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{{DISPLAYTITLE:Dynamics And Logic}}
 
{{DISPLAYTITLE:Dynamics And Logic}}
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'''Note.'''  Many problems with the sucky MathJax on this page.  The parser apparently reads 4 tildes inside math brackets the way it would in the external wiki environment, in other words, as signature tags. [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 18:00, 5 December 2014 (UTC)
 +
 
==Note 1==
 
==Note 1==
  
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:* DATA.  http://forum.wolframscience.com/showthread.php?threadid=228
 
:* DATA.  http://forum.wolframscience.com/showthread.php?threadid=228
  
One of the first things that you can do, once you have a moderately functional calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.
+
One of the first things that you can do, once you have a moderately efficient 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:
 
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" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
<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>".
+
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>
 
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>
Line 33: Line 26:
 
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.
 
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" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph Existential True.jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|                                                |
 
|                        @                        |
 
|                                                |
 
o-------------------------------------------------o
 
|                     true                      |
 
o-------------------------------------------------o
 
</pre>
 
 
|}
 
|}
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph Existential False.jpg|500px]]
<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:
+
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 in the following Figure:
  
 
{| align="center" cellpadding="10"
 
{| align="center" cellpadding="10"
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Don't think about it &mdash; just compute:
 
Don't think about it &mdash; just compute:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
<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:
+
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax.  This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph (P,dP).jpg|500px]]
<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:
+
Next question:  What is the difference between the value of the
 +
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course?  Here is the appropriate formulation:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|                    p    dp                    |
 
|                      o---o                      |
 
|                      \ /                      |
 
|                        @                        |
 
|                                                |
 
o-------------------------------------------------o
 
|                    (p, dp)                     |
 
o-------------------------------------------------o
 
</pre>
 
 
|}
 
|}
  
Next questionWhat is the difference between the value of the
+
There is one thing that I ought to mention at this pointComputed 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.
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%"
+
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" |
 
<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.
+
{| align="center" cellpadding="10"
 
+
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
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:
 
And this is equivalent to the following graph:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                 |
 
|                    dp  dq                    |
 
|                      o  o                      |
 
|                      \ /                      |
 
|                        o                        |
 
|                        |                        |
 
|                        @                        |
 
|                                                |
 
o-------------------------------------------------o
 
|                  ((dp) (dq))                   |
 
o-------------------------------------------------o
 
</pre>
 
 
|}
 
|}
  
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We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
 
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" cellpadding="10" style="text-align:center; width:90%"
| align="center" |
+
|
<math>p ~\operatorname{and}~ q \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q</math>
+
<math>\begin{matrix}
|-
+
p ~\operatorname{and}~ q
| align="center" |
+
& \quad &
<pre>
+
\xrightarrow{\quad\operatorname{Diff}\quad}
o-------------------------------------------------o
+
& \quad &
|                                                |
+
\operatorname{d}p ~\operatorname{or}~ \operatorname{d}q
|                                    dp  dq      |
+
\end{matrix}\!</math>
|                                     o  o      |
+
|}
|                                      \ /        |
+
 
|                                      o        |
+
{| align="center" cellpadding="10"
|        p q                            |        |
+
| [[Image:Cactus Graph PQ Diff ((dP)(dQ)).jpg|500px]]
|        @          --Diff-->          @        |
 
|                                                |
 
o-------------------------------------------------o
 
|        p q        --Diff-->    ((dp) (dq))   |
 
o-------------------------------------------------o
 
</pre>
 
 
|}
 
|}
  
Line 244: Line 110:
 
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.
 
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%"
+
{| align="center" cellpadding="10" 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>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \operatorname{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>
 
|-
 
|-
Line 256: Line 122:
 
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 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%"
+
{| align="center" cellpadding="10" width="90%"
 
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
 
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
 
|-
 
|-
Line 266: Line 132:
 
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:
 
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%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{array}{rcc}
 
<math>\begin{array}{rcc}
 
\operatorname{E}X & = & X \times \operatorname{d}X
 
\operatorname{E}X & = & X \times \operatorname{d}X
\end{array}</math>
+
\end{array}\!</math>
 
|}
 
|}
  
 
where:
 
where:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{array}{rcc}
 
<math>\begin{array}{rcc}
Line 293: Line 159:
 
& = &
 
& = &
 
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
 
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
\end{array}</math>
+
\end{array}\!</math>
 
|}
 
|}
  
The interpretations of these new symbols can be diverse, but the easiest
+
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>".
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.
 
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.
Line 312: Line 177:
 
following formula:
 
following formula:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" style="text-align:center"
| align="center" |
+
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
Line 325: Line 190:
 
In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is computed as follows:
 
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" cellpadding="10" style="text-align:center"
| align="center" |
+
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
 
\operatorname{E}f(p, q, \operatorname{d}p, \operatorname{d}q)
Line 335: Line 200:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
<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:
 
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" cellpadding="10" style="text-align:center"
| align="center" |
+
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
 
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
Line 367: Line 218:
 
In the example <math>f(p, q) = pq,\!</math> the difference <math>\operatorname{D}f</math> is computed as follows:
 
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" cellpadding="10" style="text-align:center"
| align="center" |
+
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
 
\operatorname{D}f(p, q, \operatorname{d}p, \operatorname{d}q)
Line 377: Line 228:
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
<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:
 
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" cellpadding="10" style="text-align:center"
| 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>
 
<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]]
[[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph PQ Difference Conj At Conj.jpg|500px]]
[[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:
 
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" cellpadding="10" style="text-align:center"
| align="center" |
+
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
 
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
 
\operatorname{d}p ~\texttt{(} \operatorname{d}q \texttt{)}
Line 439: Line 267:
 
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:
 
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" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph Df = ((P,dP)(Q,dQ),PQ).jpg|500px]]
<pre>
+
|-
o-------------------------------------------------o
+
| [[Image:Cactus Graph Df@PQ = ((dP)(dQ)).jpg|500px]]
|                                                |
 
|            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" |
+
| [[Image:Cactus Graph Df@P(Q) = (dP)dQ.jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|                dp    dq                        |
 
|            o---o o---o                        |
 
|              \  | |  /                          |
 
|              \ | | /                          |
 
|                \| |/                            |
 
|                o=o-----------o                |
 
|                  \          /                  |
 
|                  \        /                  |
 
|                    \      /                    |
 
|                    \    /                    |
 
|                      \  /                      |
 
|                      \ /                      |
 
|                        @                       |
 
|                                                |
 
o-------------------------------------------------o
 
| Df|pq =          ((dp) (dq))                   |
 
o-------------------------------------------------o
 
</pre>
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Df@(P)Q = dP(dQ).jpg|500px]]
<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" |
+
| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]
<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:
 
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" cellpadding="10"
| align="center" |
+
| [[Image:Cactus Graph Lobe Rule.jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|  e                                              |
 
| o-o-o-...-o-o-o                                |
 
|  \          /                                |
 
|    \        /                                  |
 
|    \      /                                  |
 
|      \    /                          e        |
 
|      \  /                          o        |
 
|        \ /                            |        |
 
|        @              =              @        |
 
|                                                |
 
o-------------------------------------------------o
 
|  (e, , ... , , )      =            (e)        |
 
o-------------------------------------------------o
 
</pre>
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Spike Rule.jpg|500px]]
<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>
 
 
|}
 
|}
  
Line 610: Line 295:
 
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::
 
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"
+
{| align="center" cellpadding="10"
 
|
 
|
 
<math>\begin{array}{rcccc}
 
<math>\begin{array}{rcccc}
Line 635: Line 320:
 
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.
 
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"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
 
| [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
|}
+
|-
 
 
{| align="center" cellpadding="10"
 
 
|
 
|
 
<math>\begin{array}{rcccccc}
 
<math>\begin{array}{rcccccc}
Line 664: Line 347:
 
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
 
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
  
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{array}{lccl}
 
<math>\begin{array}{lccl}
Line 689: Line 372:
 
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:
 
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%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 704: Line 387:
 
In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\operatorname{E}f</math> is formulated as follows:
 
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%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{array}{l}
 
<math>\begin{array}{l}
Line 717: Line 400:
 
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:
 
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%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
Line 734: Line 417:
 
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:
 
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" cellpadding="10" style="text-align:center"
| align="center" |
+
| [[Image:Cactus Graph Ef = (P,dP)(Q,dQ).jpg|500px]]
<pre>
+
|-
o-------------------------------------------------o
+
| [[Image:Cactus Graph Ef@PQ = (dP)(dQ).jpg|500px]]
|                                                 |
 
|                  p  dp q  dq                  |
 
|                  o---o o---o                  |
 
|                    \  | |  /                    |
 
|                    \ | | /                    |
 
|                      \| |/                      |
 
|                      @=@                      |
 
|                                                 |
 
o-------------------------------------------------o
 
| Ef =           (p, dp) (q, dq)                 |
 
o-------------------------------------------------o
 
</pre>
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Ef@P(Q) = (dP)dQ.jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|                      dp    dq                  |
 
|                  o---o o---o                  |
 
|                    \  | |  /                    |
 
|                    \ | | /                    |
 
|                      \| |/                      |
 
|                      @=@                       |
 
|                                                |
 
o-------------------------------------------------o
 
| Ef|pq =            (dp) (dq)                   |
 
o-------------------------------------------------o
 
</pre>
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Ef@(P)Q = dP(dQ).jpg|500px]]
<pre>
 
o-------------------------------------------------o
 
|                                                |
 
|                        o                      |
 
|                      dp |  dq                  |
 
|                  o---o o---o                  |
 
|                    \  | |  /                    |
 
|                    \ | | /                    |
 
|                      \| |/                      |
 
|                      @=@                       |
 
|                                                |
 
o-------------------------------------------------o
 
| Ef|p(q) =         (dp) dq                    |
 
o-------------------------------------------------o
 
</pre>
 
 
|-
 
|-
| align="center" |
+
| [[Image:Cactus Graph Ef@(P)(Q) = dP dQ.jpg|500px]]
<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>
 
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%"
+
{| align="center" cellpadding="10" width="90%"
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
Line 838: Line 448:
 
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>
 
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"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
 
| [[Image:Directed Graph PQ Enlargement Conj.jpg|500px]]
|}
+
|-
 
 
{| align="center" cellpadding="10"
 
 
|
 
|
 
<math>\begin{array}{rcccccc}
 
<math>\begin{array}{rcccccc}
Line 966: Line 574:
 
(p)~q~
 
(p)~q~
 
\\[4pt]
 
\\[4pt]
(p)~~~
+
(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
 
\\[4pt]
 
\\[4pt]
 
~p~(q)
 
~p~(q)
 
\\[4pt]
 
\\[4pt]
~~~(q)
+
[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)
 
\\[4pt]
 
\\[4pt]
 
(p,~q)
 
(p,~q)
Line 1,073: Line 681:
 
((p,~q))
 
((p,~q))
 
\\[4pt]
 
\\[4pt]
~~~~~q~~
+
17:54, 5 December 2014 (UTC)q~~
 
\\[4pt]
 
\\[4pt]
 
~(p~(q))
 
~(p~(q))
 
\\[4pt]
 
\\[4pt]
~~p~~~~~
+
~~p17:54, 5 December 2014 (UTC)
 
\\[4pt]
 
\\[4pt]
 
((p)~q)~
 
((p)~q)~
Line 2,785: Line 2,393:
 
|}
 
|}
  
<pre>
+
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:
Therefore, if we evaluate Ef at particular values of dp and dq,
 
for example, dp = i and dq = j, where i, j are in B, we obtain:
 
  
  E_ij : (X -> B) ->  (X -> B)
+
{| 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>
 +
|}
  
  E_ij :   f     ->   E_ij f
+
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:
  
  E_ij f
+
<br>
  
  = Ef | <dp = i, dq = j>
+
{| 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>
 +
|}
  
  =  f<p + i, q + j>
+
<br>
  
  = f<(p, i), (q, j)>
+
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.
  
