |       |
|       |                 |%%%%%%%|                 |       |
|       o                 o%%%%%%%o                 o       |
|        \                 \%%%%%/                 /        |
|         \                 \%%%/                 /         |
|          \                 \%/                 /          |
|           \                 o                 /           |
|            \               / \               /            |
|             o-------------o   o-------------o             |
|                                                           |
|                                                           |
o-----------------------------------------------------------o
Figure 11.  Boundary Form (p, q, r)

In relation to the center cell indicated by the conjunction pqr
the region indicated by (p, q, r) is comprised of the "adjacent"
or the "bordering" cells.  Thus they are the cells that are just
across the boundary of the center cell, as if reached by way of
Leibniz's "minimal changes" from the point of origin, here, pqr.

More generally speaking, in a k-dimensional universe of discourse
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
the same form of boundary relationship is manifested for any cell
of origin that one might choose to indicate, say, by means of the
conjunction of positive and negative basis features "u_1 ... u_k",
where u_j = x_j or u_j = (x_j), for j = 1 to k.  The proposition
(u_1, ..., u_k) indicates the disjunctive region consisting of
the cells that are "just next door" to the cell u_1 ... u_k.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 12

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| Consider what effects that might conceivably have
| practical bearings you conceive the objects of your
| conception to have.  Then, your conception of those
| effects is the whole of your conception of the object.
|
| C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438

One other subject that it would be opportune to mention at this point,
while we have an object example of a mathematical group fresh in mind,
is the relationship between the pragmatic maxim and what are commonly
known in mathematics as "representation principles".  As it turns out,
with regard to its formal characteristics, the pragmatic maxim unites
the aspects of a representation principle with the attributes of what
would ordinarily be known as a "closure principle".  We will consider
the form of closure that is invoked by the pragmatic maxim on another
occasion, focusing here and now on the topic of group representations.

Let us return to the example of the so-called "four-group" V_4.
We encountered this group in one of its concrete representations,
namely, as a "transformation group" that acts on a set of objects,
in this particular case a set of sixteen functions or propositions.
Forgetting about the set of objects that the group transforms among
themselves, we may take the abstract view of the group's operational
structure, say, in the form of the group operation table copied here:

o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   *   %   e   |   f   |   g   |   h   |
|       %       |       |       |       |
o=======o=======o=======o=======o=======o
|       %       |       |       |       |
|   e   %   e   |   f   |   g   |   h   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   f   %   f   |   e   |   h   |   g   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   g   %   g   |   h   |   e   |   f   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   h   %   h   |   g   |   f   |   e   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o

This table is abstractly the same as, or isomorphic to, the versions
with the E_ij operators and the T_ij transformations that we took up
earlier.  That is to say, the story is the same, only the names have
been changed.  An abstract group can have a variety of significantly
and superficially different representations.  But even after we have
long forgotten the details of any particular representation there is
a type of concrete representations, called "regular representations",
that are always readily available, as they can be generated from the
mere data of the abstract operation table itself.

For example, select a group element from the top margin of the Table,
and "consider its effects" on each of the group elements as they are
listed along the left margin.  We may record these effects as Peirce
usually did, as a logical "aggregate" of elementary dyadic relatives,
that is to say, a disjunction or a logical sum whose terms represent
the ordered pairs of <input : output> transactions that are produced
by each group element in turn.  This yields what is usually known as
one of the "regular representations" of the group, specifically, the
"first", the "post-", or the "right" regular representation.  It has
long been conventional to organize the terms in the form of a matrix:

Reading "+" as a logical disjunction:

   G  =  e  +  f  +  g  + h,

And so, by expanding effects, we get:

   G =

   e:e  +  f:f  +  g:g  +  h:h  +

   e:f  +  f:e  +  g:h  +  h:g  +

   e:g  +  f:h  +  g:e  +  h:f  +

   e:h  +  f:g  +  g:f  +  h:e

More on the pragmatic maxim as a representation principle later.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 13

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The above-mentioned fact about the regular representations
of a group is universally known as "Cayley's Theorem".  It
is usually stated in the form:  "Every group is isomorphic
to a subgroup of Aut(X), where X is a suitably chosen set
and Aut(X) is the group of its automorphisms".  There is
in Peirce's early papers a considerable generalization
of the concept of regular representations to a broad
class of relational algebraic systems.  The crux of
the whole idea can be summed up as follows:

   Contemplate the effects of the symbol
   whose meaning you wish to investigate
   as they play out on all the stages of
   conduct on which you have the ability
   to imagine that symbol playing a role.

