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Directory:Jon Awbrey/Papers/Dynamics And Logic (view source)

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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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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" cellspacing="10"
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{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
 
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
 
|}
 
|}
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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" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph Existential True.jpg|500px]]
 
| [[Image:Cactus Graph Existential True.jpg|500px]]
 
|}
 
|}
   −
{| align="center" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph Existential False.jpg|500px]]
 
| [[Image:Cactus Graph Existential False.jpg|500px]]
 
|}
 
|}
   −
Back to the proposition <math>pq.\!</math>  Imagine yourself standing in a fixed cell the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
+
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" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Venn Diagram P And Q.jpg|500px]]
 
| [[Image:Venn Diagram P And Q.jpg|500px]]
 
|}
 
|}
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Don't think about it &mdash; just compute:
 
Don't think about it &mdash; just compute:
   −
{| align="center" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
 
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
 
|}
 
|}
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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:
 
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" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph (P,dP).jpg|500px]]
 
| [[Image:Cactus Graph (P,dP).jpg|500px]]
 
|}
 
|}
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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:
 
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" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
 
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
 
|}
 
|}
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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:
 
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" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
 
| [[Image:Cactus Graph (( ,dP)( ,dQ), ).jpg|500px]]
 
|}
 
|}
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And this is equivalent to the following graph:
 
And this is equivalent to the following graph:
   −
{| align="center" cellspacing="10"
+
{| align="center" cellpadding="10"
 
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
 
| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]
 
|}
 
|}
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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%"
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{| 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>
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<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>
   
|}
 
|}
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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>
 
|-
 
|-
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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%"
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{| 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>
 
|-
 
|-
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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}
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& = &
 
& = &
 
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
 
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
−
\end{array}</math>
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\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.
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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)
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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)
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\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)
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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)
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\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{)}
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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
  −
|                                                |
  −
|            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@PQ = ((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" |
+
| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]
−
<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>
  −
|}
      
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
 
Laying out the arrows on the augmented venn diagram, one gets a picture of a ''differential vector field''.
Line 479: 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 504: 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 533: 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 558: 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 573: 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 586: 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 603: 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
  −
|                                                |
  −
|                  p  dp q  dq                  |
  −
|                  o---o o---o                  |
  −
|                    \  | |  /                    |
  −
|                    \ | | /                    |
  −
|                      \| |/                      |
  −
|                      @=@                      |
  −
|                                                |
  −
o-------------------------------------------------o
  −
| Ef =           (p, dp) (q, dq)                 |
  −
o-------------------------------------------------o
  −
</pre>
   
|-
 
|-
−
| align="center" |
+
| [[Image:Cactus Graph Ef@PQ = (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" |
+
| [[Image:Cactus Graph Ef@(P)(Q) = dP dQ.jpg|500px]]
−
<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 707: 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 835: 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 942: 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 2,894: Line 2,633:  
|}
 
|}
   −
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> and
+
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
 
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 <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:
   Line 2,994: Line 2,733:  
|}
 
|}
   −
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
+
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.
    
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".
 
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".
Line 3,234: Line 2,973:  
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
 