The notation is a little bit awkward, but the data of the Table should
+
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 groupIn 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!
make the sense clear.  The important thing to observe is that E_ij has
 
the effect of transforming each proposition f : X -> B into some other
 
proposition f' : X -> BAs it happens, the action is one-to-one and
 
onto for each E_ij, so the gang of four operators {E_ij : i, j in B}
 
is an example of what is called a "transformation group" on the set
 
of sixteen propositions.  Bowing to a longstanding linear and local
 
tradition, I will therefore redub the four elements of this group
 
as T_00, T_01, T_10, T_11, to bear in mind their transformative
 
character, or nature, as the case may beAbstractly viewed,
 
this group of order four has the following operation table:
 
  
o----------o----------o----------o----------o----------o
+
==Note 11==
|          %          |          |          |          |
 
|    *    %  T_00  |  T_01  |  T_10  |  T_11  |
 
|          %          |          |          |          |
 
o==========o==========o==========o==========o==========o
 
|          %          |          |          |          |
 
|  T_00  %  T_00  |  T_01  |  T_10  |  T_11  |
 
|          %          |          |          |          |
 
o----------o----------o----------o----------o----------o
 
|          %          |          |          |          |
 
|  T_01  %  T_01  |  T_00  |  T_11  |  T_10  |
 
|          %          |          |          |          |
 
o----------o----------o----------o----------o----------o
 
|          %          |          |          |          |
 
|  T_10  %  T_10  |  T_11  |  T_00  |  T_01  |
 
|          %          |          |          |          |
 
o----------o----------o----------o----------o----------o
 
|          %          |          |          |          |
 
|  T_11  %  T_11  |  T_10  |  T_01  |  T_00  |
 
|          %          |          |          |          |
 
o----------o----------o----------o----------o----------o
 
  
It happens that there are just two possible groups of 4 elements.
+
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.
One is the cyclic group Z_4 (German "Zyklus"), which this is not.
 
The other is Klein's four-group V_4 (German "Vier"), which it is.
 
  
More concretely viewed, the group as a whole pushes the set
+
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 lightSoon 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''.
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 groupThere
 
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 groupIn this instance, T_00 operates as the
 
group identity, fixing all 16 propositions, while the other
 
three group elements fix 4 propositions each, and so we get:
 
Number of orbits  =  (4 + 4 + 4 + 16) / 4  =  7. -- Amazing!
 
</pre>
 
  
==Note 11==
+
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.
  
<pre>
+
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:
We have been contemplating functions of the type f : X -> B,
 
studying the action of the operators E and D on this family.
 
These functions, that we may identify for our present aims
 
with propositions, inasmuch as they capture their abstract
 
forms, are logical analogues of "scalar potential fields".
 
These are the sorts of fields that are so picturesquely
 
presented in elementary calculus and physics textbooks
 
by images of snow-covered hills and parties of skiers
 
who trek down their slopes like least action heroes.
 
The analogous scene in propositional logic presents
 
us with forms more reminiscent of plateaunic idylls,
 
being all plains at one of two levels, the mesas of
 
verity and falsity, as it were, with nary a niche
 
to inhabit between them, restricting our options
 
for a sporting gradient of downhill dynamics to
 
just one of two, standing still on level ground
 
or falling off a bluff.
 
  
We are still working well within the logical analogue of the
+
{| align="center" cellpadding="10"
classical finite difference calculus, taking in the novelties
+
| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]
that the logical transmutation of familiar elements is able to
+
|}
bring to light.  Soon we will take up several different notions
 
of approximation relationships that may be seen to organize the
 
space of propositions, and these will allow us to define several
 
different forms of differential analysis applying to propositions.
 
In time we will find reason to consider more general types of maps,
 
having concrete types of the form X_1 x ... x X_k -> Y_1 x ... x Y_n
 
and abstract types B^k -> B^n.  We will think of these mappings as
 
transforming universes of discourse into themselves or into others,
 
in short, as "transformations of discourse".
 
  
Before we continue with this intinerary, however, I would like
+
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>
to highlight another sort of "differential aspect" that concerns
 
the "boundary operator" or the "marked connective" that serves as
 
one of a pair of basic connectives in the cactus language for ZOL.
 
  
Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B
+
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>
and abstract type f : B^3 -> B that is written as "(p, q, r)" in the
 
cactus syntax.  Taken as an assertion in what C.S. Peirce called the
 
"existential interpretation", the so-called boundary form "(p, q, r)"
 
asserts that one and only one of the propositions p, q, r is false.
 
It is instructive to consider this assertion in relation to the
 
conjunction "p q r" of the same propositions.  A venn diagram
 
for the boundary form (p, q, r) is shown in Figure 11.
 
 
 
o-----------------------------------------------------------o
 
|                                                          |
 
|                                                          |
 
|                      o-------------o                      |
 
|                    /              \                    |
 
|                    /                \                    |
 
|                  /                  \                  |
 
|                  /                    \                  |
 
|                /                      \                |
 
|                o                        o                |
 
|                |                        |                |
 
|                |            P            |                |
 
|                |                        |                |
 
|                |                        |                |
 
|                |                        |                |
 
|            o--o----------o  o----------o--o            |
 
|            /    \%%%%%%%%%%\ /%%%%%%%%%%/    \            |
 
|          /      \%%%%%%%%%%o%%%%%%%%%%/      \          |
 
|          /        \%%%%%%%%/ \%%%%%%%%/        \          |
 
|        /          \%%%%%%/  \%%%%%%/          \        |
 
|        /            \%%%%/    \%%%%/            \        |
 
|      o              o--o-------o--o              o      |
 
|      |                |%%%%%%%|                |      |
 
|      |                |%%%%%%%|                |      |
 
|      |                |%%%%%%%|                |      |
 
|      |        Q        |%%%%%%%|        R        |      |
 
|      |                |%%%%%%%|                |      |
 
|      o                o%%%%%%%o                o      |
 
|        \                \%%%%%/                /        |
 
|        \                \%%%/                /        |
 
|          \                \%/                /          |
 
|          \                o                /          |
 
|            \              / \              /            |
 
|            o-------------o  o-------------o            |
 
|                                                          |
 
|                                                          |
 
o-----------------------------------------------------------o
 
Figure 11.  Boundary Form (p, q, r)
 
 
 
In relation to the center cell indicated by the conjunction pqr
 
the region indicated by (p, q, r) is comprised of the "adjacent"
 
or the "bordering" cells.  Thus they are the cells that are just
 
across the boundary of the center cell, as if reached by way of
 
Leibniz's "minimal changes" from the point of origin, here, pqr.
 
 
 
More generally speaking, in a k-dimensional universe of discourse
 
that is based on the "alphabet" of features !X! = {x_1, ..., x_k},
 
the same form of boundary relationship is manifested for any cell
 
of origin that one might choose to indicate, say, by means of the
 
conjunction of positive and negative basis features "u_1 ... u_k",
 
where u_j = x_j or u_j = (x_j), for j = 1 to k.  The proposition
 
(u_1, ..., u_k) indicates the disjunctive region consisting of
 
the cells that are "just next door" to the cell u_1 ... u_k.
 
</pre>
 
  
 
==Note 12==
 
==Note 12==
  
<pre>
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
| Consider what effects that might conceivably have
 
| practical bearings you conceive the objects of your
 
| conception to have.  Then, your conception of those
 
| effects is the whole of your conception of the object.
 
 
|
 
|
| C.S. Peirce, "Maxim of Pragmaticism", 'Collected Papers', CP 5.438
+
<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,
+
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.
while we have an object example of a mathematical group fresh in mind,
 
is the relationship between the pragmatic maxim and what are commonly
 
known in mathematics as "representation principles".  As it turns out,
 
with regard to its formal characteristics, the pragmatic maxim unites
 
the aspects of a representation principle with the attributes of what
 
would ordinarily be known as a "closure principle".  We will consider
 
the form of closure that is invoked by the pragmatic maxim on another
 
occasion, focusing here and now on the topic of group representations.
 
  
Let us return to the example of the so-called "four-group" V_4.
+
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:
We encountered this group in one of its concrete representations,
 
namely, as a "transformation group" that acts on a set of objects,
 
in this particular case a set of sixteen functions or propositions.
 