This idea of definition by way of context transforming operators
is basically the same as Jeremy Bentham's notion of "paraphrasis",
a "method of accounting for fictions by explaining various purported
terms away" (Quine, in Van Heijenoort, 'From Frege to Gödel', p. 216).
Today we'd call these constructions "term models".  This, again, is
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
of lambda calculus, and I reckon you all know where that leads.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 14

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The next few excursions in this series will provide
a scenic tour of various ideas in group theory that
will turn out to be of constant guidance in several
of the settings that are associated with our topic.

Let me return to Peirce's early papers on the algebra of relatives
to pick up the conventions that he used there, and then rewrite my
account of regular representations in a way that conforms to those.

Peirce expresses the action of an "elementary dual relative" like so:

| [Let] A:B be taken to denote
| the elementary relative which
| multiplied into B gives A.
|
| Peirce, 'Collected Papers', CP 3.123.

Peirce is well aware that it is not at all necessary to arrange the
elementary relatives of a relation into arrays, matrices, or tables,
but when he does so he tends to prefer organizing 2-adic relations
in the following manner:

   a:b  +  a:b  +  a:c  +

   b:a  +  b:b  +  b:c  +

   c:a  +  c:b  +  c:c

For example, given the set X = {a, b, c}, suppose that
we have the 2-adic relative term m = "marker for" and
the associated 2-adic relation M c X x X, the general
pattern of whose common structure is represented by
the following matrix:

   M  =

   M_aa a:a  +  M_ab a:b  +  M_ac a:c  +

   M_ba b:a  +  M_bb b:b  +  M_bc b:c  +

   M_ca c:a  +  M_cb c:b  +  M_cc c:c

It has long been customary to omit the implicit plus signs
in these matrical displays, but I have restored them here
simply as a way of separating terms in this blancophage
web format.

For at least a little while, I will make explicit
the distinction between a "relative term" like m
and a "relation" like M c X x X, but it is best
to think of both of these entities as involving
different applications of the same information,
and so we could just as easily write this form:

   m  =

   m_aa a:a  +  m_ab a:b  +  m_ac a:c  +

   m_ba b:a  +  m_bb b:b  +  m_bc b:c  +

   m_ca c:a  +  m_cb c:b  +  m_cc c:c

By way of making up a concrete example,
let us say that M is given as follows:

   a is a marker for a

   a is a marker for b

   b is a marker for b

   b is a marker for c

   c is a marker for c

   c is a marker for a

In sum, we have this matrix:

   M  =

   1 a:a  +  1 a:b  +  0 a:c  +

   0 b:a  +  1 b:b  +  1 b:c  +

   1 c:a  +  0 c:b  +  1 c:c

I think that will serve to fix notation
and set up the remainder of the account.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 15

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients".  When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:

   M  =

   1  1  0

   0  1  1

   1  0  1

However the specification may come to be written, this
is all just convenient schematics for stipulating that:

   M  =  a:a  +  b:b  +  c:c  +  a:b  +  b:c  +  c:a

Recognizing !1! = a:a + b:b + c:c as the identity transformation,
the 2-adic relative term m = "marker for" can be represented as an
element !1! + a:b + b:c + c:a of the so-called "group ring", all of
which makes this element just a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives
of the form i:j in the way that Peirce customarily read them in
logical contexts:  i is the relate, j is the correlate, and in
our current example we reading i:j, or more exactly, m_ij = 1,
to say that i is a marker for j.  This is the mode of reading
that we call "multiplying on the left".

