Difference between revisions of "Directory:Jon Awbrey/Papers/Differential Logic : Introduction"

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A simple example of a differential logical calculus is furnished by a ''[[differential propositional calculus]]''.  A differential propositional calculus is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.  This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
 
A simple example of a differential logical calculus is furnished by a ''[[differential propositional calculus]]''.  A differential propositional calculus is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.  This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
 +
 +
==Note 1==
 +
 +
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 <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      p q                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                    p and q                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
Written as a string, this is just the concatenation "<math>p~q</math>".
 +
 +
The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>
 +
 +
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                                |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                      true                      |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                        |
 +
|                        |                        |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                      false                      |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
Back to the proposition <math>pq.\!</math>  Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown here:
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Venn Diagram P And Q.jpg|500px]]
 +
|}
 +
 +
Now ask yourself:  What is the value of the proposition <math>pq\!</math> at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> from the cell <math>pq\!</math> where you are standing?
 +
 +
Don't think about it &mdash; just compute:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  dp o  o dq                  |
 +
|                    / \ / \                    |
 +
|                  p o---@---o q                  |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
However you draw it, these expressions follow because the expression <math>p + \operatorname{d}p,</math> where the plus sign indicates addition in <math>\mathbb{B},</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    p    dp                    |
 +
|                      o---o                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                    (p, dp)                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
Next question:  What is the difference between the value of the
 +
proposition <math>pq\!</math> "over there" and the value of the proposition <math>pq\!</math> where you are, all expressed in the form of
 +
a general formula, of course?  Here is the appropriate formulation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
There is one thing that I ought to mention at this point:  Computed over <math>\mathbb{B},</math> 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 <math>pq\!</math> is true?  Well, substituting <math>1\!</math> for <math>p\!</math> and <math>1\!</math> for <math>q\!</math> in the graph amounts to erasing the labels <math>p\!</math> and <math>q\!,</math> as shown here:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/                            |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|              (( , dp)( , dq),    )              |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
And this is equivalent to the following graph:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                    dp  dq                    |
 +
|                      o  o                      |
 +
|                      \ /                      |
 +
|                        o                        |
 +
|                        |                        |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|                  ((dp) (dq))                  |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
==Note 2==
 +
 +
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                                    dp  dq      |
 +
|                                    o  o      |
 +
|                                      \ /        |
 +
|                                      o        |
 +
|        p q                            |        |
 +
|        @          --Diff-->          @        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|        p q        --Diff-->    ((dp) (dq))    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.
 +
 +
If the form of the above statement reminds you of De&nbsp;Morgan'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&ndash;De&nbsp;Morgan 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 adequate to handle the complexity of 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 <math>f(p, q) = pq.\!</math>
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Venn Diagram F = P And Q.jpg|500px]]
 +
|-
 +
| [[Image:Cactus Graph F = P And Q.jpg|500px]]
 +
|}
 +
 +
A function like this has an abstract type and a concrete type.  The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math>  The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
 +
|-
 +
| Let <math>Q\!</math> be the set of values <math>\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \operatorname{not}~ q,~ q \} ~\cong~ \mathbb{B}.</math>
 +
|}
 +
 +
Then interpret the usual propositions about <math>p, q\!</math> as functions of the concrete type <math>f : P \times Q \to \mathbb{B}.</math>
 +
 +
We are going to consider various ''operators'' on these functions.  Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
 +
 +
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:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
 +
|-
 +
| The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math>
 +
|}
 +
 +
These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.
 +
 +
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse.  Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{rcc}
 +
\operatorname{E}X & = & X \times \operatorname{d}X
 +
\end{array}</math>
 +
|}
 +
 +
where:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{rcc}
 +
X
 +
& = &
 +
P \times Q
 +
\\[4pt]
 +
\operatorname{d}X
 +
& = &
 +
\operatorname{d}P \times \operatorname{d}Q
 +
\\[4pt]
 +
\operatorname{d}P
 +
& = &
 +
\{ \texttt{(} \operatorname{d}p \texttt{)},~ \operatorname{d}p \}
 +
\\[4pt]
 +
\operatorname{d}Q
 +
& = &
 +
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
 +
\end{array}</math>
 +
|}
 +
 +
The interpretations of these new symbols can be diverse, but the easiest
 +
option for now is just to say that <math>\operatorname{d}p</math> means "change <math>p\!</math>" and <math>\operatorname{d}q</math> means "change <math>q\!</math>".
 +
 +
Drawing a venn diagram for the differential extension <math>\operatorname{E}X = X \times \operatorname{d}X</math> requires four logical dimensions, <math>P, Q, \operatorname{d}P, \operatorname{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\operatorname{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\operatorname{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Venn Diagram P Q dP dQ.jpg|500px]]
 +
|}
 +
 +
Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone.  For example, the proposition <math>\texttt{(} \operatorname{d}p \texttt{(} \operatorname{d}q \texttt{))}</math> says the same thing as  <math>\operatorname{d}p \Rightarrow \operatorname{d}q,</math> in other words, that there is no change in <math>p\!</math> without a change in <math>q.\!</math>
 +
 +
==Note 3==
 +
 +
Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> over the differential extension <math>\operatorname{E}X</math> that is defined by the
 +
following formula:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
f(p + \operatorname{d}p,~ q + \operatorname{d}q)
 +
& = &
 +
f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} )
 +
\end{matrix}</math>
 +
|}
 +
 +
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
(p + \operatorname{d}p)(q + \operatorname{d}q)
 +
& = &
 +
\texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}
 +
\end{matrix}</math>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef =            (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> over <math>\operatorname{E}X</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> or, written out in full:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
f(p + \operatorname{d}p,~ q + \operatorname{d}q) - f(p, q)
 +
& = &
 +
\texttt{(} f( \texttt{(} p, \operatorname{d}p \texttt{)},~ \texttt{(} q, \operatorname{d}q \texttt{)} ),~ f(p, q) \texttt{)}
 +
\end{matrix}</math>
 +
|}
 +
 +
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
(p + \operatorname{d}p)(q + \operatorname{d}q) - pq
 +
& = &
 +
\texttt{((} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}, pq \texttt{)}
 +
\end{matrix}</math>
 +
|-
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
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 <math>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math>  This evaluation is written in the form <math>\operatorname{D}f|_{pq}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>f(p, q) ~=~ pq ~=~ p ~\operatorname{and}~ q \quad \Rightarrow \quad \operatorname{D}f|_{pq} ~=~ \texttt{((} \operatorname{dp} \texttt{)(} \operatorname{d}q \texttt{))} ~=~ \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
 +
|-
 +
| align="center" |
 +
[[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
 +
|-
 +
| align="center" |
 +
[[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
 +
|}
 +
 +
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{))}</math> into the following exclusive disjunction:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
 +
& + &
 +
\texttt{(} \operatorname{d}p \texttt{)}~ \operatorname{d}q
 +
& + &
 +
\operatorname{d}p ~\operatorname{d}q
 +
\end{matrix}</math>
 +
|}
 +
 +
The differential proposition that results may be interpreted to say "change <math>p\!</math> or change <math>q\!</math> 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.
 +
 +
==Note 4==
 +
 +
Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>
 +
 +
In the universe <math>X = P \times Q,</math> the four propositions <math>pq,~ p \texttt{(} q \texttt{)},~ \texttt{(} p \texttt{)} q,~ \texttt{(} p \texttt{)(} q \texttt{)}</math> 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 <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
 +
 +
Thus we can write <math>\operatorname{D}f_x = \operatorname{D}f|x = \operatorname{D}f|(1, 1) = \operatorname{D}f|pq,</math> so long as we know the frame of reference in force.
 +
 +
In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\operatorname{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /                          |
 +
|                \| |/                            |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|pq =          ((dp) (dq))                  |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o                            |
 +
|                dp |  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /        o                |
 +
|                \| |/          |                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|p(q) =          (dp) dq                      |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            o                                  |
 +
|            |  dp    dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /        o                |
 +
|                \| |/          |                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|(p)q =            dp (dq)                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|            o    o                            |
 +
|            |  dp |  dq                        |
 +
|            o---o o---o                        |
 +
|              \  | |  /                          |
 +
|              \ | | /      o  o              |
 +
|                \| |/        \ /                |
 +
|                o=o-----------o                |
 +
|                  \          /                  |
 +
|                  \        /                  |
 +
|                    \      /                    |
 +
|                    \    /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        @                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Df|(p)(q) =          dp dq                      |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|  e                                              |
 +
|  o-o-o-...-o-o-o                                |
 +
|  \          /                                |
 +
|    \        /                                  |
 +
|    \      /                                  |
 +
|      \    /                          e        |
 +
|      \  /                          o        |
 +
|        \ /                            |        |
 +
|        @              =              @        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
|  (e, , ... , , )      =            (e)        |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
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
 +
</pre>
 +
|}
 +
 +
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Venn Diagram PQ Difference Conj.jpg|500px]]
 +
|}
 +
 +
The Figure shows the points of the extended universe <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q</math> that are indicated by the difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::
 +
 +
{| align="center" cellpadding="6"
 +
|
 +
<math>\begin{array}{rcccc}
 +
1. &  p  &  q  &  \operatorname{d}p  &  \operatorname{d}q
 +
\\
 +
2. &  p  &  q  &  \operatorname{d}p  & (\operatorname{d}q)
 +
\\
 +
3. &  p  &  q  & (\operatorname{d}p) &  \operatorname{d}q
 +
\\
 +
4. &  p  & (q) & (\operatorname{d}p) &  \operatorname{d}q
 +
\\
 +
5. & (p) &  q  &  \operatorname{d}p  & (\operatorname{d}q)
 +
\\
 +
6. & (p) & (q) &  \operatorname{d}p  &  \operatorname{d}q
 +
\end{array}</math>
 +
|}
 +
 +
The information borne by <math>\operatorname{D}f</math> should be clear enough from a survey of these six points &mdash; they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.