Forgetting about the set of objects that the group transforms among
 
themselves, we may take the abstract view of the group's operational
 
structure, say, in the form of the group operation table copied here:
 
  
o-------o-------o-------o-------o-------o
+
<br>
|      %      |      |      |      |
 
|  *  %  e  |  f  |  g  |  h  |
 
|      %      |      |      |      |
 
o=======o=======o=======o=======o=======o
 
|      %      |      |      |      |
 
|  e  %  e  |  f  |  g  |  h  |
 
|      %      |      |      |      |
 
o-------o-------o-------o-------o-------o
 
|      %      |      |      |      |
 
|  f  %  f  |  e  |  h  |  g  |
 
|      %      |      |      |      |
 
o-------o-------o-------o-------o-------o
 
|      %      |      |      |      |
 
|  g  %  g  |  h  |  e  |  f  |
 
|      %      |      |      |      |
 
o-------o-------o-------o-------o-------o
 
|      %      |      |      |      |
 
|  h  %  h  |  g  |  f  |  e  |
 
|      %      |      |      |      |
 
o-------o-------o-------o-------o-------o
 
  
This table is abstractly the same as, or isomorphic to, the versions
+
{| 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%"
with the E_ij operators and the T_ij transformations that we took up
+
|- style="height:50px"
earlier.  That is to say, the story is the same, only the names have
+
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
been changed.  An abstract group can have a variety of significantly
+
| width="22%" style="border-bottom:1px solid black" |
and superficially different representations.  But even after we have
+
<math>\operatorname{e}</math>
long forgotten the details of any particular representation there is
+
| width="22%" style="border-bottom:1px solid black" |
a type of concrete representations, called "regular representations",
+
<math>\operatorname{f}</math>
that are always readily available, as they can be generated from the
+
| width="22%" style="border-bottom:1px solid black" |
mere data of the abstract operation table itself.
+
<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>
 +
|}
  
For example, select a group element from the top margin of the Table,
+
<br>
and "consider its effects" on each of the group elements as they are
 
listed along the left margin.  We may record these effects as Peirce
 
usually did, as a logical "aggregate" of elementary dyadic relatives,
 
that is to say, a disjunction or a logical sum whose terms represent
 
the ordered pairs of <input : output> transactions that are produced
 
by each group element in turn.  This yields what is usually known as
 
one of the "regular representations" of the group, specifically, the
 
"first", the "post-", or the "right" regular representation.  It has
 
long been conventional to organize the terms in the form of a matrix:
 
  
Reading "+" as a logical disjunction:
+
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.
  
  G = e +  f  +  g  + h,
+
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:
  
And so, by expanding effects, we get:
+
Reading "<math>+\!</math>" as a logical disjunction:
  
  G =
+
{| align="center" cellpadding="6" width="90%"
 +
| align="center" |
 +
<math>\begin{matrix}
 +
\operatorname{G}
 +
& = & \operatorname{e}
 +
& + & \operatorname{f}
 +
& + & \operatorname{g}
 +
& + & \operatorname{h}
 +
\end{matrix}</math>
 +
|}
  
  e:e  +  f:f  +  g:g  +  h:h  +
+
And so, by expanding effects, we get:
  
  e:f + f:e + g:h + h:g +
+
{| align="center" cellpadding="6" width="90%"
 
+
| align="center" |
  e:g + f:h + g:e + h:f +
+
<math>\begin{matrix}
 
+
\operatorname{G}
  e:h + f:g + g:f + h:e
+
& = & \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.
 
More on the pragmatic maxim as a representation principle later.
</pre>
 
  
 
==Note 13==
 
==Note 13==
  
<pre>
+
The above-mentioned fact about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:
The above-mentioned fact about the regular representations
 
of a group is universally known as "Cayley's Theorem".  It
 
is usually stated in the form:  "Every group is isomorphic
 
to a subgroup of Aut(X), where X is a suitably chosen set
 
and Aut(X) is the group of its automorphisms".  There is
 
in Peirce's early papers a considerable generalization
 
of the concept of regular representations to a broad
 
class of relational algebraic systems.  The crux of
 
the whole idea can be summed up as follows:
 
  
  Contemplate the effects of the symbol
+
{| align="center" cellpadding="6" width="90%"
  whose meaning you wish to investigate
+
| 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>.
  as they play out on all the stages of
+
|}
  conduct on which you have the ability
 
  to imagine that symbol playing a role.
 
  
This idea of definition by way of context transforming operators
+
There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papersThe crux of the whole idea is this:
is basically the same as Jeremy Bentham's notion of "paraphrasis",
 
a "method of accounting for fictions by explaining various purported
 
terms away" (Quine, in Van Heijenoort, 'From Frege to Gödel', p. 216).
 
Today we'd call these constructions "term models"This, again, is
 
the big idea behind Schönfinkel's combinators {S, K, I}, and hence
 
of lambda calculus, and I reckon you all know where that leads.
 
</pre>
 
  
==Note 14==
+
{| 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.
 +
|}
  
<pre>
+
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.
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
+
==Note 14==
to pick up the conventions that he used there, and then rewrite my
 
account of regular representations in a way that conforms to those.
 
  
Peirce expresses the action of an "elementary dual relative" like so:
+
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] A:B be taken to denote
+
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.
| the elementary relative which
 
| multiplied into B gives A.
 
|
 
| Peirce, 'Collected Papers', CP 3.123.
 
  
Peirce is well aware that it is not at all necessary to arrange the
+
Peirce describes the action of an "elementary dual relative" in this way:
elementary relatives of a relation into arrays, matrices, or tables,
 
but when he does so he tends to prefer organizing 2-adic relations
 
in the following manner:
 
  
  a:b  +  a:b  +  a:c  +
+
{| 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.
 +
|}
  
  b:a +  b:b  +  b:c  +
+
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:
  
  c:a c:b c:c
+
{| 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 X = {a, b, c}, suppose that
+
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\prime\prime}</math> and
we have the 2-adic relative term m = "marker for" 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:
the associated 2-adic relation M c X x X, the general
 
pattern of whose common structure is represented by
 
the following matrix:
 
  
  M =
+
{| 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>
 +
|}
  
  M_aa a:a +  M_ab a:b  +  M_ac a:c  +
+
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:
  
  M_ba b:a +  M_bb b:b +  M_bc b:c +
+
{| 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>
 +
|}
  
  M_ca c:a +  M_cb c:b  +  M_cc c:c
+
By way of making up a concrete example, let us say that <math>\mathit{m}\!</math> or <math>M\!</math> is given as follows:
  
It has long been customary to omit the implicit plus signs
+
{| align="center" cellpadding="6" width="90%"
in these matrical displays, but I have restored them here
+
| align="center" |
simply as a way of separating terms in this blancophage
+
<math>\begin{array}{l}
web format.
+
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>
 +
|}
  
For at least a little while, I will make explicit
+
In sum, then, the relative term <math>\mathit{m}\!</math> and the relation <math>M\!</math> are both represented by the following matrix:
the distinction between a "relative term" like m
 
and a "relation" like M c X x X, but it is best
 
to think of both of these entities as involving
 
different applications of the same information,
 
and so we could just as easily write this form:
 
  
  m  =
+
{| align="center" cellpadding="6" width="90%"
 
+
| align="center" |
  m_aa a:a +  m_ab a:b +  m_ac a:c +
+
<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>
 +
|}
  
  m_ba b:a  +  m_bb b:b  +  m_bc b:c  +
+
I think this much will serve to fix notation and set up the remainder of the discussion.
  
  m_ca c:a  +  m_cb c:b  +  m_cc c:c
+
==Note 15==
  
By way of making up a concrete example,
+
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:
let us say that M is given as follows:
 
  
  a is a marker for a
+
{| 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>
 +
|}
  
  a is a marker for b
+
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:
  
  b is a marker for b
+
{| 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>
 +
|}
  
  b is a marker for c
+
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{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 17:54, 5 December 2014 (UTC)}\, {}^{\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.
  
  c is a marker for c
+
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".
  
  c is a marker for a
+
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:
  
In sum, we have this matrix:
+
{| align="center" cellpadding="6"
 +
|
 +
<math>\begin{Bmatrix}
 +
a & b & c
 +
\\
 +
b & c & a
 +
\end{Bmatrix}</math>
 +
|}
  
  M  =
+
This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327).
  
  1 a:a  +  1 a:b  +  0 a:c  +
+
==Note 16==
  
  0 b:+  1 b:b  +  1 b:c  +
+
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.
  
  1 c:+  0 c:b  +  1 c:c
+
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.
  
I think that will serve to fix notation
+
Here is is the operation table of <math>V_4\!</math> once again:
and set up the remainder of the account.
 
</pre>
 
  
==Note 15==
+
<br>
  
<pre>
+
{| 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%"
In Peirce's time, and even in some circles of mathematics today,
+
|+ <math>\text{Klein Four-Group}~ V_4</math>
the information indicated by the elementary relatives (i:j), as
+
|- style="height:50px"
i, j range over the universe of discourse, would be referred to
+
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
as the "umbral elements" of the algebraic operation represented
+
| width="22%" style="border-bottom:1px solid black" |
by the matrix, though I seem to recall that Peirce preferred to
+
<math>\operatorname{e}</math>
call these terms the "ingredients".  When this ordered basis is
+
| width="22%" style="border-bottom:1px solid black" |
understood well enough, one will tend to drop any mention of it
+
<math>\operatorname{f}</math>
from the matrix itself, leaving us nothing but these bare bones:
+
| 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>
 +
|}
  
  M  =
+
<br>
  
  1  1  0
+
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>
  
  0  1  1
+
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:
  
  1  0  1
+
{| 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>
 +
|}
  
However the specification may come to be written, this
+
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:
is all just convenient schematics for stipulating that:
 
  
  M  = a:a  +  b:b  +  c:c  +  a:b  +  b:c  +  c:a
+
{| 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>
 +
|}
  
Recognizing !1! = a:a + b:b + c:c as the identity transformation,
+
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:
the 2-adic relative term m = "marker for" can be represented as an
 
element !1! + a:b + b:c + c:a of the so-called "group ring", all of
 
which makes this element just a special sort of linear transformation.
 
  
Up to this point, we are still reading the elementary relatives
+
{| align="center" cellpadding="6" width="90%"
of the form i:j in the way that Peirce customarily read them in
+
| align="center" |
logical contexts: i is the relate, j is the correlate, and in
+
<math>\begin{matrix}
our current example we reading i:j, or more exactly, m_ij = 1,
+
\operatorname{e}
to say that i is a marker for j.  This is the mode of reading
+
& = & \operatorname{e}\!:\!\operatorname{e}
that we call "multiplying on the left".
+
& + & \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 the algebraic, permutational, or transformational contexts of
+
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:
application, however, Peirce converts to the alternative mode of
 
reading, although still calling i the relate and j the correlate,
 
the elementary relative i:j now means that i gets changed into j.
 
In this scheme of reading, the transformation a:b + b:c + c:a is
 
a permutation of the aggregate $1$ = a + b + c, or what we would
 
now call the set {a, b, c}, in particular, it is the permutation
 
that is otherwise notated as:
 
  
  ( a b c )
+
{| align="center" cellpadding="6" width="90%"
  <       >
+
| align="center" |
  ( b c a )
+
<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>
 +
|}
  
This is consistent with the convention that Peirce uses in
+
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.
the paper "On a Class of Multiple Algebras" (CP 3.324-327).
 