In the algebraic, permutational, or transformational contexts of
application, however, Peirce converts to the alternative mode of
reading, although still calling i the relate and j the correlate,
the elementary relative i:j now means that i gets changed into j.
In this scheme of reading, the transformation a:b + b:c + c:a is
a permutation of the aggregate $1$ = a + b + c, or what we would
now call the set {a, b, c}, in particular, it is the permutation
that is otherwise notated as:

   ( a b c )
   <       >
   ( b c a )

This is consistent with the convention that Peirce uses in
the paper "On a Class of Multiple Algebras" (CP 3.324-327).

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 16

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

We've been exploring the applications of a certain technique
for clarifying abstruse concepts, a rough-cut version of the
pragmatic maxim that I've been accustomed to refer to as the
"operationalization" of ideas.  The basic idea is to replace
the question of "What it is", which modest people comprehend
is far beyond their powers to answer any time soon, with the
question of "What it does", which most people know at least
a modicum about.

In the case of regular representations of groups we found
a non-plussing surplus of answers to sort our way through.
So let us track back one more time to see if we can learn
any lessons that might carry over to more realistic cases.

Here is is the operation table of V_4 once again:

o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   *   %   e   |   f   |   g   |   h   |
|       %       |       |       |       |
o=======o=======o=======o=======o=======o
|       %       |       |       |       |
|   e   %   e   |   f   |   g   |   h   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   f   %   f   |   e   |   h   |   g   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   g   %   g   |   h   |   e   |   f   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o
|       %       |       |       |       |
|   h   %   h   |   g   |   f   |   e   |
|       %       |       |       |       |
o-------o-------o-------o-------o-------o

A group operation table is really just a device for recording
a certain 3-adic relation, specifically, the set of 3-tuples
of the form <x, y, z> that satisfy the equation x * y = z,
where the sign '*' that indicates the group operation is
frequently omitted in contexts where it is understood.

In the case of V_4 = (G, *), where G is the "underlying set"
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
whose triples are listed below:

   e:e:e
   e:f:f
   e:g:g
   e:h:h

   f:e:f
   f:f:e
   f:g:h
   f:h:g

   g:e:g
   g:f:h
   g:g:e
   g:h:f

   h:e:h
   h:f:g
   h:g:f
   h:h:e

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.

Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
    the empty slot of the rheme <_, y>, with
    the effect of producing the saturated
    rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
    the empty slot of the rheme <y, _>, with
    the effect of producing the saturated
    rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {<y : xy> : y in G}.
The pairs <y : xy> can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run across the top margin.  This aspect of
pragmatic definition we recognize as the regular
ante-representation:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  e:f  +  f:e  +  g:h  +  h:g

   g  =  e:g  +  f:h  +  g:e  +  h:f

   h  =  e:h  +  f:g  +  g:f  +  h:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {<y : yx> : y in G}.
The pairs <y : yx> can be found by picking an x
from the top margin of the group operation table
and considering its effects on each y in turn as
these run down the left margin.  This aspect of
pragmatic definition we recognize as the regular
post-representation:

   e  =  e:e  +  f:f  +  g:g  +  h:h

   f  =  e:f  +  f:e  +  g:h  +  h:g

   g  =  e:g  +  f:h  +  g:e  +  h:f

   h  =  e:h  +  f:g  +  g:f  +  h:e

If the ante-rep looks the same as the post-rep,
now that I'm writing them in the same dialect,
that is because V_4 is abelian (commutative),
and so the two representations have the very
same effects on each point of their bearing.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 17

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

So long as we're in the neighborhood, we might as well take in
some more of the sights, for instance, the smallest example of
a non-abelian (non-commutative) group.  This is a group of six
elements, say, G = {e, f, g, h, i, j}, with no relation to any
other employment of these six symbols being implied, of course,
and it can be most easily represented as the permutation group
on a set of three letters, say, X = {a, b, c}, usually notated
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
Here are the permutation (= substitution) operations in Sym(X):