 +
 +
==Note 5==
 +
 +
We have been studying the action of the difference operator <math>\operatorname{D}</math> on propositions of the form <math>f : P \times Q \to \mathbb{B},</math> as illustrated by the example <math>f(p, q) = pq\!</math> that is known in logic as the conjunction of <math>p\!</math> and <math>q.\!</math>  The resulting difference map <math>\operatorname{D}f</math> is a (first order) differential proposition, that is, a proposition of the form <math>\operatorname{D}f : P \times Q \times \operatorname{d}P \times \operatorname{d}Q \to \mathbb{B}.</math>
 +
 +
Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute over the extended universe of discourse <math>\operatorname{E}X = P \times Q \times \operatorname{d}P \times \operatorname{d}Q,</math> the difference map <math>\operatorname{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X =  P \times Q</math> and whose arrows are labeled with the elements of <math>\operatorname{d}X = \operatorname{d}P \times \operatorname{d}Q,</math> as shown in the following Figure.
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
 +
|}
 +
 +
{| align="center" cellpadding="10"
 +
|
 +
<math>\begin{array}{rcccccc}
 +
f
 +
& = & p  & \cdot & q
 +
\\[4pt]
 +
\operatorname{D}f
 +
& = &  p  & \cdot &  q  & \cdot & ((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
& + &  p  & \cdot & (q) & \cdot & ~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
& + & (p) & \cdot &  q  & \cdot & ~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
& + & (p) & \cdot & (q) & \cdot & ~~\operatorname{d}p~~\operatorname{d}q~~
 +
\end{array}</math>
 +
|}
 +
 +
Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning.  We will encounter more and more of these alternative readings as we go.
 +
 +
==Note 6==
 +
 +
The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(p, q) = pq.\!</math>
 +
 +
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lccl}
 +
\text{Let} &
 +
X
 +
& = &
 +
X_1 \times \ldots \times X_k.
 +
\\[6pt]
 +
\text{Let} &
 +
\operatorname{d}X
 +
& = &
 +
\operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.
 +
\\[6pt]
 +
\text{Then} &
 +
\operatorname{E}X
 +
& = &
 +
X \times \operatorname{d}X
 +
\\[6pt]
 +
&
 +
& = & X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k
 +
\end{array}</math>
 +
|}
 +
 +
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B}</math> that is defined by the following equation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{l}
 +
\operatorname{E}f(x_1, \ldots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k)
 +
\\[6pt]
 +
= \quad f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k)
 +
\\[6pt]
 +
= \quad f( \texttt{(} x_1, \operatorname{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \operatorname{d}x_k \texttt{)} )
 +
\end{array}</math>
 +
|}
 +
 +
The ''differential variables'' <math>\operatorname{d}x_j</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math>  Although it is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\operatorname{d}</math>" this way of notating differential variables is entirely optional.  It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
 +
 +
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{l}
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
\\[6pt]
 +
= \quad (p + \operatorname{d}p)(q + \operatorname{d}q)
 +
\\[6pt]
 +
= \quad \texttt{(} p, \operatorname{d}p \texttt{)(} q, \operatorname{d}q \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
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 following result:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
p~q
 +
& + &
 +
p~\operatorname{d}q
 +
& + &
 +
q~\operatorname{d}p
 +
& + &
 +
\operatorname{d}p~\operatorname{d}q
 +
\end{matrix}</math>
 +
|}
 +
 +
To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math>  Toward that end, the value of <math>\operatorname{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  p  dp q  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef =            (p, dp) (q, dq)                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                      dp    dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|pq =            (dp) (dq)                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                        o                      |
 +
|                      dp |  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|p(q) =          (dp)  dq                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o                            |
 +
|                  |  dp    dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|(p)q =          dp  (dq)                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|-
 +
| align="center" |
 +
<pre>
 +
o-------------------------------------------------o
 +
|                                                |
 +
|                  o    o                      |
 +
|                  |  dp |  dq                  |
 +
|                  o---o o---o                  |
 +
|                    \  | |  /                    |
 +
|                    \ | | /                    |
 +
|                      \| |/                      |
 +
|                      @=@                      |
 +
|                                                |
 +
o-------------------------------------------------o
 +
| Ef|(p)(q) =        dp  dq                    |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\operatorname{E}f.</math>
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{E}f
 +
& = &
 +
pq \cdot \operatorname{E}f_{pq}
 +
& + &
 +
p(q) \cdot \operatorname{E}f_{p(q)}
 +
& + &
 +
(p)q \cdot \operatorname{E}f_{(p)q}
 +
& + &
 +
(p)(q) \cdot \operatorname{E}f_{(p)(q)}
 +
\end{matrix}</math>
 +
|}
 +
 +
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}p)(\operatorname{d}q)</math> is drawn as a loop at the point <math>p~q.</math>
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
 +
|}
 +
 +
{| align="center" cellpadding="10"
 +
|
 +
<math>\begin{array}{rcccccc}
 +
f
 +
& = & p  & \cdot & q
 +
\\[4pt]
 +
\operatorname{E}f
 +
& = &  p  & \cdot &  q  & \cdot & (\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
& + &  p  & \cdot & (q) & \cdot & (\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
& + & (p) & \cdot &  q  & \cdot & ~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
& + & (p) & \cdot & (q) & \cdot & ~\operatorname{d}p~~\operatorname{d}q~\end{array}</math>
 +
|}
 +
 +
We may understand the enlarged proposition <math>\operatorname{E}f</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
 +
 +
==Note 7==
 +
 +
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
 +
 +
Tables&nbsp;A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_0
 +
\\[4pt]
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_3
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_5
 +
\\[4pt]
 +
f_6
 +
\\[4pt]
 +
f_7
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}
 +
\\[4pt]
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0011}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{0101}
 +
\\[4pt]
 +
f_{0110}
 +
\\[4pt]
 +
f_{0111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[4pt]
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~0~1~1
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
0~1~0~1
 +
\\[4pt]
 +
0~1~1~0
 +
\\[4pt]
 +
0~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)~~~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~~~(q)
 +
\\[4pt]
 +
(p,~q)
 +
\\[4pt]
 +
(p~~q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\[4pt]
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
\text{not}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q
 +
\\[4pt]
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
\text{not both}~ p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\[4pt]
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
\lnot p
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
\lnot q
 +
\\[4pt]
 +
p \ne q
 +
\\[4pt]
 +
\lnot p \lor \lnot q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_8
 +
\\[4pt]
 +
f_9
 +
\\[4pt]
 +
f_{10}
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~p~~q~~
 +
\\[4pt]
 +
((p,~q))
 +
\\[4pt]
 +
~~~~~q~~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
~~p~~~~~
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{and}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \land q
 +
\\[4pt]
 +
p = q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\\[4pt]
 +
1
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1</math></p>
 +
<p><math>\text{Decimal}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2</math></p>
 +
<p><math>\text{Binary}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3</math></p>
 +
<p><math>\text{Vector}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4</math></p>
 +
<p><math>\text{Cactus}</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5</math></p>
 +
<p><math>\text{English}</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6</math></p>
 +
<p><math>\text{Ordinary}</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0</math>
 +
| <math>(~)</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
p ~\text{and}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
p \land q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p
 +
\\[4pt]
 +
p
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
p \ne q
 +
\\[4pt]
 +
p = q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot q
 +
\\[4pt]
 +
q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ p ~\text{and}~ q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \lor \lnot q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1</math>
 +
| <math>((~))</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 +
<br>
 +
 +
==Note 8==
 +
 +
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> 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.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{11} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{10} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{01} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{00} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{\operatorname{d}p~\operatorname{d}q}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{\operatorname{d}p(\operatorname{d}q)}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{(\operatorname{d}p)\operatorname{d}q}</math>
 +
| width="18%" |
 +
<math>\operatorname{D}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|}
 +
 +
<br>
 +
 +
==Note 9==
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\operatorname{E}f|_{xy}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{p(q)}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{(p)q}</math>
 +
| width="18%" | <math>\operatorname{E}f|_{(p)(q)}</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)(\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p)~\operatorname{d}q~
 +
\\[4pt]
 +
~\operatorname{d}p~(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}p~~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~
 +
\\[4pt]
 +