</pre>
 
  
==Note 16==
+
==Note 17==
  
<pre>
+
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) groupThis 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>
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 ideasThe basic idea is to replace
 
the question of "What it is", which modest people comprehend
 
is far beyond their powers to answer any time soon, with the
 
question of "What it does", which most people know at least
 
a modicum about.
 
  
In the case of regular representations of groups we found
+
<br>
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:
+
{| 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>
o-------o-------o-------o-------o-------o
+
|- style="background:#f0f0ff"
|       %       |      |      |      |
+
| width="16%" | <math>\operatorname{e}</math>
|   *  %   e  |   f   |  g  |  h  |
+
| width="16%" | <math>\operatorname{f}</math>
|       %       |       |       |      |
+
| width="16%" | <math>\operatorname{g}</math>
o=======o=======o=======o=======o=======o
+
| width="16%" | <math>\operatorname{h}</math>
|      %       |      |      |      |
+
| width="16%" | <math>\operatorname{i}</math>
|   e  %   e  |  f  |  g  |  h  |
+
| width="16%" | <math>\operatorname{j}</math>
|       %       |       |      |       |
+
|-
o-------o-------o-------o-------o-------o
+
|
|      %      |      |      |      |
+
<math>\begin{matrix}
|  f  %  f  |  e  |  h  |  g  |
+
a & b & c
|      %      |      |      |      |
+
\\[3pt]
o-------o-------o-------o-------o-------o
+
\downarrow & \downarrow & \downarrow
|      %      |      |      |      |
+
\\[6pt]
|  g  %  g  |  h  |  e  |  f  |
+
a & b & c
|      %      |      |      |      |
+
\end{matrix}</math>
o-------o-------o-------o-------o-------o
+
|
|       %      |      |      |      |
+
<math>\begin{matrix}
|  h  %  h  |  g  |  f  |  e  |
+
a & b & c
|      %      |      |      |      |
+
\\[3pt]
o-------o-------o-------o-------o-------o
+
\downarrow & \downarrow & \downarrow
 
+
\\[6pt]
A group operation table is really just a device for recording
+
c & a & b
a certain 3-adic relation, specifically, the set of 3-tuples
+
\end{matrix}</math>
of the form <x, y, z> that satisfy the equation x * y = z,
+
|
where the sign '*' that indicates the group operation is
+
<math>\begin{matrix}
frequently omitted in contexts where it is understood.
+
a & b & c
 
+
\\[3pt]
In the case of V_4 = (G, *), where G is the "underlying set"
+
\downarrow & \downarrow & \downarrow
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
+
\\[6pt]
whose triples are listed below:
+
b & c & a
 
+
\end{matrix}</math>
  e:e:e
+
|
  e:f:f
+
<math>\begin{matrix}
  e:g:g
+
a & b & c
  e:h:h
+
\\[3pt]
 
+
\downarrow & \downarrow & \downarrow
  f:e:f
+
\\[6pt]
  f:f:e
+
a & c & b
  f:g:h
+
\end{matrix}</math>
  f:h:g
+
|
 
+
<math>\begin{matrix}
  g:e:g
+
a & b & c
  g:f:h
+
\\[3pt]
  g:g:e
+
\downarrow & \downarrow & \downarrow
  g:h:f
+
\\[6pt]
 
+
c & b & a
  h:e:h
+
\end{matrix}</math>
  h:f:g
+
|
  h:g:f
+
<math>\begin{matrix}
  h:h:e
+
a & b & c
 +
\\[3pt]
 +
\downarrow & \downarrow & \downarrow
 +
\\[6pt]
 +
b & a & c
 +
\end{matrix}</math>
 +
|}
  
It is part of the definition of a group that the 3-adic
+
<br>
relation L c G^3 is actually a function L : G x G -> G.
 
It is from this functional perspective that we can see
 
an easy way to derive the two regular representations.
 
  
Since we have a function of the type L : G x G -> G,
+
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
we can define a couple of substitution operators:
 
  
1.  Sub(x, <_, y>) puts any specified x into
+
{| align="center" cellpadding="10" style="text-align:center"
    the empty slot of the rheme <_, y>, with
+
| <math>\text{Symmetric Group}~ S_3</math>
    the effect of producing the saturated
+
|-
    rheme <x, y> that evaluates to xy.
+
| [[Image:Symmetric Group S(3).jpg|500px]]
 +
|}
  
2.  Sub(x, <y, _>) puts any specified x into
+
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).
    the empty slot of the rheme <y, _>, with
 
    the effect of producing the saturated
 
    rheme <y, x> that evaluates to yx.
 
  
In (1), we consider the effects of each x in its
+
==Note 18==
practical bearing on contexts of the form <_, y>,
 
as y ranges over G, and the effects are such that
 
x takes <_, y> into xy, for y in G, all of which
 
is summarily notated as x = {<y : xy> : y in G}.
 
The pairs <y : xy> can be found by picking an x
 
from the left margin of the group operation table
 
and considering its effects on each y in turn as
 
these run across the top margin.  This aspect of
 
pragmatic definition we recognize as the regular
 
ante-representation:
 
  
  e  =  e:e + f:f  +  g:g  +  h:h
+
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.
  
  f  =  e:f  +  f:e  +  g:h  +  h:g
+
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>
  
  g  = e:+ f:+ g:+ h:f
+
{| 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>
 +
|}
  
  h  =  e:h  +  f:g  + g:f  +  h:e
+
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.
  
In (2), we consider the effects of each x in its
+
==Note 19==
practical bearing on contexts of the form <y, _>,
 
as y ranges over G, and the effects are such that
 
x takes <y, _> into yx, for y in G, all of which
 
is summarily notated as x = {<y : yx> : y in G}.
 
The pairs <y : yx> can be found by picking an x
 
from the top margin of the group operation table
 
and considering its effects on each y in turn as
 
these run down the left margin.  This aspect of
 
pragmatic definition we recognize as the regular
 
post-representation:
 
  
  e  =  e:e  +  f:f  +  g:g  +  h:h
+
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
  
  f  = e:f  +  f:e  +  g:h  +  h:g
+
{| align="center" cellpadding="10" style="text-align:center"
 +
| <math>\text{Symmetric Group}~ S_3</math>
 +
|-
 +
| [[Image:Symmetric Group S(3).jpg|500px]]
 +
|}
  
  g  =  e:g  +  f:h  +  g:e  +  h:f
+
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
  
  h  =  e:h  +  f:g  + g:f  +  h:e
+
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.
  
If the ante-rep looks the same as the post-rep,
+
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
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.
 
</pre>
 
  
==Note 17==
+
{| align="center" cellpadding="10" 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>
 +
|}
  
<pre>
+
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:
So long as we're in the neighborhood, we might as well take in
 
some more of the sights, for instance, the smallest example of
 
a non-abelian (non-commutative) group.  This is a group of six
 
elements, say, G = {e, f, g, h, i, j}, with no relation to any
 
other employment of these six symbols being implied, of course,
 
and it can be most easily represented as the permutation group
 
on a set of three letters, say, X = {a, b, c}, usually notated
 
as G = Sym(X) or more abstractly and briefly, as Sym(3) or S_3.
 
Here are the permutation (= substitution) operations in Sym(X):
 
  
Table 17-a.  Permutations or Substitutions in Sym_{a, b, c}
+
{| align="center" cellpadding="10" style="text-align:center"
o---------o---------o---------o---------o---------o---------o
+
|
|        |        |        |        |        |        |
+
<math>\begin{array}{*{13}{c}}
|    e   |    f   |    g   |    h   |    i   |    j   |
+
\operatorname{e}
|        |        |        |        |        |        |
+
& = & \operatorname{e}\!:\!\operatorname{e}
o=========o=========o=========o=========o=========o=========o
+
& + & \operatorname{f}\!:\!\operatorname{f}
|        |        |        |        |        |        |
+
& + & \operatorname{g}\!:\!\operatorname{g}
|  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |  a b c  |
+
& + & \operatorname{h}\!:\!\operatorname{h}
|        |        |        |        |        |        |
+
& + & \operatorname{i}\!:\!\operatorname{i}
|  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |  | | |  |
+
& + & \operatorname{j}\!:\!\operatorname{j}
|  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |  v v v  |
+
\\[4pt]
|        |        |        |        |        |        |
+
\operatorname{f}
| a b c  |  c a b  |  b c a  |  a c b  |  c b a  |  b a c  |
+
& = & \operatorname{e}\!:\!\operatorname{f}
|        |        |        |        |        |        |
+
& + & \operatorname{f}\!:\!\operatorname{g}
o---------o---------o---------o---------o---------o---------o
+
& + & \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>
 +
|}
  
Here is the operation table for S_3, given in abstract fashion:
+
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:
  