Table 17-a.  Permutations or Substitutions in Sym_{a, b, c}
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |
|         |         |         |         |         |         |
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
|         |         |         |         |         |         |
|  a b c  |  c a b  |  b c a  |  a c b  |  c b a  |  b a c  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

Here is the operation table for S_3, given in abstract fashion:

Table 17-b.  Symmetric Group S_3
o-------------------------------------------------o
|                                                 |
|                        o                        |
|                     e / \ e                     |
|                      /   \                      |
|                     /  e  \                     |
|                  f / \   / \ f                  |
|                   /   \ /   \                   |
|                  /  f  \  f  \                  |
|               g / \   / \   / \ g               |
|                /   \ /   \ /   \                |
|               /  g  \  g  \  g  \               |
|            h / \   / \   / \   / \ h            |
|             /   \ /   \ /   \ /   \             |
|            /  h  \  e  \  e  \  h  \            |
|         i / \   / \   / \   / \   / \ i         |
|          /   \ /   \ /   \ /   \ /   \          |
|         /  i  \  i  \  f  \  j  \  i  \         |
|      j / \   / \   / \   / \   / \   / \ j      |
|       /   \ /   \ /   \ /   \ /   \ /   \       |
|      o  j  \  j  \  j  \  i  \  h  \  j  o      |
|       \   / \   / \   / \   / \   / \   /       |
|        \ /   \ /   \ /   \ /   \ /   \ /        |
|         \  h  \  h  \  e  \  j  \  i  /         |
|          \   / \   / \   / \   / \   /          |
|           \ /   \ /   \ /   \ /   \ /           |
|            \  i  \  g  \  f  \  h  /            |
|             \   / \   / \   / \   /             |
|              \ /   \ /   \ /   \ /              |
|               \  f  \  e  \  g  /               |
|                \   / \   / \   /                |
|                 \ /   \ /   \ /                 |
|                  \  g  \  f  /                  |
|                   \   / \   /                   |
|                    \ /   \ /                    |
|                     \  e  /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o

I think that the NKS reader can guess how we might apply
this group to the space of propositions of type B^3 -> B.

By the way, we will meet with the symmetric group S_3 again
when we return to take up the study of Peirce's early paper
"On a Class of Multiple Algebras" (CP 3.324-327), and also
his late unpublished work "The Simplest Mathematics" (1902)
(CP 4.227-323), with particular reference to the section
that treats of "Trichotomic Mathematics" (CP 4.307-323).

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 18

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

By way of collecting a short-term pay-off for all the work that we
did on the regular representations of the Klein 4-group V_4, let us
write out as quickly as possible in "relative form" a minimal budget
of representations for the symmetric group on three letters, Sym(3).
After doing the usual bit of compare and contrast among the various
representations, we will have enough concrete material beneath our
abstract belts to tackle a few of the presently obscured details
of Peirce's early "Algebra + Logic" papers.

Writing the permutations or substitutions of Sym {a, b, c}
in relative form generates what is generally thought of as
a "natural representation" of S_3.

   e  =  a:a + b:b + c:c

   f  =  a:c + b:a + c:b

   g  =  a:b + b:c + c:a

   h  =  a:a + b:c + c:b

   i  =  a:c + b:b + c:a

   j  =  a:b + b:a + c:c

I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as x:y
constitutes the turning of x into y.  It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 19

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

To construct the regular representations of S_3,
we pick up from the data of its operation table,
DAL 17, Table 17-b, at either one of these sites:

http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321

Just by way of staying clear about what we are doing,
let's return to the recipe that we worked out before:

It is part of the definition of a group that the 3-adic
relation L c G^3 is actually a function L : G x G -> G.
It is from this functional perspective that we can see
an easy way to derive the two regular representations.

Since we have a function of the type L : G x G -> G,
we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into
    the empty slot of the rheme <_, y>, with
    the effect of producing the saturated
    rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
    the empty slot of the rheme <y, _>, with
    the effect of producing the saturated
    rheme <y, x> that evaluates to yx.