(\operatorname{d}p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}p~
 +
\\[4pt]
 +
(\operatorname{d}p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)
 +
\\[4pt]
 +
~\operatorname{d}p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p)
 +
\\[4pt]
 +
~\operatorname{d}p~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\\[4pt]
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p,~\operatorname{d}q)~
 +
\\[4pt]
 +
((\operatorname{d}p,~\operatorname{d}q))
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~\operatorname{d}q~
 +
\\[4pt]
 +
(\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}q)
 +
\\[4pt]
 +
~\operatorname{d}q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\operatorname{d}p~~\operatorname{d}q~)
 +
\\[4pt]
 +
(~\operatorname{d}p~(\operatorname{d}q))
 +
\\[4pt]
 +
((\operatorname{d}p)~\operatorname{d}q~)
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ p, q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\operatorname{D}f|_{xy}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{p(q)}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(p)q}</math>
 +
| width="18%" | <math>\operatorname{D}f|_{(p)(q)}</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}p
 +
\\[4pt]
 +
\operatorname{d}p
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\\[4pt]
 +
(\operatorname{d}p,~\operatorname{d}q)
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\operatorname{d}q
 +
\\[4pt]
 +
\operatorname{d}q
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\operatorname{d}p~~\operatorname{d}q~~
 +
\\[4pt]
 +
~~\operatorname{d}p~(\operatorname{d}q)~
 +
\\[4pt]
 +
~(\operatorname{d}p)~\operatorname{d}q~~
 +
\\[4pt]
 +
((\operatorname{d}p)(\operatorname{d}q))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|}
 +
 +
<br>
 +
 +
==Note 10==
 +
 +
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 <math>\operatorname{E}</math> and <math>\operatorname{D}</math> 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&nbsp;A3.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}p, \operatorname{d}q \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{11} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p~\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{10} f</math></p>
 +
<p><math>\operatorname{E}f|_{\operatorname{d}p(\operatorname{d}q)}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{01} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)\operatorname{d}q}</math></p>
 +
| width="18%" |
 +
<p><math>\operatorname{T}_{00} f</math></p>
 +
<p><math>\operatorname{E}f|_{(\operatorname{d}p)(\operatorname{d}q)}</math></p>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
| <math>(~)</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
| <math>((~))</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
 +
 +
<br>
 +
 +
The shift operator <math>\operatorname{E}</math> can be understood as enacting a substitution operation on the propositional form <math>f(p, q)\!</math> that is given as its argument.  In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lcl}
 +
\operatorname{E} ~:~ (X \to \mathbb{B})
 +
& \to &
 +
(\operatorname{E}X \to \mathbb{B})
 +
\\[6pt]
 +
\operatorname{E} ~:~ f(p, q)
 +
& \mapsto &
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
\\[6pt]
 +
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 +
& = &
 +
f(p + \operatorname{d}p, q + \operatorname{d}q)
 +
\\[6pt]
 +
& = &
 +
f( \texttt{(} p, \operatorname{d}p \texttt{)}, \texttt{(} q, \operatorname{d}q \texttt{)} )
 +
\end{array}</math>
 +
|}
 +
 +
Evaluating <math>\operatorname{E}f</math> at particular values of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> for example, <math>\operatorname{d}p = i</math> and <math>\operatorname{d}q = j,</math> where <math>i\!</math> and <math>j\!</math> are values in <math>\mathbb{B},</math> produces the following result:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<math>\begin{array}{lclcl}
 +
\operatorname{E}_{ij}
 +
& : &
 +
(X \to \mathbb{B})
 +
& \to &
 +
(X \to \mathbb{B})
 +
\\[6pt]
 +
\operatorname{E}_{ij}
 +
& : &
 +
f
 +
& \mapsto &
 +
\operatorname{E}_{ij}f
 +
\\[6pt]
 +
\operatorname{E}_{ij}f
 +
& = &
 +
\operatorname{E}f|_{\operatorname{d}p = i, \operatorname{d}q = j}
 +
& = &
 +
f(p + i, q + j)
 +
\\[6pt]
 +
&  &
 +
& = &
 +
f( \texttt{(} p, i \texttt{)}, \texttt{(} q, j \texttt{)} )
 +
\end{array}</math>
 +
|}
 +
 +
The notation is a little awkward, but the data of Table&nbsp;A3 should make the sense clear.  The important thing to observe is that <math>\operatorname{E}_{ij}</math> has the effect of transforming each proposition <math>f : X \to \mathbb{B}</math> into a proposition <math>f^\prime : X \to \mathbb{B}.</math>  As it happens, the action of each <math>\operatorname{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \operatorname{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions.  Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\operatorname{T}_{00}, \operatorname{T}_{01}, \operatorname{T}_{10}, \operatorname{T}_{11},</math> 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:
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|- style="height:50px"
 +
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{T}_{00}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{T}_{01}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{T}_{10}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{T}_{11}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{11}</math>
 +
| <math>\operatorname{T}_{10}</math>
 +
| <math>\operatorname{T}_{01}</math>
 +
| <math>\operatorname{T}_{00}</math>
 +
|}
 +
 +
<br>
 +
 +
It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this 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, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:  <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math>  Amazing!
 +
 +
==Note 11==
 +
 +
We have been contemplating functions of the type <math>f : X \to \mathbb{B}</math> and studying the action of the operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> 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 <math>X_1 \times \ldots \times X_k \to Y_1 \times \ldots \times Y_n</math> and abstract types <math>\mathbb{B}^k \to \mathbb{B}^n.</math>  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 the two basic connectives in the cactus language for ZOL.
 +
 +
For example, consider the proposition <math>f\!</math> of concrete type <math>f : P \times Q \times R \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} p, q, r \texttt{)}</math> in cactus syntax.  Taken as an assertion in what Peirce called the ''existential interpretation'', the proposition <math>\texttt{(} p, q, r \texttt{)}</math> says that just one of <math>p, q, r\!</math> is false.  It is instructive to consider this assertion in relation to the logical conjunction <math>pqr\!</math> of the same propositions.  A venn diagram of <math>\texttt{(} p, q, r \texttt{)}</math> looks like this:
 +
 +
{| align="center" cellpadding="10"
 +
| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]
 +
|}
 +
 +
In relation to the center cell indicated by the conjunction <math>pqr,\!</math> the region indicated by <math>\texttt{(} p, q, r \texttt{)}</math> is comprised of the adjacent or bordering cells.  Thus they are the cells that are just across the boundary of the center cell, reached as if by way of Leibniz's ''minimal changes'' from the point of origin, in this case, <math>pqr.\!</math>
 +
 +
More generally speaking, in a <math>k\!</math>-dimensional universe of discourse that is based on the ''alphabet'' of features <math>\mathcal{X} = \{ x_1, \ldots, x_k \},</math> the same form of boundary relationship is manifested for any cell of origin that one chooses to indicate.  One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form <math>e_1 \cdot \ldots \cdot e_k,</math> where <math>e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},</math> for <math>j = 1 ~\text{to}~ k.</math>  The proposition <math>\texttt{(} e_1, \ldots, e_k \texttt{)}</math> indicates the disjunctive region consisting of the cells that are just next door to <math>e_1 \cdot \ldots \cdot e_k.</math>
 +
 +
==Note 12==
 +
 +
{| align="center" cellpadding="0" cellspacing="0" width="90%"
 +
|
 +
<p>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.</p>
 +
|-
 +
| align="right" | &mdash; Charles Sanders Peirce, "Issues of Pragmaticism", (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 ''four-group'' <math>V_4.\!</math>  We encountered this group in one of its concrete representations, namely, as a ''transformation group'' that acts on a set of objects, in this 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, for example, in the form of the group operation table copied here:
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|- style="height:50px"
 +
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{e}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{f}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{g}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
|}
 +
 +
<br>
 +
 +
This table is abstractly the same as, or isomorphic to, the versions with the <math>\operatorname{E}_{ij}</math> operators and the <math>\operatorname{T}_{ij}</math> 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.
 +
 +
To see how a regular representation is constructed from the abstract operation table, 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, as a logical disjunction or boolean sum whose terms represent the ordered pairs of <math>\operatorname{input} : \operatorname{output}</math> transactions that are produced by each group element in turn.  This forms one of the two possible ''regular representations'' of the group, in this case the one that is called the ''post-regular representation'' or the ''right regular representation''.  It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:
 +
 +
Reading "<math>+\!</math>" as a logical disjunction:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{G}
 +
& = & \operatorname{e}
 +
& + & \operatorname{f}
 +
& + & \operatorname{g}
 +
& + & \operatorname{h}
 +
\end{matrix}</math>
 +
|}
 +
 +
And so, by expanding effects, we get:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{G}
 +
& = & \operatorname{e}:\operatorname{e}
 +
& + & \operatorname{f}:\operatorname{f}
 +
& + & \operatorname{g}:\operatorname{g}
 +
& + & \operatorname{h}:\operatorname{h}
 +
\\[4pt]
 +
& + & \operatorname{e}:\operatorname{f}
 +
& + & \operatorname{f}:\operatorname{e}
 +
& + & \operatorname{g}:\operatorname{h}
 +
& + & \mathrm{h}:\mathrm{g}
 +
\\[4pt]
 +
& + & \operatorname{e}:\operatorname{g}
 +
& + & \operatorname{f}:\operatorname{h}
 +
& + & \operatorname{g}:\operatorname{e}
 +
& + & \operatorname{h}:\operatorname{f}
 +
\\[4pt]
 +
& + & \operatorname{e}:\operatorname{h}
 +
& + & \operatorname{f}:\operatorname{g}
 +
& + & \operatorname{g}:\operatorname{f}
 +
& + & \operatorname{h}:\operatorname{e}
 +
\end{matrix}</math>
 +
|}
 +
 +
More on the pragmatic maxim as a representation principle later.