Table 17-b.  Symmetric Group S_3
+
{| align="center" cellpadding="10" style="text-align:center"
o-------------------------------------------------o
+
|
|                                                |
+
<math>\begin{array}{*{13}{c}}
|                        o                        |
+
\operatorname{e}
|                    e / \ e                     |
+
& = & \operatorname{e}\!:\!\operatorname{e}
|                      /  \                     |
+
& + & \operatorname{f}\!:\!\operatorname{f}
|                    /  e \                     |
+
& + & \operatorname{g}\!:\!\operatorname{g}
|                  f / \   / \ f                 |
+
& + & \operatorname{h}\!:\!\operatorname{h}
|                  /  \ \                   |
+
& + & \operatorname{i}\!:\!\operatorname{i}
|                  /  f  \ \                 |
+
& + & \operatorname{j}\!:\!\operatorname{j}
|              g / \   / \   / \ g              |
+
\\[4pt]
|                /  \ \ \               |
+
\operatorname{f}
|              /  g  \ \ \               |
+
& = & \operatorname{e}\!:\!\operatorname{f}
|            h / \   / \   / \   / \ h            |
+
& + & \operatorname{f}\!:\!\operatorname{g}
|            /  \ \ \ \             |
+
& + & \operatorname{g}\!:\!\operatorname{e}
|            /  h  \ \ e \ h \           |
+
& + & \operatorname{h}\!:\!\operatorname{i}
|        i / \   / \   / \   / \   / \ i        |
+
& + & \operatorname{i}\!:\!\operatorname{j}
|          /  \ \ \ \ \         |
+
& + & \operatorname{j}\!:\!\operatorname{h}
|        /  i  \ i  \  f \ \ \         |
+
\\[4pt]
|      j / \   / \   / \   / \   / \   / \ j     |
+
\operatorname{g}
|      /  \ \ \ \ \ /  \      |
+
& = & \operatorname{e}\!:\!\operatorname{g}
|      o  j \ \ \ i \ h \  j  o      |
+
& + & \operatorname{f}\!:\!\operatorname{e}
|      \   / \   / \   / \   / \   / \   /      |
+
& + & \operatorname{g}\!:\!\operatorname{f}
|        \ \ \ \ \ \ /        |
+
& + & \operatorname{h}\!:\!\operatorname{j}
|        \ h \ \ e \ \ i  /        |
+
& + & \operatorname{i}\!:\!\operatorname{h}
|          \   / \   / \   / \   / \   /          |
+
& + & \operatorname{j}\!:\!\operatorname{i}
|          \ \ \ \ \ /          |
+
\\[4pt]
|            \ \ \ \ h /            |
+
\operatorname{h}
|            \   / \   / \   / \   /            |
+
& = & \operatorname{e}\!:\!\operatorname{h}
|              \ \ \ \ /              |
+
& + & \operatorname{f}\!:\!\operatorname{j}
|              \ \ e \ g /              |
+
& + & \operatorname{g}\!:\!\operatorname{i}
|                \   / \   / \   /                |
+
& + & \operatorname{h}\!:\!\operatorname{e}
|                \ \ \ /                |
+
& + & \operatorname{i}\!:\!\operatorname{g}
|                  \ g \ f  /                  |
+
& + & \operatorname{j}\!:\!\operatorname{f}
|                  \   / \   /                  |
+
\\[4pt]
|                    \ \ /                    |
+
\operatorname{i}
|                    \ e /                    |
+
& = & \operatorname{e}\!:\!\operatorname{i}
|                      \   /                     |
+
& + & \operatorname{f}\!:\!\operatorname{h}
|                       \ /                      |
+
& + & \operatorname{g}\!:\!\operatorname{j}
|                        o                        |
+
& + & \operatorname{h}\!:\!\operatorname{f}
|                                                |
+
& + & \operatorname{i}\!:\!\operatorname{e}
o-------------------------------------------------o
+
& + & \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>
 +
|}
  
I think that the NKS reader can guess how we might apply
+
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.
this group to the space of propositions of type B^3 -> B.
 
  
By the way, we will meet with the symmetric group S_3 again
+
==Note 20==
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).
 
</pre>
 
  
==Note 18==
+
{| cellpadding="2" cellspacing="2" width="100%"
 
+
| width="60%" | &nbsp;
<pre>
+
| width="40%" |
By way of collecting a short-term pay-off for all the work that we
+
the way of heaven and earth<br>
did on the regular representations of the Klein 4-group V_4, let us
+
is to be long continued<br>
write out as quickly as possible in "relative form" a minimal budget
+
in their operation<br>
of representations for the symmetric group on three letters, Sym(3).
+
without stopping
After doing the usual bit of compare and contrast among the various
+
|-
representations, we will have enough concrete material beneath our
+
| height="50px" | &nbsp;
abstract belts to tackle a few of the presently obscured details
+
| valign="top"  | &mdash; i ching, hexagram 32
of Peirce's early "Algebra + Logic" papers.
+
|}
  
Writing the permutations or substitutions of Sym {a, b, c}
+
The Reader may be wondering what happened to the announced subject of ''Dynamics And Logic''.  What happened was a bit like this:
in relative form generates what is generally thought of as
 
a "natural representation" of S_3.
 
  
  e  =  a:a + b:b + c:c
+
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.
  
  f  = a:c + b:a + c:b
+
==Note 21==
  
  g  = a:b + b:c + c:a
+
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.
  
  h  = a:a + b:c + c:b
+
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>
  
  i  =  a:c + b:b + c:a
+
<br>
  
  j  = a:b + b:a + c:c
+
{| 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>
I have without stopping to think about it written out this natural
+
|- style="background:#f0f0ff"
representation of S_3 in the style that comes most naturally to me,
+
| width="16%" | <math>\operatorname{e}</math>
to wit, the "right" way, whereby an ordered pair configured as x:y
+
| width="16%" | <math>\operatorname{f}</math>
constitutes the turning of x into y.  It is possible that the next
+
| width="16%" | <math>\operatorname{g}</math>
time we check in with CSP that we will have to adjust our sense of
+
| width="16%" | <math>\operatorname{h}</math>
direction, but that will be an easy enough bridge to cross when we
+
| width="16%" | <math>\operatorname{i}</math>
come to it.
+
| width="16%" | <math>\operatorname{j}</math>
</pre>
+
|-
 +
|
 +
<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>
 +
|}
  
==Note 19==
+
<br>
  
<pre>
+
These permutations were then converted to relative form as logical sums of elementary relatives:
To construct the regular representations of S_3,
 
we pick up from the data of its operation table,
 
DAL 17, Table 17-b, at either one of these sites:
 
  
http://stderr.org/pipermail/inquiry/2004-May/001419.html
+
{| align="center" cellpadding="10" width="90%"
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
+
| 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>
 +
|}
  
Just by way of staying clear about what we are doing,
+
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:
let's return to the recipe that we worked out before:
 
  
It is part of the definition of a group that the 3-adic
+
<br>
relation L c G^3 is actually a function L : G x G -> G.
 
It is from this functional perspective that we can see
 
an easy way to derive the two regular representations.
 
  
Since we have a function of the type L : G x G -> G,
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
we can define a couple of substitution operators:
+
|+ <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>
 +
|}
  
1.  Sub(x, <_, y>) puts any specified x into
+
<br>
    the empty slot of the rheme <_, y>, with
 
    the effect of producing the saturated
 
    rheme <x, y> that evaluates to xy.
 
  
2.  Sub(x, <y, _>) puts any specified x into
+
The 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:
    the empty slot of the rheme <y, _>, with
 
    the effect of producing the saturated
 
    rheme <y, x> that evaluates to yx.
 
  
In (1), we consider the effects of each x in its
+
{| align="center" cellpadding="6" width="90%"
practical bearing on contexts of the form <_, y>,
+
| align="center" |
as y ranges over G, and the effects are such that
+
<math>\begin{bmatrix}
x takes <_, y> into xy, for y in G, all of which
+
a\!:\!a &
is summarily notated as x = {<y : xy> : y in G}.
+
a\!:\!b &
The pairs <y : xy> can be found by picking an x
+
a\!:\!c
from the left margin of the group operation table
+
\\
and considering its effects on each y in turn as
+
b\!:\!a &
these run along the right margin.  This produces
+
b\!:\!b &
the regular ante-representation of S_3, like so:
+
b\!:\!c
 +
\\
 +
c\!:\!a &
 +
c\!:\!b &
 +
c\!:\!c
 +
\end{bmatrix}</math>
 +
|}
  
  e  =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  +  j:j
+
==Note 22==
  
  f  =  e:f + f:g  +  g:e  +  h:j  +  i:h  +  j:i
+
Let us 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, <math>X^\circ,</math> to considering a larger universe of discourse, <math>\operatorname{E}X^\circ.</math> An operator <math>\operatorname{W}</math> of this general type, namely,<math>\operatorname{W} : X^\circ \to \operatorname{E}X^\circ,</math> acts on each proposition <math>f : X \to \mathbb{B}</math> of the source universe <math>X^\circ</math> to produce a proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}</math> of the target universe <math>\operatorname{E}X^\circ.</math>
  
  g  =  e:g  + f:e  +  g:f +  h:i  +  i:j  +  j:h
+
The two main operators that we've examined so far are the enlargement or shift operator <math>\operatorname{E} : X^\circ \to \operatorname{E}X^\circ</math> and the difference operator <math>\operatorname{D} : X^\circ \to \operatorname{E}X^\circ.</math> The operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\operatorname{E}f, \operatorname{D}f : \operatorname{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>
  
  h  =  e:h  +  f:i  +  g:j  +  h:e  +  i:f  +  j:g
+
At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture higher into the ever more rarefied air of abstractions.
  
  i  =  e:i + f:j  +  g:h  +  h:g  +  i:e  +  j:f
+
One good picture comes to us by way of the ''field'' concept. Given a space <math>X,\!</math> a ''field'' of a specified type <math>Y\!</math> over <math>X\!</math> is formed by associating with each point of <math>X\!</math> an object of type <math>Y.\!</math> If that sounds like the same thing as a function from <math>X\!</math> to the space of things of type <math>Y\!</math> &mdash; it is nothing but &mdash; and yet 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.
  
  j  =  e:j  +  f:h  +  g:i  +  h:f +  i:g  +  j:e
+
In the field picture, a proposition <math>f : X \to \mathbb{B}</math> becomes a ''scalar field'', that is, a field of values in <math>\mathbb{B}.</math>
  
In (2), we consider the effects of each x in its
+
Let us take a moment to view an old proposition in this new light, for example, the logical conjunction <math>pq : X \to \mathbb{B}</math> pictured in Figure&nbsp;22-a.
practical bearing on contexts of the form <y, _>,
 
as y ranges over G, and the effects are such that
 
x takes <y, _> into yx, for y in G, all of which
 
is summarily notated as x = {<y : yx> : y in G}.
 
