Directory:Jon Awbrey/Papers/Dynamics And Logic - Revision history

Jon Awbrey: add cats

2014-12-10T13:28:31Z

add cats

Jon Awbrey: '''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.

2014-12-05T18:00:37Z

'''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 at 17:58, 5 December 2014

2014-12-05T17:58:51Z

Jon Awbrey: /* Note 2 */

2014-12-05T17:54:18Z

Note 2

Jon Awbrey: /* Note 23 */ tweak figure

2009-06-22T18:02:51Z

Note 23: tweak figure

Jon Awbrey: /* Note 19 */ convert graphics

2009-06-22T17:30:09Z

Note 19: convert graphics

Jon Awbrey: /* Note 17 */ convert graphics

2009-06-22T17:24:04Z

Note 17: convert graphics

Jon Awbrey: adjust spacing

2009-06-22T15:00:20Z

adjust spacing

Show changes

Jon Awbrey: /* Note 6 */ convert graphics

2009-06-22T14:14:47Z

Note 6: convert graphics

Show changes

Jon Awbrey: /* Note 5 */ spacing

2009-06-22T14:08:36Z

Note 5: spacing

← Older revision		Revision as of 13:28, 10 December 2014
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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
		+
		+	[[Category:Artificial Intelligence]]
		+	[[Category:Boolean Algebra]]
		+	[[Category:Boolean Functions]]
		+	[[Category:Charles Sanders Peirce]]
		+	[[Category:Combinatorics]]
		+	[[Category:Computational Complexity]]
		+	[[Category:Computer Science]]
		+	[[Category:Cybernetics]]
		+	[[Category:Differential Logic]]
		+	[[Category:Equational Reasoning]]
		+	[[Category:Formal Languages]]
		+	[[Category:Formal Systems]]
		+	[[Category:Graph Theory]]
		+	[[Category:Inquiry]]
		+	[[Category:Inquiry Driven Systems]]
		+	[[Category:Knowledge Representation]]
		+	[[Category:Logic]]
		+	[[Category:Logical Graphs]]
		+	[[Category:Mathematics]]
		+	[[Category:Philosophy]]
		+	[[Category:Propositional Calculus]]
		+	[[Category:Semiotics]]
		+	[[Category:Visualization]]

@@ Line 1: / Line 1: @@
 {{DISPLAYTITLE:Dynamics And Logic}}
-'''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.
+'''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==

← Older revision		Revision as of 17:58, 5 December 2014
Line 1:		Line 1:
	{{DISPLAYTITLE:Dynamics And Logic}}		{{DISPLAYTITLE:Dynamics And Logic}}
		+
		+	'''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.
		+
	==Note 1==		==Note 1==

@@ Line 84: / Line 84: @@
 & \quad &
 \operatorname{d}p ~\operatorname{or}~ \operatorname{d}q
-\end{matrix}</math>
+\end{matrix}\!</math>
 |}
@@ Line 133: / Line 133: @@
 <math>\begin{array}{rcc}
 \operatorname{E}X & = & X \times \operatorname{d}X
-\end{array}</math>
+\end{array}\!</math>
 |}
@@ Line 156: / Line 156: @@
 & = &
 \{ \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>".
 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 572: / Line 571: @@
 (p)~q~
 \\[4pt]
-(p)~~~
+(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
 \\[4pt]
 ~p~(q)
 \\[4pt]
-~~~(q)
+[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)
 \\[4pt]
 (p,~q)
@@ Line 679: / Line 678: @@
 ((p,~q))
 \\[4pt]
-~~~~~q~~
+:54, 5 December 2014 (UTC)q~~
 \\[4pt]
 ~(p~(q))
 \\[4pt]
-~~p~~~~~
+~~p17:54, 5 December 2014 (UTC)
 \\[4pt]
 ((p)~q)~
@@ Line 2,631: / Line 2,630: @@
 |}
-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:
@@ Line 2,731: / Line 2,730: @@
 |}
-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".

@@ Line 3,516: / Line 3,516: @@
 |      /                               \                              |
 |     /                                 \                             |
-|    o                                   o                            |
+|    o                 G                 o                            |
 |    |                                   |                            |
 |    |                                   |                            |
 |    |                                   |                            |
-|    |                 G                 |                            |
+|    |                        o<---------T---------o                  |
 |    |                                   |                            |
 |    |                                   |                            |
@@ Line 3,527: / Line 3,527: @@
 |     \                                 /                             |
 |      \                               /                              |
-|       \                           T /                               |
+|       \                             /                               |
-|        \             o<------------/-------------o                  |
+|        \                           /                                |
 |         \                         /                                 |
 |          \                       /                                  |

@@ Line 3,026: / Line 3,026: @@
 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]]
 |}
@@ Line 3,080: / Line 3,038: @@
 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.
 | <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 3,090: / Line 3,048: @@
 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}}
 \operatorname{e}
@@ Line 3,145: / Line 3,103: @@
 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}}
 \operatorname{e}

@@ Line 318: / Line 318: @@
 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" cellspacing="20" style="text-align:center"
 | [[Image:Directed Graph PQ Difference Conj.jpg|500px]]
-|}
+|-
+|
 <math>\begin{array}{rcccccc}