 +
 +
==Note 13==
 +
 +
The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| Every group is isomorphic to a subgroup of <math>\operatorname{Aut}(X),</math> the group of automorphisms of a suitably chosen set <math>X\!</math>.
 +
|}
 +
 +
There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers.  The crux of the whole idea is this:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| Contemplate the effects of the symbol whose meaning you wish to investigate as they play out on all the stages of conduct where you can imagine that symbol playing a role.
 +
|}
 +
 +
This idea of contextual definition by way of conduct 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.&nbsp;216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators <math>\operatorname{S}, \operatorname{K}, \operatorname{I},</math> and hence of lambda calculus, and I reckon you know where that leads.
 +
 +
==Note 14==
 +
 +
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 describes the action of an "elementary dual relative" in this way:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| Elementary simple relatives are connected together in systems of four.  For if <math>\mathrm{A}\!:\!\mathrm{B}</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}\!</math> gives <math>\mathrm{A},\!</math> then this relation existing as elementary, we have the four elementary relatives
 +
|-
 +
| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}.</math>
 +
|-
 +
| C.S. Peirce, ''Collected Papers'', CP&nbsp;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:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{bmatrix}
 +
a\!:\!a & a\!:\!b & a\!:\!c
 +
\\
 +
b\!:\!a & b\!:\!b & b\!:\!c
 +
\\
 +
c\!:\!a & c\!:\!b & c\!:\!c
 +
\end{bmatrix}</math>
 +
|}
 +
 +
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> and
 +
the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>
 +
M \quad = \quad
 +
\begin{bmatrix}
 +
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)
 +
\end{bmatrix}
 +
</math>
 +
|}
 +
 +
For at least a little while longer, I will keep explicit the distinction between a ''relative term'' like <math>\mathit{m}\!</math> and a ''relation'' like <math>M \subseteq X \times X,</math> but it is best to view both these entities as involving different applications of the same information, and so we could just as easily write the following form:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>
 +
m \quad = \quad
 +
\begin{bmatrix}
 +
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)
 +
\end{bmatrix}
 +
</math>
 +
|}
 +
 +
By way of making up a concrete example, let us say that <math>\mathit{m}\!</math> or <math>M\!</math> is given as follows:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{l}
 +
a ~\text{is a marker for}~ a
 +
\\
 +
a ~\text{is a marker for}~ b
 +
\\
 +
b ~\text{is a marker for}~ b
 +
\\
 +
b ~\text{is a marker for}~ c
 +
\\
 +
c ~\text{is a marker for}~ c
 +
\\
 +
c ~\text{is a marker for}~ a
 +
\end{array}</math>
 +
|}
 +
 +
In sum, then, the relative term <math>\mathit{m}\!</math> and the relation <math>M\!</math> are both represented by the following matrix:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{bmatrix}
 +
1 \cdot (a\!:\!a) & 1 \cdot (a\!:\!b) & 0 \cdot (a\!:\!c)
 +
\\
 +
0 \cdot (b\!:\!a) & 1 \cdot (b\!:\!b) & 1 \cdot (b\!:\!c)
 +
\\
 +
1 \cdot (c\!:\!a) & 0 \cdot (c\!:\!b) & 1 \cdot (c\!:\!c)
 +
\end{bmatrix}</math>
 +
|}
 +
 +
I think this much will serve to fix notation and set up the remainder of the discussion.
 +
 +
==Note 15==
 +
 +
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(i\!:\!j),</math> as the indices <math>i, j\!</math> 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:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>
 +
M \quad = \quad
 +
\begin{bmatrix}
 +
1 & 1 & 0
 +
\\
 +
0 & 1 & 1
 +
\\
 +
1 & 0 & 1
 +
\end{bmatrix}
 +
</math>
 +
|}
 +
 +
The various representations of <math>M\!</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following aggregate of elementary relatives:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{*{13}{c}}
 +
M
 +
& = & a\!:\!a
 +
& + & b\!:\!b
 +
& + & c\!:\!c
 +
& + & a\!:\!b
 +
& + & b\!:\!c
 +
& + & c\!:\!a
 +
\end{array}</math>
 +
|}
 +
 +
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> 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 <math>i\!:\!j</math> in the way that Peirce read them in logical contexts:  <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math>  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 <math>i\!</math> the relate and <math>j\!</math> the correlate, the elementary relative <math>i\!:\!j</math> now means that <math>i\!</math> gets changed into <math>j.\!</math>  In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
 +
 +
{| align="center" cellpadding="6"
 +
|
 +
<math>\begin{Bmatrix}
 +
a & b & c
 +
\\
 +
b & c & a
 +
\end{Bmatrix}</math>
 +
|}
 +
 +
This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327).
 +
 +
==Note 16==
 +
 +
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 definitively 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 <math>V_4\!</math> once again:
 +
 +
<br>
 +
 +
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
 +
|+ <math>\text{Klein Four-Group}~ V_4</math>
 +
|- style="height:50px"
 +
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{e}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{f}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{g}</math>
 +
| width="22%" style="border-bottom:1px solid black" |
 +
<math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{e}</math>
 +
| <math>\operatorname{f}</math>
 +
|- style="height:50px"
 +
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
 +
| <math>\operatorname{h}</math>
 +
| <math>\operatorname{g}</math>
 +
| <math>\operatorname{f}</math>
 +
| <math>\operatorname{e}</math>
 +
|}
 +
 +
<br>
 +
 +
A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.</math>
 +
 +
In the case of <math>V_4 = (G, \cdot),</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h} \},</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G</math> whose triples are listed below:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
(\operatorname{e}, \operatorname{e}, \operatorname{e}) &
 +
(\operatorname{e}, \operatorname{f}, \operatorname{f}) &
 +
(\operatorname{e}, \operatorname{g}, \operatorname{g}) &
 +
(\operatorname{e}, \operatorname{h}, \operatorname{h})
 +
\\[6pt]
 +
(\operatorname{f}, \operatorname{e}, \operatorname{f}) &
 +
(\operatorname{f}, \operatorname{f}, \operatorname{e}) &
 +
(\operatorname{f}, \operatorname{g}, \operatorname{h}) &
 +
(\operatorname{f}, \operatorname{h}, \operatorname{g})
 +
\\[6pt]
 +
(\operatorname{g}, \operatorname{e}, \operatorname{g}) &
 +
(\operatorname{g}, \operatorname{f}, \operatorname{h}) &
 +
(\operatorname{g}, \operatorname{g}, \operatorname{e}) &
 +
(\operatorname{g}, \operatorname{h}, \operatorname{f})
 +
\\[6pt]
 +
(\operatorname{h}, \operatorname{e}, \operatorname{h}) &
 +
(\operatorname{h}, \operatorname{f}, \operatorname{g}) &
 +
(\operatorname{h}, \operatorname{g}, \operatorname{f}) &
 +
(\operatorname{h}, \operatorname{h}, \operatorname{e})
 +
\end{matrix}</math>
 +
|}
 +
 +
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math>  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 <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.\!</math>
 +
|}
 +
 +
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin.  This aspect of pragmatic definition we recognize as the regular ante-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
\end{matrix}</math>
 +
|}
 +
 +
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin.  This aspect of pragmatic definition we recognize as the regular post-representation:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
\end{matrix}</math>
 +
|}
 +
 +
If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.
 +
 +
==Note 17==
 +
 +
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, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> 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, <math>X = \{ a, b, c \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math>  The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\operatorname{e}</math>
 +
| width="16%" | <math>\operatorname{f}</math>
 +
| width="16%" | <math>\operatorname{g}</math>
 +
| width="16%" | <math>\operatorname{h}</math>
 +
| width="16%" | <math>\operatorname{i}</math>
 +
| width="16%" | <math>\operatorname{j}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & b & c
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & a & b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & c & a
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & c & b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & b & a
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & a & c
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
Symmetric Group S_3
 +
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      |
 +
|      /  \ /  \ /  \ /  \ /  \ /  \      |
 +
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
 +
|      \  / \  / \  / \  / \  / \  /      |
 +
|        \ /  \ /  \ /  \ /  \ /  \ /        |
 +
|        \  h  \  h  \  e  \  j  \  i  /        |
 +
|          \  / \  / \  / \  / \  /          |
 +
|          \ /  \ /  \ /  \ /  \ /          |
 +
|            \  i  \  g  \  f  \  h  /            |
 +
|            \  / \  / \  / \  /            |
 +
|              \ /  \ /  \ /  \ /              |
 +
|              \  f  \  e  \  g  /              |
 +
|                \  / \  / \  /                |
 +
|                \ /  \ /  \ /                |
 +
|                  \  g  \  f  /                  |
 +
|                  \  / \  /                  |
 +
|                    \ /  \ /                    |
 +
|                    \  e  /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        v                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227&ndash;323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307&ndash;323).
