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{{DISPLAYTITLE:Dynamics And Logic}}

{{DISPLAYTITLE:Dynamics And Logic}}

==Note 1==

==Note 1==

& \quad &

& \quad &

\operatorname{d}p ~\operatorname{or}~ \operatorname{d}q

\operatorname{d}p ~\operatorname{or}~ \operatorname{d}q

\end{matrix}</math>

\end{matrix}\!</math>

|}

|}

<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>

|}

|}

& = &

& = &

\{ \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.

(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)

((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)~

|}

|}

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:

|}

|}

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".

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]]

|                    e / \ e                    |

|                     /  \                      |

|                     /  e  \                    |

|      ( j  \  j  \  j  \  i  \  h  \  j  )     |

|        \  h  \  h  \  e  \  j  \  i  /        |

|}

|}

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>

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}

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}

|      /                              \                              |

|      /                              \                              |

|    /                                \                            |

|    /                                \                            |

|    o                                   o                            |

|    o                 G                o                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                 G                |                            |

|    |                       o<---------T---------o                  |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    |                                  |                            |

|    \                                /                            |

|    \                                /                            |

|      \                              /                              |

|      \                              /                              |

|      \                           T /                              |

|      \                             /                              |

|        \             o<------------/-------------o                  |

|        \                           /                               |

|        \                        /                                |

|        \                        /                                |

|          \                      /                                  |

|          \                      /                                  |

# 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