The pairs <y : yx> can be found by picking an x
 
on the right margin of the group operation table
 
and considering its effects on each y in turn as
 
these run along the left margin. This generates
 
the regular post-representation of S_3, like so:
 
  
  e  =   e:e  +  f:f  +  g:g  +  h:h  +  i:i  + j:j
+
{| align="center" cellpadding="10" style="text-align:center"
 +
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
 +
|-
 +
| <math>\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}</math>
 +
|}
  
  f  =  e:f +  f:g  +  g:e  +  h:i  +  i:j  +  j:h
+
Each of the operators <math>\operatorname{E}, \operatorname{D} : X^\circ \to \operatorname{E}X^\circ</math> takes us from considering propositions <math>f : X \to \mathbb{B},</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' over <math>X.\!</math>
  
  g  =  e:g  + f:e  + g:f  +  h:j  +  i:h + j:i
+
The structure of these differential fields can be described this way. With each point of <math>X\!</math> there is associated an object of the following typea proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \operatorname{d}X \to \mathbb{B}.</math> In this frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\operatorname{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \operatorname{d}p, \operatorname{d}q \}.</math> These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.
  
  h  =  e:h  +  f:j +  g:i  +  h:e  +  i:g  +  j:f
+
A differential operator <math>\operatorname{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}.</math> In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\operatorname{W}f : \operatorname{E}X \to \mathbb{B}</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>
  
  i  =  e:i  +  f:h  +  g:j  +  h:f  +  i:e  +  j:g
+
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure&nbsp;22-b.
  
  j  =   e:j  +  f:i +  g:+ h:g  + i:f  + j:e
+
{| align="center" cellpadding="10" style="text-align:center"
 
+
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
If the ante-rep looks different from the post-rep,
+
|-
it is just as it should be, as S_3 is non-abelian
+
| <math>\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
(non-commutative), and so the two representations
+
|-
differ in the details of their practical effects,
+
|
though, of course, being representations of the
+
<math>\begin{array}{rcccccc}
same abstract group, they must be isomorphic.
+
\operatorname{E}(pq)
</pre>
+
& = &
 +
p
 +
& \cdot &
 +
q
 +
& \cdot &
 +
\texttt{(} \operatorname{d}p \texttt{)}
 +
\texttt{(} \operatorname{d}q \texttt{)}
 +
\\[4pt]
 +
& + &
 +
p
 +
& \cdot &
 +
\texttt{(} q \texttt{)}
 +
& \cdot &
 +
\texttt{(} \operatorname{d}p \texttt{)}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)}
 +
& \cdot &
 +
q
 +
& \cdot &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{(} \operatorname{d}q \texttt{)}
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)}
 +
& \cdot &
 +
\texttt{(} q \texttt{)}
 +
& \cdot &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\end{array}</math>
 +
|}
  
==Note 20==
+
The differential field <math>\operatorname{E}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to reach one of the models of the proposition <math>pq,\!</math> that is, in order to satisfy the proposition <math>pq.\!</math>
  
<pre>
+
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure&nbsp;22-c.
You may be wondering what happened to the announced subject
 
of "Dynamics And Logic". What occurred was a bit like this:
 
  
We happened to make the observation that the shift operators {E_ij}
+
{| align="center" cellpadding="10" style="text-align:center"
form a transformation group that acts on the set of propositions of
+
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
the form f : B x B -> B.  Group theory is a very attractive subject,
+
|-
but it did not draw us so far from our intended course as one might
+
| <math>\text{Figure 22-c.  Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
initially think.  For one thing, groups, especially the groups that
+
|-
are named after the Norwegian mathematician Marius Sophus Lie, turn
+
|
out to be of critical importance in solving differential equations.
+
<math>\begin{array}{rcccccc}
For another thing, group operations provide us with an ample supply
+
\operatorname{D}(pq)
of triadic relations that have been extremely well-studied over the
+
& = &
years, and thus they give us no small measure of useful guidance in
+
p
the study of sign relations, another brand of 3-adic relations that
+
& \cdot &
have significance for logical studies, and in our acquaintance with
+
q
which we have scarcely begun to break the ice.  Finally, I couldn't
+
& \cdot &
resist taking up the links between group representations, amounting
+
\texttt{(}
to the very archetypes of logical models, and the pragmatic maxim.
+
\texttt{(} \operatorname{d}p \texttt{)}
 
+
\texttt{(} \operatorname{d}q \texttt{)}
Biographical Data for Marius Sophus Lie (1842-1899):
+
\texttt{)}
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html
+
\\[4pt]
</pre>
+
& + &
 +
p
 +
& \cdot &
 +
\texttt{(} q \texttt{)}
 +
& \cdot &
 +
\texttt{~}
 +
\texttt{(} \operatorname{d}p \texttt{)}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\texttt{~}
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)}
 +
& \cdot &
 +
q
 +
& \cdot &
 +
\texttt{~}
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{(} \operatorname{d}q \texttt{)}
 +
\texttt{~}
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)}
 +
& \cdot &
 +
\texttt{(}q \texttt{)}
 +
& \cdot &
 +
\texttt{~}
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\texttt{~}
 +
\end{array}</math>
 +
|}
  
==Note 21==
+
The differential field <math>\operatorname{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>
  
<pre>
+
==Note 23==
We have seen a couple of groups, V_4 and S_3, represented in
 
several different ways, and we have seen each of these types
 
of representation presented in several different fashions.
 
Let us look at one other stylistic variant for presenting
 
a group representation that is often used, the so-called
 
"matrix representation" of a group.
 
  
Returning to the example of Sym(3), we first encountered
+
I want to continue developing the basic tools of differential logic, which arose from exploring the connections between dynamics and logic, 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.
this group in concrete form as a set of permutations or
 
substitutions acting on a set of letters X = {a, b, c}.
 
This set of permutations was displayed in Table 17-a,
 
copies of which can be found here:
 
  
http://stderr.org/pipermail/inquiry/2004-May/001419.html
+
A cybernetic system has goals and actions for reaching them.  It has a state space <math>X,\!</math> giving us all of the states that the system can be in, plus it has a goal space <math>G \subseteq X,</math> 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 <math>\mathcal{T}</math> of all possible actions, each of which is a transformation of the form <math>T : X \to X,</math> 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 <math>X,\!</math> 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.
http://forum.wolframscience.com/showthread.php?postid=1321#post1321
 
  
These permutations were then converted to "relative form":
+
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&nbsp;23.
 
 
  e  =  a:a + b:b + c:c
 
 
 
  f  =  a:c + b:a + c:b
 
 
 
  g  =  a:b + b:c + c:a
 
 
 
  h  =  a:a + b:c + c:b
 
 
 
  i  =  a:c + b:b + c:a
 
 
 
  j  =  a:b + b:a + c:c
 
 
 
From this relational representation of Sym {a, b, c} ~=~ S_3,
 
one easily derives a "linear representation", regarding each
 
permutation as a linear transformation that maps the elements
 
of a suitable vector space into each other, and representing
 
each of these linear transformations by means of a matrix,
 
resulting in the following set of matrices for the group:
 
 
 
Table 21.  Matrix Representations of the Permutations in S_3
 
o---------o---------o---------o---------o---------o---------o
 
|        |        |        |        |        |        |
 
|    e    |    f    |    g    |    h    |    i    |    j    |
 
|        |        |        |        |        |        |
 
o=========o=========o=========o=========o=========o=========o
 
|        |        |        |        |        |        |
 
|  1 0 0  |  0 0 1  |  0 1 0  |  1 0 0  |  0 0 1  |  0 1 0  |
 
|  0 1 0  |  1 0 0  |  0 0 1  |  0 0 1  |  0 1 0  |  1 0 0  |
 
|  0 0 1  |  0 1 0  |  1 0 0  |  0 1 0  |  1 0 0  |  0 0 1  |
 
|        |        |        |        |        |        |
 
o---------o---------o---------o---------o---------o---------o
 
 
 
The key to the mysteries of these matrices is revealed by
 
observing that their coefficient entries are arrayed and
 
overlayed on a place mat that's marked like so:
 
 
 
o-----o-----o-----o
 
| a:a | a:b | a:c |
 
o-----o-----o-----o
 
| b:a | b:b | b:c |
 
o-----o-----o-----o
 
| c:a | c:b | c:c |
 
o-----o-----o-----o
 
</pre>
 
 
 
==Note 22==
 
  
 +
{| align="center" cellpadding="10" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<pre>
It would be good to summarize, in rough but intuitive terms,
+
o---------------------------------------------------------------------o
the outlook on differential logic that we have reached so far.
+
|                                                                    |
 
+
|  X                                                                |
We've been considering a class of operators on universes
+
|            o-------------------o                                    |
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                                    |
 
 
|          /                    \                                  |
 
|          /                    \                                  |
 
|          /                      \                                  |
 
|          /                      \                                  |
Line 3,892: Line 3,518:
 
|      /                              \                              |
 
|      /                              \                              |
 
|    /                                \                            |
 
|    /                                \                            |
|    o                                   o                            |
+
|    o                 G                o                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
|    |                 G                |                            |
+
|    |                       o<---------T---------o                  |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
Line 3,903: Line 3,529:
 
|    \                                /                            |
 
|    \                                /                            |
 
|      \                              /                              |
 
|      \                              /                              |
|      \                           T /                              |
+
|      \                             /                              |
|        \             o<------------/-------------o                  |
+
|        \                           /                               |
 
|        \                        /                                |
 
|        \                        /                                |
 
|          \                      /                                  |
 
|          \                      /                                  |
Line 3,914: Line 3,540:
 
Figure 23.  Elements of a Cybernetic System
 
Figure 23.  Elements of a Cybernetic System
 
</pre>
 
</pre>
 +
|}
  
 
==Note 24==
 
==Note 24==
  
<pre>
+
Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\operatorname{E}f : \operatorname{E}X \to \mathbb{B},</math> and its difference map <math>\operatorname{D}f : \operatorname{E}X \to \mathbb{B}</math> can now be drawn.
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
+
To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.
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
+
Figure&nbsp;24-1 shows the proposition <math>pq\!</math> once again, which we now view as a scalar field &mdash; analogous to a ''potential hill'' in physics, but in logic tantamount to a ''potential plateau'' &mdash; where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.
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
+
{| align="center" cellpadding="10" style="text-align:center"
|                                                                     |
+
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
|  X                                                                |
+
|-
|           o-------------------o  o-------------------o            |
+
| <math>\text{Figure 24-1.  Proposition}~ pq : X \to \mathbb{B}</math>
|           /                    \ /                    \           |
+
|}
|          /                      o                      \         |
+
 
|        /                       /%\                      \        |
+
Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\operatorname{E}X</math> is denoted <math>\varepsilon f : \operatorname{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universeTacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.
|       /                      /%%%\                      \        |
 
|      /                      /%%%%%\                      \      |
 
|      /                      /%%%%%%%\                       \     |
 
|    /                       /%%%%%%%%%\                       \    |
 
|    o                      o%%%%%%%%%%%o                      o    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    |          P            |%%%%%%%%%%%|            Q          |    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    |                      |%%%%%%%%%%%|                      |    |
 
|    o                      o%%%%%%%%%%%o                      o    |
 
|    \                      \%%%%%%%%%/                       /    |
 
|      \                       \%%%%%%%/                       /      |
 
|      \                       \%%%%%/                      /      |
 
|        \                       \%%%/                       /        |
 
|        \                       \%/                       /        |
 
|          \                      o                      /          |
 
|          \                    / \                    /          |
 
|            o-------------------o  o-------------------o            |
 
|                                                                    |
 
|                                                                    |
 
o---------------------------------------------------------------------o
 
Figure 24-1Proposition pq : X -> B
 
  
Given any proposition f : X -> B, the "tacit extension" of f to EX
+
Figure&nbsp;24-2 shows the tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
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
+
{| align="center" cellpadding="10" style="text-align:center"
is related to an old function in such a way that it obeys the same
+
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
constraints on the old variables, with a "don't care" condition on
+
|-
the new variables.
+
| <math>\text{Figure 24-2.  Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}</math>
 
