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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)

Revision as of 20:42, 21 June 2007

2,706 bytes added ,  20:42, 21 June 2007
→‎Analytic Series : Coordinate Method: wiki table
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Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
 
Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
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<pre>
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{| align="center" border="2" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
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Table 50.  Computation of an Analytic Series in Terms of Coordinates
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|+ Table 50.  Computation of an Analytic Series in Terms of Coordinates
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o-----------o-------------o-------------oo-------------o---------o-------------o
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|
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| u     v | du    dv  | u'     v' || !e!J    EJ  |   DJ    | dJ  d^2.J |
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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o-----------o-------------o-------------oo-------------o---------o-------------o
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|
−
|           |             |             ||             |         |             |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
−
| 0     0 | 0     0   | 0     0   || 0     0   |   0   | 0     0   |
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| ''u''
−
|           |             |             ||             |         |             |
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| ''v''
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|           | 0     1  | 0     1   ||         0  |   0    | 0      0  |
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|}
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|           |             |             ||             |         |             |
+
|
−
|           | 1      0   | 1      0   ||         0   |   0   | 0      0  |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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|           |             |             ||             |         |             |
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| d''u''
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|           | 1      1  | 1      1  ||         1   |   1    | 0     1  |
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| d''v''
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|           |             |             ||             |         |             |
+
|}
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o-----------o-------------o-------------oo-------------o---------o-------------o
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|
−
|           |             |             ||             |         |             |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| 0    1  | 0     0   | 0     1  || 0     0   |   0    | 0      0  |
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| ''u''<font face="courier new">’</font>
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|           |             |             ||             |         |             |
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| ''v''<font face="courier new">’</font>
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|           | 0     1   | 0     0   ||         0   |   0   | 0     0   |
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|}
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|           |             |             ||             |         |             |
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|-
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|           | 1      0   | 1     1   ||         1   |   1    | 1      0   |
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|
−
|           |             |             ||             |         |             |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|           | 1      1  | 1      0   ||         0   |   0   | 1     1   |
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| 0 || 0
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|           |             |             ||             |         |             |
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|-
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o-----------o-------------o-------------oo-------------o---------o-------------o
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| &nbsp; || &nbsp;
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|           |             |             ||             |         |             |
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|-
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| 1    0 | 0     0  | 1     0  || 0     0   |   0   | 0     0  |
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| &nbsp; || &nbsp;
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|           |             |             ||             |         |             |
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|-
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|           | 0      1   | 1      1  ||         1  |   1    | 1     0   |
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| &nbsp; || &nbsp;
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|           |             |             ||             |         |             |
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|}
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|           | 1      0   | 0      0  ||         0   |    0   | 0     0  |
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|
−
|           |             |             ||             |         |             |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|           | 1      1  | 0     1  ||         0   |   0   | 1     1  |
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| 0 || 0
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|           |             |             ||             |         |             |
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|-
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o-----------o-------------o-------------oo-------------o---------o-------------o
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| 0 || 1
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|           |             |             ||             |         |             |
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|-
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| 1    1  | 0     0  | 1      1  || 1      1  |   0   | 0     0   |
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| 1 || 0
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|           |             |             ||             |         |             |
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|-
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|           |  0     1  | 1      0   ||         0   |   1   | 1      0   |
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| 1 || 1
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|           |             |             ||             |         |             |
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|}
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|           | 1      0   | 0      1   ||         0   |   1    | 1      0   |
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|
−
|           |             |             ||             |         |             |
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|           | 1     1   | 0     0   ||         0   |   1   | 0     1  |
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| 0 || 0
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|           |             |             ||             |         |             |
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|-
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o-----------o-------------o-------------oo-------------o---------o-------------o
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| 0 || 1
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</pre>
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|-
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| 1 || 0
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|-
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| 1 || 1
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|}
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|-
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 1
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 1
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|-
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| 1 || 0
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|-
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| 1 || 1
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 1
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|-
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| 0 || 0
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|-
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| 1 || 1
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|-
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| 1 || 0
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 0
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 1
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|-
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| 1 || 0
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|-
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| 1 || 1
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 0
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|-
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| 1 || 1
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|-
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| 0 || 0
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|-
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| 0 || 1
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|-
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| &nbsp; || &nbsp;
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 1 || 1
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|-
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| 1 || 0
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|-
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| 0 || 1
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|-
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| 0 || 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 1
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|-
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| 1 || 0
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|-
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| 1 || 1
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|}
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|}
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|
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| <math>\epsilon</math>''J''
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| E''J''
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| D''J''
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
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| d''J''
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| d<sup>2</sup>''J''
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0      || 0
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 1
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|}
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|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0
 +
|-
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| 0
 +
|-
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| 0
 +
|-
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| 1
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|}
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|
 +
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 0
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|-
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| 0 || 0
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|-
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| 0 || 1
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0     || 0
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 1
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|-
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| &nbsp; || 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0
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|-
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| 0
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|-
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| 1
 +
|-
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| 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 0 || 0
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|-
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| 1 || 0
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|-
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| 1 || 1
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0      || 0
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|-
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| &nbsp; || 1
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0
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|-
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| 1
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|-
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| 0
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|-
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| 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 1 || 0
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|-
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| 0 || 0
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|-
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| 1 || 1
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|}
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|-
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0      || 1
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 0
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|-
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| &nbsp; || 0
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0
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|-
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| 1
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|-
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| 1
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|-
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| 1
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|}
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|
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{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
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| 0 || 0
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|-
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| 1 || 0
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|-
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| 1 || 0
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| 0 || 1
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|}
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|}
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|}
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<br>
    
The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the ''contingent universe'' [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;], or the bundle of ''contingency spaces'' [d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;] over the universe [''u'',&nbsp;''v''].  Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described
 
The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the ''contingent universe'' [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;], or the bundle of ''contingency spaces'' [d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;] over the universe [''u'',&nbsp;''v''].  Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described
Jon Awbrey
12,122

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