 +
 +
==Note 18==
 +
 +
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 <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations for the symmetric group on three letters, <math>\operatorname{Sym}(3).</math>  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 <math>\operatorname{Sym} \{ a, b, c \}</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.\!</math>
 +
 +
{| align="center" cellpadding="10" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{e}
 +
& = & a\!:\!a
 +
& + & b\!:\!b
 +
& + & c\!:\!c
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & a\!:\!c
 +
& + & b\!:\!a
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & a\!:\!b
 +
& + & b\!:\!c
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & a\!:\!a
 +
& + & b\!:\!c
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & a\!:\!c
 +
& + & b\!:\!b
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & a\!:\!b
 +
& + & b\!:\!a
 +
& + & c\!:\!c
 +
\end{matrix}</math>
 +
|}
 +
 +
I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>x\!:\!y</math> constitutes the turning of <math>x\!</math> into <math>y.\!</math>  It is possible that the next time we check in with CSP we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
 +
 +
==Note 19==
 +
 +
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<pre>
 +
Symmetric Group S_3
 +
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      |
 +
|      /  \ /  \ /  \ /  \ /  \ /  \      |
 +
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
 +
|      \  / \  / \  / \  / \  / \  /      |
 +
|        \ /  \ /  \ /  \ /  \ /  \ /        |
 +
|        \  h  \  h  \  e  \  j  \  i  /        |
 +
|          \  / \  / \  / \  / \  /          |
 +
|          \ /  \ /  \ /  \ /  \ /          |
 +
|            \  i  \  g  \  f  \  h  /            |
 +
|            \  / \  / \  / \  /            |
 +
|              \ /  \ /  \ /  \ /              |
 +
|              \  f  \  e  \  g  /              |
 +
|                \  / \  / \  /                |
 +
|                \ /  \ /  \ /                |
 +
|                  \  g  \  f  /                  |
 +
|                  \  / \  /                  |
 +
|                    \ /  \ /                    |
 +
|                    \  e  /                    |
 +
|                      \  /                      |
 +
|                      \ /                      |
 +
|                        v                        |
 +
|                                                |
 +
o-------------------------------------------------o
 +
</pre>
 +
|}
 +
 +
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 <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math>  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 <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| valign="top" | 1.
 +
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>xy.\!</math>
 +
|-
 +
| valign="top" | 2.
 +
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>yx.\!</math>
 +
|}
 +
 +
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin.  This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{*{13}{c}}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
& + & \operatorname{i}\!:\!\operatorname{i}
 +
& + & \operatorname{j}\!:\!\operatorname{j}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{j}
 +
& + & \operatorname{i}\!:\!\operatorname{h}
 +
& + & \operatorname{j}\!:\!\operatorname{i}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{i}
 +
& + & \operatorname{i}\!:\!\operatorname{j}
 +
& + & \operatorname{j}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{i}
 +
& + & \operatorname{g}\!:\!\operatorname{j}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
& + & \operatorname{i}\!:\!\operatorname{f}
 +
& + & \operatorname{j}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & \operatorname{e}\!:\!\operatorname{i}
 +
& + & \operatorname{f}\!:\!\operatorname{j}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
& + & \operatorname{i}\!:\!\operatorname{e}
 +
& + & \operatorname{j}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & \operatorname{e}\!:\!\operatorname{j}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{i}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
& + & \operatorname{i}\!:\!\operatorname{g}
 +
& + & \operatorname{j}\!:\!\operatorname{e}
 +
\end{array}</math>
 +
|}
 +
 +
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin.  This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{array}{*{13}{c}}
 +
\operatorname{e}
 +
& = & \operatorname{e}\!:\!\operatorname{e}
 +
& + & \operatorname{f}\!:\!\operatorname{f}
 +
& + & \operatorname{g}\!:\!\operatorname{g}
 +
& + & \operatorname{h}\!:\!\operatorname{h}
 +
& + & \operatorname{i}\!:\!\operatorname{i}
 +
& + & \operatorname{j}\!:\!\operatorname{j}
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & \operatorname{e}\!:\!\operatorname{f}
 +
& + & \operatorname{f}\!:\!\operatorname{g}
 +
& + & \operatorname{g}\!:\!\operatorname{e}
 +
& + & \operatorname{h}\!:\!\operatorname{i}
 +
& + & \operatorname{i}\!:\!\operatorname{j}
 +
& + & \operatorname{j}\!:\!\operatorname{h}
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & \operatorname{e}\!:\!\operatorname{g}
 +
& + & \operatorname{f}\!:\!\operatorname{e}
 +
& + & \operatorname{g}\!:\!\operatorname{f}
 +
& + & \operatorname{h}\!:\!\operatorname{j}
 +
& + & \operatorname{i}\!:\!\operatorname{h}
 +
& + & \operatorname{j}\!:\!\operatorname{i}
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & \operatorname{e}\!:\!\operatorname{h}
 +
& + & \operatorname{f}\!:\!\operatorname{j}
 +
& + & \operatorname{g}\!:\!\operatorname{i}
 +
& + & \operatorname{h}\!:\!\operatorname{e}
 +
& + & \operatorname{i}\!:\!\operatorname{g}
 +
& + & \operatorname{j}\!:\!\operatorname{f}
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & \operatorname{e}\!:\!\operatorname{i}
 +
& + & \operatorname{f}\!:\!\operatorname{h}
 +
& + & \operatorname{g}\!:\!\operatorname{j}
 +
& + & \operatorname{h}\!:\!\operatorname{f}
 +
& + & \operatorname{i}\!:\!\operatorname{e}
 +
& + & \operatorname{j}\!:\!\operatorname{g}
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & \operatorname{e}\!:\!\operatorname{j}
 +
& + & \operatorname{f}\!:\!\operatorname{i}
 +
& + & \operatorname{g}\!:\!\operatorname{h}
 +
& + & \operatorname{h}\!:\!\operatorname{g}
 +
& + & \operatorname{i}\!:\!\operatorname{f}
 +
& + & \operatorname{j}\!:\!\operatorname{e}
 +
\end{array}</math>
 +
|}
 +
 +
If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> 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.
 +
 +
==Note 20==
 +
 +
{| cellpadding="2" cellspacing="2" width="100%"
 +
| width="60%" | &nbsp;
 +
| width="40%" |
 +
the way of heaven and earth<br>
 +
is to be long continued<br>
 +
in their operation<br>
 +
without stopping
 +
|-
 +
| height="50px" | &nbsp;
 +
| valign="top"  | &mdash; i ching, hexagram 32
 +
|}
 +
 +
The Reader may be wondering what happened to the announced subject of ''Dynamics And Logic''.  What happened was a bit like this:
 +
 +
We made the observation that the shift operators <math>\{ \operatorname{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  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 [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842&ndash;1899)], have turned out to be of critical utility in the solution of 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 barely 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.
 +
 +
==Note 21==
 +
 +
We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.
 +
 +
Recalling the manner of our acquaintance with the symmetric group <math>S_3,\!</math> we began with the ''bigraph'' (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set <math>X = \{ a, b, c \}.\!</math>
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\operatorname{e}</math>
 +
| width="16%" | <math>\operatorname{f}</math>
 +
| width="16%" | <math>\operatorname{g}</math>
 +
| width="16%" | <math>\operatorname{h}</math>
 +
| width="16%" | <math>\operatorname{i}</math>
 +
| width="16%" | <math>\operatorname{j}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & b & c
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & a & b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & c & a
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
a & c & b
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
c & b & a
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & a & c
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
These permutations were then converted to relative form as logical sums of elementary relatives:
 +
 +
{| align="center" cellpadding="10" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{e}
 +
& = & a\!:\!a
 +
& + & b\!:\!b
 +
& + & c\!:\!c
 +
\\[4pt]
 +
\operatorname{f}
 +
& = & a\!:\!c
 +
& + & b\!:\!a
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\operatorname{g}
 +
& = & a\!:\!b
 +
& + & b\!:\!c
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\operatorname{h}
 +
& = & a\!:\!a
 +
& + & b\!:\!c
 +
& + & c\!:\!b
 +
\\[4pt]
 +
\operatorname{i}
 +
& = & a\!:\!c
 +
& + & b\!:\!b
 +
& + & c\!:\!a
 +
\\[4pt]
 +
\operatorname{j}
 +
& = & a\!:\!b
 +
& + & b\!:\!a
 +
& + & c\!:\!c
 +
\end{matrix}</math>
 +
|}
 +
 +
From the relational representation of <math>\operatorname{Sym} \{ a, b, c \} \cong S_3,</math> one easily derives a ''linear representation'' of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other.  Each of these linear transformations is in turn represented by a 2-dimensional array of coefficients in <math>\mathbb{B},</math> resulting in the following set of matrices for the group:
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
 +
|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
 +
|- style="background:#f0f0ff"
 +
| width="16%" | <math>\operatorname{e}</math>
 +
| width="16%" | <math>\operatorname{f}</math>
 +
| width="16%" | <math>\operatorname{g}</math>
 +
| width="16%" | <math>\operatorname{h}</math>
 +
| width="16%" | <math>\operatorname{i}</math>
 +
| width="16%" | <math>\operatorname{j}</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
1 & 0 & 0
 +
\\
 +
0 & 1 & 0
 +
\\
 +
0 & 0 & 1
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 0 & 1
 +
\\
 +
1 & 0 & 0
 +
\\
 +
0 & 1 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 1 & 0
 +
\\
 +
0 & 0 & 1
 +
\\
 +
1 & 0 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1 & 0 & 0
 +
\\
 +
0 & 0 & 1
 +
\\
 +
0 & 1 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 0 & 1
 +
\\
 +
0 & 1 & 0
 +
\\
 +
1 & 0 & 0
 +
\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
0 & 1 & 0
 +
\\
 +
1 & 0 & 0
 +
\\
 +
0 & 0 & 1
 +
\end{matrix}</math>
 +
|}
 +
 +
<br>
 +
 +
The key to the mysteries of these matrices is revealed by observing that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{bmatrix}
 +
a\!:\!a &
 +
a\!:\!b &
 +
a\!:\!c
 +
\\
 +
b\!:\!a &
 +
b\!:\!b &
 +
b\!:\!c
 +
\\
 +
c\!:\!a &
 +
c\!:\!b &
 +
c\!:\!c
 +
\end{bmatrix}</math>
 +
|}
 +
 +
==Note 22==
 +
 +
<pre>
 +
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.