+
|-
Figure 24-2 illustrates the "tacit extension" of the proposition
+
|
or scalar field f = pq : X -> B to give the extended proposition
+
<math>\begin{array}{rcccccc}
or differential field that we notate as !e!f = !e![pq] : EX -> B.
+
\varepsilon (pq)
 
+
& = &
o---------------------------------------------------------------------o
+
p & \cdot & q & \cdot &
|                                                                    |
+
\texttt{(} \operatorname{d}p \texttt{)}
|   X                                                                |
+
\texttt{(} \operatorname{d}q \texttt{)}
|            o-------------------o  o-------------------o            |
+
\\[4pt]
|           /                    \ /                    \           |
+
& + &
|          /  P                    o                    Q  \         |
+
p & \cdot & q & \cdot &
|        /                      / \                       \         |
+
\texttt{(} \operatorname{d}p \texttt{)}
|       /                      /  \                      \        |
+
\texttt{~} \operatorname{d}q \texttt{~}
|      /                      /    \                      \      |
+
\\[4pt]
|      /                      /      \                       \      |
+
& + &
|    /                      /        \                      \     |
+
p & \cdot & q & \cdot &
|    o                      o (dp) (dq) o                      o    |
+
\texttt{~} \operatorname{d}p \texttt{~}
|    |                      |  o-->--o  |                      |    |
+
\texttt{(} \operatorname{d}q \texttt{)}
|    |                      |  \   /  |                      |    |
+
\\[4pt]
|    |            (dp) dq  |    \ /    |  dp (dq)             |    |
+
& + &
|    |          o<-----------------o----------------->o          |    |
+
p & \cdot & q & \cdot &
|    |                      |    |    |                      |    |
+
\texttt{~} \operatorname{d}p \texttt{~}
|    |                      |    |    |                      |    |
+
\texttt{~} \operatorname{d}q \texttt{~}
|    |                      |    |    |                      |    |
+
\end{array}</math>
|    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==
 
==Note 25==
  
<pre>
+
Continuing with the example <math>pq : X \to \mathbb{B},</math> Figure&nbsp;25-1 shows the enlargement or shift map <math>\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.</math>
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
+
{| align="center" cellpadding="10" style="text-align:center"
|                                                                     |
+
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
|   X                                                                |
+
|-
|           o-------------------o  o-------------------o            |
+
| <math>\text{Figure 25-1.  Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|          /                    \ /                     \          |
+
|-
|         /  P                    o                    Q  \         |
+
|
|        /                      / \                       \         |
+
<math>\begin{array}{rcccccc}
|        /                      /  \                       \       |
+
\operatorname{E}(pq)
|      /                      /    \                       \       |
+
& = &
|      /                      /      \                       \     |
+
p
|    /                      /        \                       \     |
+
& \cdot &
|    o                      o (dp) (dq) o                      o    |
+
q
|    |                      |  o-->--o  |                      |    |
+
& \cdot &
|    |                      |  \   /  |                      |    |
+
\texttt{(} \operatorname{d}p \texttt{)}
|    |            (dp) dq  |    \ /    |  dp (dq)             |    |
+
\texttt{(} \operatorname{d}q \texttt{)}
|    |          o----------------->o<-----------------o          |    |
+
\\[4pt]
|    |                      |    ^    |                      |    |
+
& + &
|    |                      |    |    |                      |    |
+
p
|    |                      |    |    |                      |    |
+
& \cdot &
|    o                      o    |    o                      o    |
+
\texttt{(} q \texttt{)}
|    \                       \   |    /                      /    |
+
& \cdot &
|      \                       \   |  /                      /      |
+
\texttt{(} \operatorname{d}p \texttt{)}
|      \                       \ |  /                      /      |
+
\texttt{~} \operatorname{d}q \texttt{~}
|        \                       \ | /                      /        |
+
\\[4pt]
|        \                       \|/                      /        |
+
& + &
|          \                       |                      /          |
+
\texttt{(} p \texttt{)}
|          \                     /|\                     /          |
+
& \cdot &
|            o-------------------o | o-------------------o            |
+
q
|                                  |                                  |
+
& \cdot &
|                              dp | dq                              |
+
\texttt{~} \operatorname{d}p \texttt{~}
|                                  |                                  |
+
\texttt{(} \operatorname{d}q \texttt{)}
|                                  |                                  |
+
\\[4pt]
|                                  o                                  |
+
& + &
|                                                                     |
+
\texttt{(} p \texttt{)}
o---------------------------------------------------------------------o
+
& \cdot &
Figure 25-1.  Enlargement E[pq] : EX -> B
+
\texttt{(} q \texttt{)}
 +
& \cdot &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\end{array}</math>
 +
|}
  
A very important conceptual transition has just occurred here,
+
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 <math>\varepsilon f</math> and <math>\operatorname{E}f,</math> both of the type <math>\operatorname{E}X \to \mathbb{B},</math> 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.
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
+
In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\operatorname{E}f</math> are in a certain sense dual to each other.  The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\operatorname{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true.  The only arc they have in common is the no-change loop <math>\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)</math> that are illustrated in Figure&nbsp;25-2.
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
+
{| align="center" cellpadding="10" style="text-align:center"
|                                                                     |
+
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
|   X                                                                 |
+
|-
|           o-------------------o  o-------------------o            |
+
| <math>\text{Figure 25-2.  Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|          /                    \ /                    \           |
+
|-
|          /  P                    o                    Q  \         |
+
|
|        /                      / \                       \         |
+
<math>\begin{array}{rcccccc}
|        /                      /  \                       \       |
+
\operatorname{D}(pq)
|      /                      /    \                       \       |
+
& = &
|      /                      /      \                       \      |
+
p
|    /                      /        \                       \     |
+
& \cdot &
|    o                      o          o                      o    |
+
q
|    |                      |          |                      |    |
+
& \cdot &
|    |                      |          |                      |    |
+
\texttt{(}
|    |            (dp) dq  |          |  dp (dq)             |    |
+
\texttt{(} \operatorname{d}p \texttt{)}
|    |          o<---------------->o<---------------->o          |    |
+
\texttt{(} \operatorname{d}q \texttt{)}
|    |                      |    ^    |                      |    |
+
\texttt{)}
|    |                      |    |    |                      |    |
+
\\[4pt]
|    |                      |    |    |                      |    |
+
& + &
|    o                      o    |    o                      o    |
+
p
|    \                       \   |    /                      /    |
+
& \cdot &
|      \                       \   |  /                      /      |
+
\texttt{(} q \texttt{)}
|      \                       \ |  /                      /      |
+
& \cdot &
|        \                       \ | /                      /        |
+
\texttt{~}
|        \                       \|/                      /        |
+
\texttt{(} \operatorname{d}p \texttt{)}
|          \                       |                      /          |
+
\texttt{~} \operatorname{d}q \texttt{~}
|          \                     /|\                     /          |
+
\texttt{~}
|            o-------------------o | o-------------------o            |
+
\\[4pt]
|                                  |                                  |
+
& + &
|                              dp | dq                              |
+
\texttt{(} p \texttt{)}
|                                  |                                  |
+
& \cdot &
|                                  v                                  |
+
q
|                                  o                                  |
+
& \cdot &
|                                                                    |
+
\texttt{~}
o---------------------------------------------------------------------o
+
\texttt{~} \operatorname{d}p \texttt{~}
Figure 25-2.  Difference Map D[pq] : EX -> B
+
\texttt{(} \operatorname{d}q \texttt{)}
 
+
\texttt{~}
The differential features of D[pq] may be collected cell by cell of
+
\\[4pt]
the underlying universe X% = [p, q] to give the following expansion:
+
& + &
 
+
\texttt{(} p \texttt{)}
  D[pq]
+
& \cdot &
 +
\texttt{(}q \texttt{)}
 +
& \cdot &
 +
\texttt{~}
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\texttt{~}
 +
\end{array}</math>
 +
|}
  
  =
+
==Note 26==
  
  p q . ((dp)(dq))
+
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 <math>f = pq : X \to \mathbb{B}</math> in the following way.
  
  +
+
Figure&nbsp;26-1 shows the differential proposition <math>\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.</math>  This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> 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 <math>\operatorname{d}f.\!</math>
  
  p (q) . (dp) dq
+
{| align="center" cellpadding="10" style="text-align:center"
 +
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
 +
|-
 +
| <math>\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}</math>
 +
|}
  
  +
+
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
  
  (p) q . dp (dq)
+
{| align="center" cellpadding="10" style="text-align:center"
 +
|
 +
<math>\begin{array}{rcccccc}
 +
\operatorname{d}(pq)
 +
& = &
 +
p & \cdot & q & \cdot &
 +
\texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)}
 +
\\[4pt]
 +
& + &
 +
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
 +
\operatorname{d}q
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
 +
\operatorname{d}p
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0
 +
\end{array}</math>
 +
|}
  
  +
+
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:
  
  (p)(q) . dp dq
+
{| align="center" cellpadding="10" style="text-align:center"
</pre>
+
|
 
+
<math>\begin{matrix}
==Note 26==
+
\texttt{(}
 
+
\operatorname{d}p
<pre>
+
\texttt{,}
If we follow the classical line that singles out linear functions
+
\operatorname{d}q
as ideals of simplicity, then we may complete the analytic series
+
\texttt{)}
of the proposition f = pq : X -> B in the following way.
+
& = &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{(} \operatorname{d}q \texttt{)}
 +
& + &
 +
\texttt{(} \operatorname{d}p \texttt{)}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\\[4pt]
 +
dp
 +
& = &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
& + &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{(} \operatorname{d}q \texttt{)}
 +
\\[4pt]
 +
\operatorname{d}q
 +
& = &
 +
\texttt{~} \operatorname{d}p \texttt{~}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
& + &
 +
\texttt{(} \operatorname{d}p \texttt{)}
 +
\texttt{~} \operatorname{d}q \texttt{~}
 +
\end{matrix}</math>
 +
|}
  
Figure 26-1 shows the differential proposition df = d[pq] : EX -> B
+
Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, Figure&nbsp;26-2 shows the remainder map <math>\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.
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
+
{| align="center" cellpadding="10" style="text-align:center"
|                                                                     |
+
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
|  X                                                                |
+
|-
|            o-------------------o  o-------------------o            |
+
| <math>\text{Figure 26-2.  Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}</math>
|          /                    \ /                    \          |
+
|}
|          /  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:
 
Reading the arrows off the map produces the following data:
  
    r[pq]
+
{| align="center" cellpadding="10" style="text-align:center"
 
+
|
    =
+
<math>\begin{array}{rcccccc}
 
+
\operatorname{r}(pq)
    p q . dp dq
+
& = &
 
+
p & \cdot & q & \cdot &
    +
+
\operatorname{d}p ~ \operatorname{d}q
 
+
\\[4pt]
    p (q) . dp dq
+
& + &
 
+
p & \cdot & \texttt{(} q \texttt{)} & \cdot &
    +
+
\operatorname{d}p ~ \operatorname{d}q
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)} & \cdot & q & \cdot &
 +
\operatorname{d}p ~ \operatorname{d}q
 +
\\[4pt]
 +
& + &
 +
\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot &
 +
\operatorname{d}p ~ \operatorname{d}q
 +
\end{array}</math>
 +
|}
  
    (p) q . dp dq
+
In short, <math>\operatorname{r}(pq)</math> is a constant field, having the value <math>\operatorname{d}p~\operatorname{d}q</math> at each cell.
  