In (1), we consider the effects of each x in its
practical bearing on contexts of the form <_, y>,
as y ranges over G, and the effects are such that
x takes <_, y> into xy, for y in G, all of which
is summarily notated as x = {<y : xy> : y in G}.
The pairs <y : xy> can be found by picking an x
from the left margin of the group operation table
and considering its effects on each y in turn as
these run along the right margin.  This produces
the regular ante-representation of S_3, like so:

   e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

   f   =   e:f  +  f:g  +  g:e  +  h:j  +  i:h  +  j:i

   g   =   e:g  +  f:e  +  g:f  +  h:i  +  i:j  +  j:h

   h   =   e:h  +  f:i  +  g:j  +  h:e  +  i:f  +  j:g

   i   =   e:i  +  f:j  +  g:h  +  h:g  +  i:e  +  j:f

   j   =   e:j  +  f:h  +  g:i  +  h:f  +  i:g  +  j:e

In (2), we consider the effects of each x in its
practical bearing on contexts of the form <y, _>,
as y ranges over G, and the effects are such that
x takes <y, _> into yx, for y in G, all of which
is summarily notated as x = {<y : yx> : y in G}.
The pairs <y : yx> can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin.  This generates
the regular post-representation of S_3, like so:

   e   =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j

   f   =   e:f  +  f:g  +  g:e  +  h:i  +  i:j  +  j:h

   g   =   e:g  +  f:e  +  g:f  +  h:j  +  i:h  +  j:i

   h   =   e:h  +  f:j  +  g:i  +  h:e  +  i:g  +  j:f

   i   =   e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g

   j   =   e:j  +  f:i  +  g:h  +  h:g  +  i:f  +  j:e

If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 20

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

You may be wondering what happened to the announced subject
of "Dynamics And Logic".  What occurred was a bit like this:

We happened to make the observation that the shift operators {E_ij}
form a transformation group that acts on the set of propositions of
the form f : B x B -> B.  Group theory is a very attractive subject,
but it did not draw us so far from our intended course as one might
initially think.  For one thing, groups, especially the groups that
are named after the Norwegian mathematician Marius Sophus Lie, turn
out to be of critical importance in solving differential equations.
For another thing, group operations provide us with an ample supply
of triadic relations that have been extremely well-studied over the
years, and thus they give us no small measure of useful guidance in
the study of sign relations, another brand of 3-adic relations that
have significance for logical studies, and in our acquaintance with
which we have scarcely begun to break the ice.  Finally, I couldn't
resist taking up the links between group representations, amounting
to the very archetypes of logical models, and the pragmatic maxim.

Biographical Data for Marius Sophus Lie (1842-1899):
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 21

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

We have seen a couple of groups, V_4 and S_3, represented in
several different ways, and we have seen each of these types
of representation presented in several different fashions.
Let us look at one other stylistic variant for presenting
a group representation that is often used, the so-called
"matrix representation" of a group.

Returning to the example of Sym(3), we first encountered
this group in concrete form as a set of permutations or
substitutions acting on a set of letters X = {a, b, c}.
This set of permutations was displayed in Table 17-a,
copies of which can be found here:

http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321

These permutations were then converted to "relative form":

   e  =  a:a + b:b + c:c

   f  =  a:c + b:a + c:b

   g  =  a:b + b:c + c:a

   h  =  a:a + b:c + c:b

   i  =  a:c + b:b + c:a

   j  =  a:b + b:a + c:c

From this relational representation of Sym {a, b, c} ~=~ S_3,
one easily derives a "linear representation", regarding each
permutation as a linear transformation that maps the elements
of a suitable vector space into each other, and representing
each of these linear transformations by means of a matrix,
resulting in the following set of matrices for the group:

Table 21.  Matrix Representations of the Permutations in S_3
o---------o---------o---------o---------o---------o---------o
|         |         |         |         |         |         |
|    e    |    f    |    g    |    h    |    i    |    j    |
|         |         |         |         |         |         |
o=========o=========o=========o=========o=========o=========o
|         |         |         |         |         |         |
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
|         |         |         |         |         |         |
o---------o---------o---------o---------o---------o---------o

The key to the mysteries of these matrices is revealed by
observing that their coefficient entries are arrayed and
overlayed on a place mat that's marked like so:

o-----o-----o-----o
| a:a | a:b | a:c |
o-----o-----o-----o
| b:a | b:b | b:c |
o-----o-----o-----o
| c:a | c:b | c:c |
o-----o-----o-----o

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 22

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

It would be good to summarize, in rough but intuitive terms,
the outlook on differential logic that we have reached so far.

We've been considering a class of operators on universes
of discourse, each of which takes us from considering one
universe of discourse, X%, to considering a larger universe
of discourse, EX%.

Each of these operators, in broad terms having the form
W : X% -> EX%, acts on each proposition f : X -> B of the
source universe X% to produce a proposition Wf : EX -> B
of the target universe EX%.

The two main operators that we have worked with up to this
point are the enlargement or shift operator E : X% -> EX%
and the difference operator D : X% -> EX%.

E and D take a proposition in X%, that is, a proposition f : X -> B
that is said to be "about" the subject matter of X, and produce the
extended propositions Ef, Df : EX -> B, which may be interpreted as
being about specified collections of changes that might occur in X.

Here we have need of visual representations,
some array of concrete pictures to anchor our
more earthy intuitions and to help us keep our
wits about us before we try to climb any higher
into the ever more rarefied air of abstractions.

One good picture comes to us by way of the "field" concept.
Given a space X, a "field" of a specified type Y over X is
formed by assigning to each point of X an object of type Y.
If that sounds like the same thing as a function from X to
the space of things of type Y -- it is -- but it does seem
helpful to vary the mental images and to take advantage of
the figures of speech that spring to mind under the emblem
of this field idea.

In the field picture, a proposition f : X -> B becomes
a "scalar" field, that is, a field of values in B, or
a "field of model indications" (FOMI).

Let us take a moment to view an old proposition
in this new light, for example, the conjunction
pq : X -> B that is depicted in Figure 22-a.

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /                 o                 \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |                 |%%%%%|                 |   |
|   |        P        |%%%%%|        Q        |   |
|   |                 |%%%%%|                 |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
Figure 22-a.  Conjunction pq : X -> B

Each of the operators E, D : X% -> EX% takes us from considering
propositions f : X -> B, here viewed as "scalar fields" over X,
to considering the corresponding "differential fields" over X,
analogous to what are usually called "vector fields" over X.

The structure of these differential fields can be described this way.
To each point of X there is attached an object of the following type:
a proposition about changes in X, that is, a proposition g : dX -> B.
In this frame, if X% is the universe that is generated by the set of
coordinate propositions {p, q}, then dX% is the differential universe
that is generated by the set of differential propositions {dp, dq}.
These differential propositions may be interpreted as indicating
"change in p" and "change in q", respectively.

A differential operator W, of the first order sort that we have
been considering, takes a proposition f : X -> B and gives back
a differential proposition Wf: EX -> B.

In the field view, we see the proposition f : X -> B as a scalar field
and we see the differential proposition Wf: EX -> B as a vector field,
specifically, a field of propositions about contemplated changes in X.

The field of changes produced by E on pq is shown in Figure 22-b.

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o.->-.o                 o   |
|   |    p(q)(dp)dq   |%\%/%|  (p)q dp(dq)    |   |
|   | o---------------|->o<-|---------------o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                  (p)(q) dp dq                   |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Ef =              p  q   (dp)(dq)               |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-b.  Enlargement E[pq] : EX -> B

The differential field E[pq] specifies the changes
that need to be made from each point of X in order
to reach one of the models of the proposition pq,
that is, in order to satisfy the proposition pq.

The field of changes produced by D on pq is shown in Figure 22-c.