 +
</pre>
 +
 +
==Note 23==
 +
 +
<pre>
 +
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
 +
</pre>
 +
 +
==Note 24==
 +
 +
<pre>
 +
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)
 +
</pre>
 +
 +
==Note 25==
 +
 +
<pre>
 +
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
 +
</pre>
 +
 +
==Note 26==
 +
 +
<pre>
 +
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
 +
</pre>
 +
 +
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Revision as of 16:06, 15 June 2009

Differential logic is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in universes of discourse that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a differential logical calculus, that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple example of a differential logical calculus is furnished by a differential propositional calculus. A differential propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe. This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Note 1

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 \(p ~\operatorname{and}~ q\) 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 \(f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\) where \(\mathbb{B} = \{ 0, 1 \}\) is read in such a way that \(0\!\) means \(\operatorname{false}\) and \(1\!\) means \(\operatorname{true}.\)

In this style of graphical representation, the value \(\operatorname{true}\) looks like a blank label and the value \(\operatorname{false}\) 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 shown here:

Venn Diagram P And Q.jpg

Now ask yourself: What is the value of the proposition \(pq\!\) at a distance of \(\operatorname{d}p\) and \(\operatorname{d}q\) 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 ASCII, I will 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 \(p + \operatorname{d}p,\) where the plus sign indicates addition in \(\mathbb{B},\) that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation 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" 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 \(\mathbb{B},\) 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 \(q\!,\) as shown here: 

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

Note 2

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

\(p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q\) 

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 that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan'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–De Morgan 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 adequate to handle the complexity of 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 \(f(p, q) = pq.\!\)

Venn Diagram F = P And Q.jpg
Cactus Graph F = P And Q.jpg

A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like \(f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\) or \(f : \mathbb{B}^2 \to \mathbb{B}.\) 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 values \(\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.\)
Let \(Q\!\) be the set of values \(\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \operatorname{not}~ q,~ q \} ~\cong~ \mathbb{B}.\)

Then interpret the usual propositions about \(p, q\!\) as functions of the concrete type \(f : P \times Q \to \mathbb{B}.\)

We are going to consider various operators on these functions. Here, an operator \(\operatorname{F}\) is a function that takes one function \(f\!\) into another function \(\operatorname{F}f.\)

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 \(\Delta,\!\) written here as \(\operatorname{D}.\)
The enlargement" operator \(\Epsilon,\!\) written here as \(\operatorname{E}.\)

These days, \(\operatorname{E}\) is more often called the shift operator.

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space \(X = P \times Q,\) its (first order) differential extension \(\operatorname{E}X\) is constructed according to the following specifications:

\(\begin{array}{rcc} \operatorname{E}X & = & X \times \operatorname{d}X \end{array}\)

where:

\(\begin{array}{rcc} X & = & P \times Q \'"`UNIQ-MathJax1-QINU`"' Amazing! =='"`UNIQ--h-10--QINU`"'Note 11== We have been contemplating functions of the type \(f : X \to \mathbb{B}\) and studying the action of the operators \(\operatorname{E}\) and \(\operatorname{D}\) 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 \(X_1 \times \ldots \times X_k \to Y_1 \times \ldots \times Y_n\) and abstract types \(\mathbb{B}^k \to \mathbb{B}^n.\) 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 the two basic connectives in the cactus language for ZOL.

For example, consider the proposition \(f\!\) of concrete type \(f : P \times Q \times R \to \mathbb{B}\) and abstract type \(f : \mathbb{B}^3 \to \mathbb{B}\) that is written \(\texttt{(} p, q, r \texttt{)}\) in cactus syntax. Taken as an assertion in what Peirce called the existential interpretation, the proposition \(\texttt{(} p, q, r \texttt{)}\) says that just one of \(p, q, r\!\) is false. It is instructive to consider this assertion in relation to the logical conjunction \(pqr\!\) of the same propositions. A venn diagram of \(\texttt{(} p, q, r \texttt{)}\) looks like this:

Minimal Negation Operator (p,q,r).jpg

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

More generally speaking, in a \(k\!\)-dimensional universe of discourse that is based on the alphabet of features \(\mathcal{X} = \{ x_1, \ldots, x_k \},\) the same form of boundary relationship is manifested for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form \(e_1 \cdot \ldots \cdot e_k,\) where \(e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},\) for \(j = 1 ~\text{to}~ k.\) The proposition \(\texttt{(} e_1, \ldots, e_k \texttt{)}\) indicates the disjunctive region consisting of the cells that are just next door to \(e_1 \cdot \ldots \cdot e_k.\)

Note 12

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.

— Charles Sanders Peirce, "Issues of Pragmaticism", (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 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 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, for example, in the form of the group operation table copied here:


\(\cdot\)

\(\operatorname{e}\)

\(\operatorname{f}\)

\(\operatorname{g}\)

\(\operatorname{h}\)

\(\operatorname{e}\) \(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\)
\(\operatorname{f}\) \(\operatorname{f}\) \(\operatorname{e}\) \(\operatorname{h}\) \(\operatorname{g}\)
\(\operatorname{g}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{e}\) \(\operatorname{f}\)
\(\operatorname{h}\) \(\operatorname{h}\) \(\operatorname{g}\) \(\operatorname{f}\) \(\operatorname{e}\)


This table is abstractly the same as, or isomorphic to, the versions with the \(\operatorname{E}_{ij}\) operators and the \(\operatorname{T}_{ij}\) 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.

To see how a regular representation is constructed from the abstract operation table, 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, as a logical disjunction or boolean sum whose terms represent the ordered pairs of \(\operatorname{input} : \operatorname{output}\) transactions that are produced by each group element in turn. This forms one of the two possible regular representations of the group, in this case the one that is called the post-regular representation or the right regular representation. It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:

Reading "\(+\!\)" as a logical disjunction:

\(\begin{matrix} \operatorname{G} & = & \operatorname{e} & + & \operatorname{f} & + & \operatorname{g} & + & \operatorname{h} \end{matrix}\)

And so, by expanding effects, we get:

\(\begin{matrix} \operatorname{G} & = & \operatorname{e}:\operatorname{e} & + & \operatorname{f}:\operatorname{f} & + & \operatorname{g}:\operatorname{g} & + & \operatorname{h}:\operatorname{h} \'"`UNIQ-MathJax2-QINU`"' is the relate, \(j\!\) is the correlate, and in our current example \(i\!:\!j,\) or more exactly, \(m_{ij} = 1,\!\) is taken 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 \(\mathbf{1} = a + b + c,\) or what we would now call the set \(\{ a, b, c \},\!\) in particular, it is the permutation that is otherwise notated as follows:

\(\begin{Bmatrix} a & b & c \\ b & c & a \end{Bmatrix}\)

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

Note 16

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 definitively 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:


\(\text{Klein Four-Group}~ V_4\)
\(\cdot\)

\(\operatorname{e}\)

\(\operatorname{f}\)

\(\operatorname{g}\)

\(\operatorname{h}\)

\(\operatorname{e}\) \(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\)
\(\operatorname{f}\) \(\operatorname{f}\) \(\operatorname{e}\) \(\operatorname{h}\) \(\operatorname{g}\)
\(\operatorname{g}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{e}\) \(\operatorname{f}\)
\(\operatorname{h}\) \(\operatorname{h}\) \(\operatorname{g}\) \(\operatorname{f}\) \(\operatorname{e}\)


A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form \((x, y, z)\!\) satisfying the equation \(x \cdot y = z.\)

In the case of \(V_4 = (G, \cdot),\) where \(G\!\) is the underlying set \(\{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h} \},\) we have the 3-adic relation \(L(V_4) \subseteq G \times G \times G\) whose triples are listed below:

\(\begin{matrix} (\operatorname{e}, \operatorname{e}, \operatorname{e}) & (\operatorname{e}, \operatorname{f}, \operatorname{f}) & (\operatorname{e}, \operatorname{g}, \operatorname{g}) & (\operatorname{e}, \operatorname{h}, \operatorname{h}) \\[6pt] (\operatorname{f}, \operatorname{e}, \operatorname{f}) & (\operatorname{f}, \operatorname{f}, \operatorname{e}) & (\operatorname{f}, \operatorname{g}, \operatorname{h}) & (\operatorname{f}, \operatorname{h}, \operatorname{g}) \\[6pt] (\operatorname{g}, \operatorname{e}, \operatorname{g}) & (\operatorname{g}, \operatorname{f}, \operatorname{h}) & (\operatorname{g}, \operatorname{g}, \operatorname{e}) & (\operatorname{g}, \operatorname{h}, \operatorname{f}) \\[6pt] (\operatorname{h}, \operatorname{e}, \operatorname{h}) & (\operatorname{h}, \operatorname{f}, \operatorname{g}) & (\operatorname{h}, \operatorname{g}, \operatorname{f}) & (\operatorname{h}, \operatorname{h}, \operatorname{e}) \end{matrix}\)

It is part of the definition of a group that the 3-adic relation \(L \subseteq G^3\) is actually a function \(L : G \times G \to G.\) 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 \(L : G \times G \to G,\) we can define a couple of substitution operators:

1. \(\operatorname{Sub}(x, (\underline{~~}, y))\) puts any specified \(x\!\) into the empty slot of the rheme \((\underline{~~}, y),\) with the effect of producing the saturated rheme \((x, y)\!\) that evaluates to \(xy.\!\)
2. \(\operatorname{Sub}(x, (y, \underline{~~}))\) puts any specified \(x\!\) into the empty slot of the rheme \((y, \underline{~~}),\) 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 \((\underline{~~}, y),\) as \(y\!\) ranges over \(G,\!\) and the effects are such that \(x\!\) takes \((\underline{~~}, y)\) into \(xy,\!\) for \(y\!\) in \(G,\!\) all of which is 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:

\(\begin{matrix} \operatorname{e} & = & \operatorname{e}\!:\!\operatorname{e} & + & \operatorname{f}\!:\!\operatorname{f} & + & \operatorname{g}\!:\!\operatorname{g} & + & \operatorname{h}\!:\!\operatorname{h} \\[4pt] \operatorname{f} & = & \operatorname{e}\!:\!\operatorname{f} & + & \operatorname{f}\!:\!\operatorname{e} & + & \operatorname{g}\!:\!\operatorname{h} & + & \operatorname{h}\!:\!\operatorname{g} \\[4pt] \operatorname{g} & = & \operatorname{e}\!:\!\operatorname{g} & + & \operatorname{f}\!:\!\operatorname{h} & + & \operatorname{g}\!:\!\operatorname{e} & + & \operatorname{h}\!:\!\operatorname{f} \\[4pt] \operatorname{h} & = & \operatorname{e}\!:\!\operatorname{h} & + & \operatorname{f}\!:\!\operatorname{g} & + & \operatorname{g}\!:\!\operatorname{f} & + & \operatorname{h}\!:\!\operatorname{e} \end{matrix}\)

In (2) we consider the effects of each \(x\!\) in its practical bearing on contexts of the form \((y, \underline{~~}),\) as \(y\!\) ranges over \(G,\!\) and the effects are such that \(x\!\) takes \((y, \underline{~~})\) into \(yx,\!\) for \(y\!\) in \(G,\!\) all of which is 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:

\(\begin{matrix} \operatorname{e} & = & \operatorname{e}\!:\!\operatorname{e} & + & \operatorname{f}\!:\!\operatorname{f} & + & \operatorname{g}\!:\!\operatorname{g} & + & \operatorname{h}\!:\!\operatorname{h} \\[4pt] \operatorname{f} & = & \operatorname{e}\!:\!\operatorname{f} & + & \operatorname{f}\!:\!\operatorname{e} & + & \operatorname{g}\!:\!\operatorname{h} & + & \operatorname{h}\!:\!\operatorname{g} \\[4pt] \operatorname{g} & = & \operatorname{e}\!:\!\operatorname{g} & + & \operatorname{f}\!:\!\operatorname{h} & + & \operatorname{g}\!:\!\operatorname{e} & + & \operatorname{h}\!:\!\operatorname{f} \\[4pt] \operatorname{h} & = & \operatorname{e}\!:\!\operatorname{h} & + & \operatorname{f}\!:\!\operatorname{g} & + & \operatorname{g}\!:\!\operatorname{f} & + & \operatorname{h}\!:\!\operatorname{e} \end{matrix}\)

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.

Note 17

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, \(G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!\) 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 \(G = \operatorname{Sym}(X)\) or more abstractly and briefly, as \(\operatorname{Sym}(3)\) or \(S_3.\!\) The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in \(\operatorname{Sym}(X).\)


\(\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}\)
\(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{i}\) \(\operatorname{j}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] a & b & c \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] c & a & b \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] b & c & a \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] a & c & b \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] c & b & a \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] b & a & c \end{matrix}\)


Here is the operation table for \(S_3,\!\) given in abstract fashion: 

Symmetric Group S_3
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      |
|       /   \ /   \ /   \ /   \ /   \ /   \       |
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
|       \   / \   / \   / \   / \   / \   /       |
|        \ /   \ /   \ /   \ /   \ /   \ /        |
|         \  h  \  h  \  e  \  j  \  i  /         |
|          \   / \   / \   / \   / \   /          |
|           \ /   \ /   \ /   \ /   \ /           |
|            \  i  \  g  \  f  \  h  /            |
|             \   / \   / \   / \   /             |
|              \ /   \ /   \ /   \ /              |
|               \  f  \  e  \  g  /               |
|                \   / \   / \   /                |
|                 \ /   \ /   \ /                 |
|                  \  g  \  f  /                  |
|                   \   / \   /                   |
|                    \ /   \ /                    |
|                     \  e  /                     |
|                      \   /                      |
|                       \ /                       |
|                        v                        |
|                                                 |
o-------------------------------------------------o

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).

Note 18

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, \(\operatorname{Sym}(3).\) 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 \(\operatorname{Sym} \{ a, b, c \}\) in relative form generates what is generally thought of as a natural representation of \(S_3.\!\)

\(\begin{matrix} \operatorname{e} & = & a\!:\!a & + & b\!:\!b & + & c\!:\!c \\[4pt] \operatorname{f} & = & a\!:\!c & + & b\!:\!a & + & c\!:\!b \\[4pt] \operatorname{g} & = & a\!:\!b & + & b\!:\!c & + & c\!:\!a \\[4pt] \operatorname{h} & = & a\!:\!a & + & b\!:\!c & + & c\!:\!b \\[4pt] \operatorname{i} & = & a\!:\!c & + & b\!:\!b & + & c\!:\!a \\[4pt] \operatorname{j} & = & a\!:\!b & + & b\!:\!a & + & c\!:\!c \end{matrix}\)

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 we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

Note 19

To construct the regular representations of \(S_3,\!\) we begin with the data of its operation table: 

Symmetric Group S_3
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      |
|       /   \ /   \ /   \ /   \ /   \ /   \       |
|      (  j  \  j  \  j  \  i  \  h  \  j  )      |
|       \   / \   / \   / \   / \   / \   /       |
|        \ /   \ /   \ /   \ /   \ /   \ /        |
|         \  h  \  h  \  e  \  j  \  i  /         |
|          \   / \   / \   / \   / \   /          |
|           \ /   \ /   \ /   \ /   \ /           |
|            \  i  \  g  \  f  \  h  /            |
|             \   / \   / \   / \   /             |
|              \ /   \ /   \ /   \ /              |
|               \  f  \  e  \  g  /               |
|                \   / \   / \   /                |
|                 \ /   \ /   \ /                 |
|                  \  g  \  f  /                  |
|                   \   / \   /                   |
|                    \ /   \ /                    |
|                     \  e  /                     |
|                      \   /                      |
|                       \ /                       |
|                        v                        |
|                                                 |
o-------------------------------------------------o

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 \(L \subseteq G^3\) is actually a function \(L : G \times G \to G.\) 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 \(L : G \times G \to G,\) we can define a couple of substitution operators:

1. \(\operatorname{Sub}(x, (\underline{~~}, y))\) puts any specified \(x\!\) into the empty slot of the rheme \((\underline{~~}, y),\) with the effect of producing the saturated rheme \((x, y)\!\) that evaluates to \(xy.\!\)
2. \(\operatorname{Sub}(x, (y, \underline{~~}))\) puts any specified \(x\!\) into the empty slot of the rheme \((y, \underline{~~}),\) 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 \((\underline{~~}, y),\) as \(y\!\) ranges over \(G,\!\) and the effects are such that \(x\!\) takes \((\underline{~~}, y)\) into \(xy,\!\) for \(y\!\) in \(G,\!\) all of which is 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:

\(\begin{array}{*{13}{c}} \operatorname{e} & = & \operatorname{e}\!:\!\operatorname{e} & + & \operatorname{f}\!:\!\operatorname{f} & + & \operatorname{g}\!:\!\operatorname{g} & + & \operatorname{h}\!:\!\operatorname{h} & + & \operatorname{i}\!:\!\operatorname{i} & + & \operatorname{j}\!:\!\operatorname{j} \\[4pt] \operatorname{f} & = & \operatorname{e}\!:\!\operatorname{f} & + & \operatorname{f}\!:\!\operatorname{g} & + & \operatorname{g}\!:\!\operatorname{e} & + & \operatorname{h}\!:\!\operatorname{j} & + & \operatorname{i}\!:\!\operatorname{h} & + & \operatorname{j}\!:\!\operatorname{i} \\[4pt] \operatorname{g} & = & \operatorname{e}\!:\!\operatorname{g} & + & \operatorname{f}\!:\!\operatorname{e} & + & \operatorname{g}\!:\!\operatorname{f} & + & \operatorname{h}\!:\!\operatorname{i} & + & \operatorname{i}\!:\!\operatorname{j} & + & \operatorname{j}\!:\!\operatorname{h} \\[4pt] \operatorname{h} & = & \operatorname{e}\!:\!\operatorname{h} & + & \operatorname{f}\!:\!\operatorname{i} & + & \operatorname{g}\!:\!\operatorname{j} & + & \operatorname{h}\!:\!\operatorname{e} & + & \operatorname{i}\!:\!\operatorname{f} & + & \operatorname{j}\!:\!\operatorname{g} \\[4pt] \operatorname{i} & = & \operatorname{e}\!:\!\operatorname{i} & + & \operatorname{f}\!:\!\operatorname{j} & + & \operatorname{g}\!:\!\operatorname{h} & + & \operatorname{h}\!:\!\operatorname{g} & + & \operatorname{i}\!:\!\operatorname{e} & + & \operatorname{j}\!:\!\operatorname{f} \\[4pt] \operatorname{j} & = & \operatorname{e}\!:\!\operatorname{j} & + & \operatorname{f}\!:\!\operatorname{h} & + & \operatorname{g}\!:\!\operatorname{i} & + & \operatorname{h}\!:\!\operatorname{f} & + & \operatorname{i}\!:\!\operatorname{g} & + & \operatorname{j}\!:\!\operatorname{e} \end{array}\)

In (2) we consider the effects of each \(x\!\) in its practical bearing on contexts of the form \((y, \underline{~~}),\) as \(y\!\) ranges over \(G,\!\) and the effects are such that \(x\!\) takes \((y, \underline{~~})\) into \(yx,\!\) for \(y\!\) in \(G,\!\) all of which is 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 produces the regular post-representation of \(S_3,\!\) like so:

\(\begin{array}{*{13}{c}} \operatorname{e} & = & \operatorname{e}\!:\!\operatorname{e} & + & \operatorname{f}\!:\!\operatorname{f} & + & \operatorname{g}\!:\!\operatorname{g} & + & \operatorname{h}\!:\!\operatorname{h} & + & \operatorname{i}\!:\!\operatorname{i} & + & \operatorname{j}\!:\!\operatorname{j} \\[4pt] \operatorname{f} & = & \operatorname{e}\!:\!\operatorname{f} & + & \operatorname{f}\!:\!\operatorname{g} & + & \operatorname{g}\!:\!\operatorname{e} & + & \operatorname{h}\!:\!\operatorname{i} & + & \operatorname{i}\!:\!\operatorname{j} & + & \operatorname{j}\!:\!\operatorname{h} \\[4pt] \operatorname{g} & = & \operatorname{e}\!:\!\operatorname{g} & + & \operatorname{f}\!:\!\operatorname{e} & + & \operatorname{g}\!:\!\operatorname{f} & + & \operatorname{h}\!:\!\operatorname{j} & + & \operatorname{i}\!:\!\operatorname{h} & + & \operatorname{j}\!:\!\operatorname{i} \\[4pt] \operatorname{h} & = & \operatorname{e}\!:\!\operatorname{h} & + & \operatorname{f}\!:\!\operatorname{j} & + & \operatorname{g}\!:\!\operatorname{i} & + & \operatorname{h}\!:\!\operatorname{e} & + & \operatorname{i}\!:\!\operatorname{g} & + & \operatorname{j}\!:\!\operatorname{f} \\[4pt] \operatorname{i} & = & \operatorname{e}\!:\!\operatorname{i} & + & \operatorname{f}\!:\!\operatorname{h} & + & \operatorname{g}\!:\!\operatorname{j} & + & \operatorname{h}\!:\!\operatorname{f} & + & \operatorname{i}\!:\!\operatorname{e} & + & \operatorname{j}\!:\!\operatorname{g} \\[4pt] \operatorname{j} & = & \operatorname{e}\!:\!\operatorname{j} & + & \operatorname{f}\!:\!\operatorname{i} & + & \operatorname{g}\!:\!\operatorname{h} & + & \operatorname{h}\!:\!\operatorname{g} & + & \operatorname{i}\!:\!\operatorname{f} & + & \operatorname{j}\!:\!\operatorname{e} \end{array}\)

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.

Note 20

 

the way of heaven and earth
is to be long continued
in their operation
without stopping

  — i ching, hexagram 32

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

We made the observation that the shift operators \(\{ \operatorname{E}_{ij} \}\) form a transformation group that acts on the set of propositions of the form \(f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\) 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 (1842–1899), have turned out to be of critical utility in the solution of 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 barely 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.

Note 21

We've seen a couple of groups, \(V_4\!\) and \(S_3,\!\) represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called matrix representation of a group.

Recalling the manner of our acquaintance with the symmetric group \(S_3,\!\) we began with the bigraph (bipartite graph) picture of its natural representation as the set of all permutations or substitutions on the set \(X = \{ a, b, c \}.\!\)


\(\text{Permutation Substitutions in}~ \operatorname{Sym} \{ a, b, c \}\)
\(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{i}\) \(\operatorname{j}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] a & b & c \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] c & a & b \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] b & c & a \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] a & c & b \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] c & b & a \end{matrix}\)

\(\begin{matrix} a & b & c \\[3pt] \downarrow & \downarrow & \downarrow \\[6pt] b & a & c \end{matrix}\)


These permutations were then converted to relative form as logical sums of elementary relatives:

\(\begin{matrix} \operatorname{e} & = & a\!:\!a & + & b\!:\!b & + & c\!:\!c \\[4pt] \operatorname{f} & = & a\!:\!c & + & b\!:\!a & + & c\!:\!b \\[4pt] \operatorname{g} & = & a\!:\!b & + & b\!:\!c & + & c\!:\!a \\[4pt] \operatorname{h} & = & a\!:\!a & + & b\!:\!c & + & c\!:\!b \\[4pt] \operatorname{i} & = & a\!:\!c & + & b\!:\!b & + & c\!:\!a \\[4pt] \operatorname{j} & = & a\!:\!b & + & b\!:\!a & + & c\!:\!c \end{matrix}\)

From the relational representation of \(\operatorname{Sym} \{ a, b, c \} \cong S_3,\) one easily derives a linear representation of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other. Each of these linear transformations is in turn represented by a 2-dimensional array of coefficients in \(\mathbb{B},\) resulting in the following set of matrices for the group:


\(\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)\)
\(\operatorname{e}\) \(\operatorname{f}\) \(\operatorname{g}\) \(\operatorname{h}\) \(\operatorname{i}\) \(\operatorname{j}\)

\(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\)

\(\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}\)

\(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}\)

\(\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}\)

\(\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}\)

\(\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\)


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

\(\begin{bmatrix} a\!:\!a & a\!:\!b & a\!:\!c \\ b\!:\!a & b\!:\!b & b\!:\!c \\ c\!:\!a & c\!:\!b & c\!:\!c \end{bmatrix}\)

Note 22

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.

Note 23

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

Note 24

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)

Note 25

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

Note 26

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

Document History

Ontology List (Apr–Jul 2002)

  1. http://suo.ieee.org/ontology/msg04040.html
  2. http://suo.ieee.org/ontology/msg04041.html
  3. http://suo.ieee.org/ontology/msg04045.html
  4. http://suo.ieee.org/ontology/msg04046.html
  5. http://suo.ieee.org/ontology/msg04047.html
  6. http://suo.ieee.org/ontology/msg04048.html
  7. http://suo.ieee.org/ontology/msg04052.html
  8. http://suo.ieee.org/ontology/msg04054.html
  9. http://suo.ieee.org/ontology/msg04055.html
  10. http://suo.ieee.org/ontology/msg04067.html
  11. http://suo.ieee.org/ontology/msg04068.html
  12. http://suo.ieee.org/ontology/msg04069.html
  13. http://suo.ieee.org/ontology/msg04070.html
  14. http://suo.ieee.org/ontology/msg04072.html
  15. http://suo.ieee.org/ontology/msg04073.html
  16. http://suo.ieee.org/ontology/msg04074.html
  17. http://suo.ieee.org/ontology/msg04077.html
  18. http://suo.ieee.org/ontology/msg04079.html
  19. http://suo.ieee.org/ontology/msg04080.html
  20. http://suo.ieee.org/ontology/msg04268.html
  21. http://suo.ieee.org/ontology/msg04269.html
  22. http://suo.ieee.org/ontology/msg04272.html
  23. http://suo.ieee.org/ontology/msg04273.html
  24. http://suo.ieee.org/ontology/msg04290.html

Inquiry List (May & Jul 2004)

  1. http://stderr.org/pipermail/inquiry/2004-May/001400.html
  2. http://stderr.org/pipermail/inquiry/2004-May/001401.html
  3. http://stderr.org/pipermail/inquiry/2004-May/001402.html
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  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
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  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 (May & Jul 2004)

  1. http://forum.wolframscience.com/showthread.php?postid=1282#post1282
  2. http://forum.wolframscience.com/showthread.php?postid=1285#post1285
  3. http://forum.wolframscience.com/showthread.php?postid=1289#post1289
  4. http://forum.wolframscience.com/showthread.php?postid=1292#post1292
  5. http://forum.wolframscience.com/showthread.php?postid=1293#post1293
  6. http://forum.wolframscience.com/showthread.php?postid=1294#post1294
  7. http://forum.wolframscience.com/showthread.php?postid=1296#post1296
  8. http://forum.wolframscience.com/showthread.php?postid=1299#post1299
  9. 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
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