    +
+
==Further Reading==
 
 
    (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:
 
A more detailed presentation of Differential Logic can be found here:
  
DLOG D.  http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
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:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
 
DLOG D.  http://stderr.org/pipermail/inquiry/2003-June/thread.html#571
 
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Latest revision as of 13:28, 10 December 2014


Note. Many problems with the sucky MathJax on this page. The parser apparently reads 4 tildes inside math brackets the way it would in the external wiki environment, in other words, as signature tags. Jon Awbrey (talk) 18:00, 5 December 2014 (UTC)

Note 1

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

One of the first things that you can do, once you have a moderately efficient 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:

Cactus Graph Existential P And Q.jpg

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.

Cactus Graph Existential True.jpg
Cactus Graph Existential False.jpg

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 in the following Figure:

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:

Cactus Graph (P,dP)(Q,dQ).jpg

The cactus formula \(\texttt{(p, dp)(q, dq)}\) and its corresponding graph arise by substituting \(p + \operatorname{d}p\) for \(p\!\) and \(q + \operatorname{d}q\) for \(q\!\) in the boolean product or logical conjunction \(pq\!\) and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum \(p + \operatorname{d}p\) is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

Cactus Graph (P,dP).jpg

Next question: What is the difference between the value of the proposition \(pq\!\) over there, at a distance of \(\operatorname{d}p\) and \(\operatorname{d}q,\) and the value of the proposition \(pq\!\) where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:

Cactus Graph ((P,dP)(Q,dQ),PQ).jpg

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:

Cactus Graph (( ,dP)( ,dQ), ).jpg

And this is equivalent to the following graph:

Cactus Graph ((dP)(dQ)).jpg

Note 2

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

\(\begin{matrix} p ~\operatorname{and}~ q & \quad & \xrightarrow{\quad\operatorname{Diff}\quad} & \quad & \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q \end{matrix}\!\)

Cactus Graph PQ Diff ((dP)(dQ)).jpg

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:

\(\text{Symmetric Group}~ S_3\)
Symmetric Group S(3).jpg

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:

\(\text{Symmetric Group}~ S_3\)
Symmetric Group S(3).jpg

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

Let us 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, \(X^\circ,\) to considering a larger universe of discourse, \(\operatorname{E}X^\circ.\) An operator \(\operatorname{W}\) of this general type, namely,\(\operatorname{W} : X^\circ \to \operatorname{E}X^\circ,\) acts on each proposition \(f : X \to \mathbb{B}\) of the source universe \(X^\circ\) to produce a proposition \(\operatorname{W}f : \operatorname{E}X \to \mathbb{B}\) of the target universe \(\operatorname{E}X^\circ.\)

The two main operators that we've examined so far are the enlargement or shift operator \(\operatorname{E} : X^\circ \to \operatorname{E}X^\circ\) and the difference operator \(\operatorname{D} : X^\circ \to \operatorname{E}X^\circ.\) The operators \(\operatorname{E}\) and \(\operatorname{D}\) act on propositions in \(X^\circ,\) that is, propositions of the form \(f : X \to \mathbb{B}\) that are said to be about the subject matter of \(X,\!\) and they produce extended propositions of the forms \(\operatorname{E}f, \operatorname{D}f : \operatorname{E}X \to \mathbb{B},\) propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in \(X.\!\)

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture 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 associating with 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 nothing but — and yet 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 \(f : X \to \mathbb{B}\) becomes a scalar field, that is, a field of values in \(\mathbb{B}.\)

Let us take a moment to view an old proposition in this new light, for example, the logical conjunction \(pq : X \to \mathbb{B}\) pictured in Figure 22-a.

Field Picture PQ Conjunction.jpg
\(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\)

Each of the operators \(\operatorname{E}, \operatorname{D} : X^\circ \to \operatorname{E}X^\circ\) takes us from considering propositions \(f : X \to \mathbb{B},\) 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. With each point of \(X\!\) there is associated an object of the following type: a proposition about changes in \(X,\!\) that is, a proposition \(g : \operatorname{d}X \to \mathbb{B}.\) In this frame of reference, if \(X^\circ\) is the universe that is generated by the set of coordinate propositions \(\{ p, q \},\!\) then \(\operatorname{d}X^\circ\) is the differential universe that is generated by the set of differential propositions \(\{ \operatorname{d}p, \operatorname{d}q \}.\) These differential propositions may be interpreted as indicating \({}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}\) and \({}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},\) respectively.

A differential operator \(\operatorname{W},\) of the first order class that we have been considering, takes a proposition \(f : X \to \mathbb{B}\) and gives back a differential proposition \(\operatorname{W}f : \operatorname{E}X \to \mathbb{B}.\) In the field view, we see the proposition \(f : X \to \mathbb{B}\) as a scalar field and we see the differential proposition \(\operatorname{W}f : \operatorname{E}X \to \mathbb{B}\) as a vector field, specifically, a field of propositions about contemplated changes in \(X.\!\)

The field of changes produced by \(\operatorname{E}\) on \(pq\!\) is shown in Figure 22-b.

Field Picture PQ Enlargement Conjunction.jpg
\(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

The differential field \(\operatorname{E}(pq)\) 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 \(\operatorname{D}\!\) on \(pq\!\) is shown in Figure 22-c.

Field Picture PQ Difference Conjunction.jpg
\(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

The differential field \(\operatorname{D}(pq)\) 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 from exploring the connections between dynamics and logic, 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 \(G \subseteq X,\) 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 \(\mathcal{T}\) of all possible actions, each of which is a transformation of the form \(T : X \to X,\) 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                 G                 o                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                        o<---------T---------o                  |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    o                                   o                            |
|     \                                 /                             |
|      \                               /                              |
|       \                             /                               |
|        \                           /                                |
|         \                         /                                 |
|          \                       /                                  |
|           \                     /                                   |
|            o-------------------o                                    |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 23.  Elements of a Cybernetic System

Note 24

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition \(f : X \to \mathbb{B},\) its enlargement or shift map \(\operatorname{E}f : \operatorname{E}X \to \mathbb{B},\) and its difference map \(\operatorname{D}f : \operatorname{E}X \to \mathbb{B}\) 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 — analogous to a potential hill in physics, but in logic tantamount to a potential plateau — where the shaded region indicates an elevation of 1 and the unshaded region indicates an elevation of 0.

Field Picture PQ Conjunction.jpg
\(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\)

Given a proposition \(f : X \to \mathbb{B},\) the tacit extension of \(f\!\) to \(\operatorname{E}X\) is denoted \(\varepsilon f : \operatorname{E}X \to \mathbb{B}\) and defined by the equation \(\varepsilon f = f,\) so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

Figure 24-2 shows the tacit extension of the scalar field \(pq : X \to \mathbb{B}\) to the differential field \(\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.\)

Field Picture PQ Tacit Extension Conjunction.jpg
\(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

Note 25

Continuing with the example \(pq : X \to \mathbb{B},\) Figure 25-1 shows the enlargement or shift map \(\operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) in the same style of differential field picture that we drew for the tacit extension \(\varepsilon (pq) : \operatorname{E}X \to \mathbb{B}.\)

Field Picture PQ Enlargement Conjunction.jpg
\(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

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 \(\varepsilon f\) and \(\operatorname{E}f,\) both of the type \(\operatorname{E}X \to \mathbb{B},\) 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 \(\varepsilon f\) and the enlargement \(\operatorname{E}f\) are in a certain sense dual to each other. The tacit extension \(\varepsilon f\) indicates all the arrows out of the region where \(f\!\) is true and the enlargement \(\operatorname{E}f\) indicates all the arrows into the region where \(f\!\) is true. The only arc they have in common is the no-change loop \(\texttt{(} \operatorname{d}p \texttt{)(} \operatorname{d}q \texttt{)}\) at \(pq.\!\) If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of \(\operatorname{D}(pq) = \varepsilon(pq) + \operatorname{E}(pq)\) that are illustrated in Figure 25-2.

Field Picture PQ Difference Conjunction.jpg
\(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

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 \(f = pq : X \to \mathbb{B}\) in the following way.

Figure 26-1 shows the differential proposition \(\operatorname{d}f = \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\) that we get by extracting the cell-wise linear approximation to the difference map \(\operatorname{D}f = \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}.\) 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 \(\operatorname{d}f.\!\)

Field Picture PQ Differential Conjunction.jpg
\(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\)

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

\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\)

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:

\(\begin{matrix} \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} & = & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} & + & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] dp & = & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} & + & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] \operatorname{d}q & = & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} & + & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \end{matrix}\)

Capping the series that analyzes the proposition \(pq\!\) in terms of succeeding orders of linear propositions, Figure 26-2 shows the remainder map \(\operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B},\) that happens to be linear in pairs of variables.

Field Picture PQ Remainder Conjunction.jpg
\(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\)

Reading the arrows off the map produces the following data:

\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\)

In short, \(\operatorname{r}(pq)\) is a constant field, having the value \(\operatorname{d}p~\operatorname{d}q\) at each cell.

Further Reading

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

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)

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NKS Forum (May & Jul 2004)

  1. http://forum.wolframscience.com/showthread.php?postid=1282#post1282
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