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |       (dp)dq    |%%%%%|    dp(dq)       |   |
|   | o<--------------|->o<-|-------------->o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                      dp dq                      |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Df =              p  q  ((dp)(dq))              |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-c.  Difference D[pq] : EX -> B

The differential field D[pq] specifies the changes
that need to be made from each point of X in order
to feel a change in the felt value of the field pq.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 23

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

I want to continue developing the basic tools of differential logic,
which arose out of many years of thinking about the connections
between dynamics and logic -- those there are and those there
ought to be -- but I also wanted to give some hint of the
applications that have motivated this work all along.
One of these applications is to cybernetic systems,
whether we see these systems as agents or cultures,
individuals or species, organisms or organizations.

A cybernetic system has goals and actions for reaching them.
It has a state space X, giving us all of the states that the
system can be in, plus it has a goal space G c X, the set of
states that the system "likes" to be in, in other words, the
distinguished subset of possible states where the system is
regarded as living, surviving, or thriving, depending on the
type of goal that one has in mind for the system in question.
As for actions, there is to begin with the full set !T! of all
possible actions, each of which is a transformation of the form
T : X -> X, but a given cybernetic system will most likely have
but a subset of these actions available to it at any given time.
And even if we begin by thinking of actions in very general and
very global terms, as arbitrarily complex transformations acting
on the whole state space X, we quickly find a need to analyze and
approximate them in terms of simple transformations acting locally.
The preferred measure of "simplicity" will of course vary from one
paradigm of research to another.

A generic enough picture at this stage of the game, and one that will
remind us of these fundamental features of the cybernetic system even
as things get far more complex, is afforded by Figure 23.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o                                    |
|           /                     \                                   |
|          /                       \                                  |
|         /                         \                                 |
|        /                           \                                |
|       /                             \                               |
|      /                               \                              |
|     /                                 \                             |
|    o                                   o                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                 G                 |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    o                                   o                            |
|     \                                 /                             |
|      \                               /                              |
|       \                           T /                               |
|        \             o<------------/-------------o                  |
|         \                         /                                 |
|          \                       /                                  |
|           \                     /                                   |
|            o-------------------o                                    |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 23.  Elements of a Cybernetic System

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 24

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Now that we've introduced the field picture for thinking about
propositions and their analytic series, a very pleasing way of
picturing the relationship among a proposition f : X -> B, its
enlargement or shift map Ef : EX -> B, and its difference map
Df : EX -> B can now be drawn.

To illustrate this possibility, let's return to the differential
analysis of the conjunctive proposition f<p, q> = pq, giving the
development a slightly different twist at the appropriate point.

Figure 24-1 shows the proposition pq once again, which we now view
as a scalar field, in effect, a potential "plateau" of elevation 1
over the shaded region, with an elevation of 0 everywhere else.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       /%\                       \         |
|        /                       /%%%\                       \        |
|       /                       /%%%%%\                       \       |
|      /                       /%%%%%%%\                       \      |
|     /                       /%%%%%%%%%\                       \     |
|    o                       o%%%%%%%%%%%o                       o    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |          P            |%%%%%%%%%%%|            Q          |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    o                       o%%%%%%%%%%%o                       o    |
|     \                       \%%%%%%%%%/                       /     |
|      \                       \%%%%%%%/                       /      |
|       \                       \%%%%%/                       /       |
|        \                       \%%%/                       /        |
|         \                       \%/                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-1.  Proposition pq : X -> B

Given any proposition f : X -> B, the "tacit extension" of f to EX
is notated !e!f : EX -> B and defined by the equation !e!f = f, so
it's really just the same proposition living in a bigger universe.

Tacit extensions formalize the intuitive idea that a new function
is related to an old function in such a way that it obeys the same
constraints on the old variables, with a "don't care" condition on
the new variables.