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
   −
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" style="text-align:center"
−
| align="center" |
+
| <math>\text{Symmetric Group}~ S_3</math>
−
<pre>
+
|-
−
Symmetric Group S_3
+
| [[Image:Symmetric Group S(3).jpg|500px]]
−
o-------------------------------------------------o
  −
|                                                |
  −
|                        ^                        |
  −
|                    e / \ e                    |
  −
|                     /  \                      |
  −
|                     /  e  \                    |
  −
|                  f / \  / \ f                  |
  −
|                  /  \ /  \                  |
  −
|                  /  f  \  f  \                  |
  −
|              g / \  / \  / \ g              |
  −
|                /  \ /  \ /  \                |
  −
|              /  g  \  g  \  g  \              |
  −
|            h / \  / \  / \  / \ h            |
  −
|            /  \ /  \ /  \ /  \            |
  −
|            /  h  \  e  \  e  \  h  \            |
  −
|        i / \  / \  / \  / \  / \ i        |
  −
|          /  \ /  \ /  \ /  \ /  \          |
  −
|        /  i  \  i  \  f  \  j  \  i  \        |
  −
|      j / \  / \  / \  / \  / \  / \ j      |
  −
|      /  \ /  \ /  \ /  \ /  \ /  \      |
  −
|      ( j  \  j  \  j  \  i  \  h  \  j  )     |
  −
|      \  / \  / \  / \  / \  / \  /      |
  −
|        \ /  \ /  \ /  \ /  \ /  \ /        |
  −
|        \  h  \  h  \  e  \  j  \  i  /        |
  −
|          \  / \  / \  / \  / \  /          |
  −
|          \ /  \ /  \ /  \ /  \ /          |
  −
|            \  i  \  g  \  f  \  h  /            |
  −
|            \  / \  / \  / \  /            |
  −
|              \ /  \ /  \ /  \ /              |
  −
|              \  f  \  e  \  g  /              |
  −
|                \  / \  / \  /                |
  −
|                \ /  \ /  \ /                |
  −
|                  \  g  \  f  /                  |
  −
|                  \  / \  /                  |
  −
|                    \ /  \ /                    |
  −
|                    \  e  /                    |
  −
|                      \  /                      |
  −
|                      \ /                      |
  −
|                        v                        |
  −
|                                                |
  −
o-------------------------------------------------o
  −
</pre>
   
|}
 
|}
   Line 3,331: Line 3,028:  
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
 
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
   −
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" style="text-align:center"
−
| align="center" |
+
| <math>\text{Symmetric Group}~ S_3</math>
−
<pre>
+
|-
−
Symmetric Group S_3
+
| [[Image:Symmetric Group S(3).jpg|500px]]
−
o-------------------------------------------------o
  −
|                                                |
  −
|                        ^                        |
  −
|                    e / \ e                    |
  −
|                     /  \                      |
  −
|                     /  e  \                    |
  −
|                  f / \  / \ f                  |
  −
|                  /  \ /  \                  |
  −
|                  /  f  \  f  \                  |
  −
|              g / \  / \  / \ g              |
  −
|                /  \ /  \ /  \                |
  −
|              /  g  \  g  \  g  \              |
  −
|            h / \  / \  / \  / \ h            |
  −
|            /  \ /  \ /  \ /  \            |
  −
|            /  h  \  e  \  e  \  h  \            |
  −
|        i / \  / \  / \  / \  / \ i        |
  −
|          /  \ /  \ /  \ /  \ /  \          |
  −
|        /  i  \  i  \  f  \  j  \  i  \        |
  −
|      j / \  / \  / \  / \  / \  / \ j      |
  −
|      /  \ /  \ /  \ /  \ /  \ /  \      |
  −
|      ( j  \  j  \  j  \  i  \  h  \  j  )     |
  −
|      \  / \  / \  / \  / \  / \  /      |
  −
|        \ /  \ /  \ /  \ /  \ /  \ /        |
  −
|        \  h  \  h  \  e  \  j  \  i  /        |
  −
|          \  / \  / \  / \  / \  /          |
  −
|          \ /  \ /  \ /  \ /  \ /          |
  −
|            \  i  \  g  \  f  \  h  /            |
  −
|            \  / \  / \  / \  /            |
  −
|              \ /  \ /  \ /  \ /              |
  −
|              \  f  \  e  \  g  /              |
  −
|                \  / \  / \  /                |
  −
|                \ /  \ /  \ /                |
  −
|                  \  g  \  f  /                  |
  −
|                  \  / \  /                  |
  −
|                    \ /  \ /                    |
  −
|                    \  e  /                    |
  −
|                      \  /                      |
  −
|                      \ /                      |
  −
|                        v                        |
  −
|                                                |
  −
o-------------------------------------------------o
  −
</pre>
   
|}
 
|}
   Line 3,385: Line 3,040:  
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
 