Figure 24-2 illustrates the "tacit extension" of the proposition
or scalar field f = pq : X -> B to give the extended proposition
or differential field that we notate as !e!f = !e![pq] : EX -> B.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o<-----------------o----------------->o          |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-2.  Tacit Extension !e![pq] : EX -> B

Thus we have a pictorial way of visualizing the following data:

   !e![pq]

    =

    p q . dp dq

    +

    p q . dp (dq)

    +

    p q . (dp) dq

    +

    p q . (dp)(dq)

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 25

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Staying with the example pq : X -> B, Figure 25-1 shows
the enlargement or shift map E[pq] : EX -> B in the same
style of differential field picture that we drew for the
tacit extension !e![pq] : EX -> B.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o----------------->o<-----------------o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-1.  Enlargement E[pq] : EX -> B

A very important conceptual transition has just occurred here,
almost tacitly, as it were.  Generally speaking, having a set
of mathematical objects of compatible types, in this case the
two differential fields !e!f and Ef, both of the type EX -> B,
is very useful, because it allows us to consider these fields
as integral mathematical objects that can be operated on and
combined in the ways that we usually associate with algebras.

In this case one notices that the tacit extension !e!f and the
enlargement Ef are in a certain sense dual to each other, with
!e!f indicating all of the arrows out of the region where f is
true, and with Ef indicating all of the arrows into the region
where f is true.  The only arc that they have in common is the
no-change loop (dp)(dq) at pq.  If we add the two sets of arcs
mod 2, then the common loop drops out, leaving the 6 arrows of
D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |             (dp) dq   |           |   dp (dq)             |    |
|    |          o<---------------->o<---------------->o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-2.  Difference Map D[pq] : EX -> B

The differential features of D[pq] may be collected cell by cell of
the underlying universe X% = [p, q] to give the following expansion:

   D[pq]

   =

   p q . ((dp)(dq))

   +

   p (q) . (dp) dq

   +

   (p) q . dp (dq)

   +

   (p)(q) . dp dq

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 26

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

If we follow the classical line that singles out linear functions
as ideals of simplicity, then we may complete the analytic series
of the proposition f = pq : X -> B in the following way.

Figure 26-1 shows the differential proposition df = d[pq] : EX -> B
that we get by extracting the cell-wise linear approximation to the
difference map Df = D[pq] : EX -> B.  This is the logical analogue
of what would ordinarily be called 'the' differential of pq, but
since I've been attaching the adjective "differential" to just
about everything in sight, the distinction tends to be lost.
For the time being, I'll resort to using the alternative
name "tangent map" for df.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   o   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |                   (dp)/ dq     dp \(dq)                   |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 v     \  dp dq  /     v                 /     |
|      \               o<--------------------->o               /      |
|       \                       \     /                       /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-1.  Differential or Tangent d[pq] : EX -> B

Just to be clear about what's being indicated here,
it's a visual way of specifying the following data:

   d[pq]

   =

   p q . (dp, dq)

   +

   p (q) . dq

   +

   (p) q . dp

   +

   (p)(q) . 0

To understand the extended interpretations, that is,
the conjunctions of basic and differential features
that are being indicated here, it may help to note
the following equivalences:

   (dp, dq)   =   dp + dq   =   dp(dq) + (dp)dq

      dp      =   dp dq  +  dp(dq)

      dq      =   dp dq  +  (dp)dq

Capping the series that analyzes the proposition pq
in terms of succeeding orders of linear propositions,
Figure 26-2 shows the remainder map r[pq] : EX -> B,
that happens to be linear in pairs of variables.

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   dp dq   |                       |    |
|    |            o<------------------------------->o            |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |     o     |                       |    |
|    o                       o     ^     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    dp | dq                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-2.  Remainder r[pq] : EX -> B

Reading the arrows off the map produces the following data:

    r[pq]

    =

    p q . dp dq

    +

    p (q) . dp dq

    +

    (p) q . dp dq

    +

    (p)(q) . dp dq

In short, r[pq] is a constant field,
having the value dp dq at each cell.

A more detailed presentation of Differential Logic can be found here:

DLOG D.  http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#571

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Dynamics And Logic

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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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