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
   −
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" width="90%"
 
| valign="top" | 1.
 
| 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>
 
| <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>
Line 3,395: Line 3,050:  
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:
 
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>xy,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : xy) ~|~ y \in G \}.</math>  The pairs <math>(y : xy)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin.  This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
   −
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" style="text-align:center"
−
| align="center" |
+
|
 
<math>\begin{array}{*{13}{c}}
 
<math>\begin{array}{*{13}{c}}
 
\operatorname{e}
 
\operatorname{e}
Line 3,450: Line 3,105:  
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:
 
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>yx,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : yx) ~|~ y \in G \}.</math>  The pairs <math>(y : yx)\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin.  This produces the ''regular post-representation'' of <math>S_3,\!</math> like so:
   −
{| align="center" cellpadding="6" width="90%"
+
{| align="center" cellpadding="10" style="text-align:center"
−
| align="center" |
+
|
 
<math>\begin{array}{*{13}{c}}
 
<math>\begin{array}{*{13}{c}}
 
\operatorname{e}
 
\operatorname{e}
Line 3,729: Line 3,384:  
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.
 
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.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
 
|-
 
|-
Line 3,743: Line 3,398:  
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure&nbsp;22-b.
 
The field of changes produced by <math>\operatorname{E}</math> on <math>pq\!</math> is shown in Figure&nbsp;22-b.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
 
|-
 
|-
Line 3,789: Line 3,444:  
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure&nbsp;22-c.
 
The field of changes produced by <math>\operatorname{D}\!</math> on <math>pq\!</math> is shown in Figure&nbsp;22-c.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
 
|-
 
|-
Line 3,849: Line 3,504:  
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.
 
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.
   −
{| align="center" cellspacing="10" style="text-align:center; width:90%"
+
{| align="center" cellpadding="10" style="text-align:center; width:90%"
 
|
 
|
 
<pre>
 
<pre>
Line 3,863: Line 3,518:  
|      /                              \                              |
 
|      /                              \                              |
 
|    /                                \                            |
 
|    /                                \                            |
−
|    o                                   o                            |
+
|    o                 G                o                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
−
|    |                 G                |                            |
+
|    |                       o<---------T---------o                  |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
 
|    |                                  |                            |
Line 3,874: Line 3,529:  
|    \                                /                            |
 
|    \                                /                            |
 
|      \                              /                              |
 
|      \                              /                              |
−
|      \                           T /                              |
+
|      \                             /                              |
−
|        \             o<------------/-------------o                  |
+
|        \                           /                               |
 
|        \                        /                                |
 
|        \                        /                                |
 
|          \                      /                                  |
 
|          \                      /                                  |
Line 3,895: Line 3,550:  
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.
 
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.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Conjunction.jpg|500px]]
 
|-
 
|-
Line 3,905: Line 3,560:  
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>
 
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>
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]
 
|-
 
|-
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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>
 
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>
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]
 
|-
 
|-
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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.
 
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.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]
 
|-
 
|-
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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>
 
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>
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]
 
|-
 
|-
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Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
 
Just to be clear about what's being indicated here, it's a visual way of summarizing the following data:
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
|
 
|
 
<math>\begin{array}{rcccccc}
 
<math>\begin{array}{rcccccc}
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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:
 
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:
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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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.
 
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.
   −
{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
 
| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]
 
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Reading the arrows off the map produces the following data:
 
Reading the arrows off the map produces the following data:
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{| align="center" cellspacing="10" style="text-align:center"
+
{| align="center" cellpadding="10" style="text-align:center"
 
|
 
|
 
<math>\begin{array}{rcccccc}
 
<math>\begin{array}{rcccccc}
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# http://forum.wolframscience.com/showthread.php?postid=1602#post1602
 
# http://forum.wolframscience.com/showthread.php?postid=1602#post1602
 
# http://forum.wolframscience.com/showthread.php?postid=1603#post1603
 
# http://forum.wolframscience.com/showthread.php?postid=1603#post1603
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