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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 01:15, 8 June 2008

No change in size ,  01:15, 8 June 2008
→‎Propositional Forms on Two Variables: reorder tables
Line 3,664: Line 3,664:     
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
+
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
−
|- style="background:ghostwhite; height:36px"
+
|- style="background:ghostwhite; height:48px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{E}f|_{xy}</math>
+
|
−
| <math>\operatorname{E}f|_{x(y)}</math>
+
{| align="center"
−
| <math>\operatorname{E}f|_{(x)y}</math>
+
|-
−
| <math>\operatorname{E}f|_{(x)(y)}</math>
+
| <math>\operatorname{T}_{11}f</math>
 +
|-
 +
| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| <math>\operatorname{T}_{10}f</math>
 +
|-
 +
| <math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| <math>\operatorname{T}_{01}f</math>
 +
|-
 +
| <math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| <math>\operatorname{T}_{00}f</math>
 +
|-
 +
| <math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
 +
|}
 
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 3,705: Line 3,729:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
+
| height="36px" | <math>(x)(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\ y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\ (y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\ y\!</math>
 
|}
 
|}
 
|-
 
|-
Line 3,764: Line 3,788:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|-
 
|-
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{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|-
 
|-
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{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|-
 
|-
Line 3,901: Line 3,925:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
+
| height="36px" | <math>(x\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
+
| height="36px" | <math>(x\ (y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
+
| height="36px" | <math>((x)\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x)(y))\!</math>
 
|}
 
|}
 
|- style="height:36px"
 
|- style="height:36px"
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| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 +
|- style="background:ghostwhite; height:36px"
 +
| colspan="2" | Fixed Point Total :
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 
|}
 
|}
 
<br>
 
<br>
    
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
+
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
 
|- style="background:ghostwhite; height:36px"
 
|- style="background:ghostwhite; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{D}f|_{xy}</math>
+
| <math>\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
−
| <math>\operatorname{D}f|_{x(y)}</math>
+
| <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
−
| <math>\operatorname{D}f|_{(x)y}</math>
+
| <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
−
| <math>\operatorname{D}f|_{(x)(y)}</math>
+
| <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
 
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 3,994: Line 4,024:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,053: Line 4,083:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,096: Line 4,126:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,139: Line 4,169:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>((~))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|-
 
|-
Line 4,190: Line 4,220:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>((x,\ y))\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(y)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>y\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(y)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>x\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(x)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(x)\!</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
+
| height="36px" | <math>(~)\!</math>
 
|-
 
|-
−
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
+
| height="36px" | <math>(~)\!</math>
 
|}
 
|}
 
|- style="height:36px"
 
|- style="height:36px"
Line 4,242: Line 4,272:     
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
+
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
−
|- style="background:ghostwhite; height:48px"
+
|- style="background:ghostwhite; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
|
+
| <math>\operatorname{E}f|_{xy}</math>
−
{| align="center"
+
| <math>\operatorname{E}f|_{x(y)}</math>
−
|-
+
| <math>\operatorname{E}f|_{(x)y}</math>
−
| <math>\operatorname{T}_{11}f</math>
+
| <math>\operatorname{E}f|_{(x)(y)}</math>
−
|-
  −
| <math>\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{10}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{01}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
  −
|}
  −
|
  −
{| align="center"
  −
|-
  −
| <math>\operatorname{T}_{00}f</math>
  −
|-
  −
| <math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
  −
|}
   
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 4,307: Line 4,313:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\ y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\ (y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>x\ y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 4,366: Line 4,372:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|-
 
|-
Line 4,409: Line 4,415:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
 
|}
 
|}
 
|-
 
|-
Line 4,452: Line 4,458:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>(\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 4,503: Line 4,509:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x\ (y))\!</math>
+
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>((x)\ y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>((x)(y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|}
 
|}
 
|- style="height:36px"
 
|- style="height:36px"
Line 4,551: Line 4,557:  
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
−
|- style="background:ghostwhite; height:36px"
  −
| colspan="2" | Fixed Point Total :
  −
| <math>4\!</math>
  −
| <math>4\!</math>
  −
| <math>4\!</math>
  −
| <math>16\!</math>
   
|}
 
|}
 
<br>
 
<br>
    
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
+
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|- style="background:ghostwhite; height:36px"
 
|- style="background:ghostwhite; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}</math>
+
| <math>\operatorname{D}f|_{xy}</math>
−
| <math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
+
| <math>\operatorname{D}f|_{x(y)}</math>
−
| <math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
+
| <math>\operatorname{D}f|_{(x)y}</math>
−
| <math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
+
| <math>\operatorname{D}f|_{(x)(y)}</math>
 
|- style="height:36px"
 
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
Line 4,602: Line 4,602:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(x,\ y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>((x,\ y))\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(y)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>y\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(x)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>x\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 
|-
 
|-
−
| height="36px" | <math>(~)\!</math>
+
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 
|}
 
|}
 
|-
 
|-
Line 4,661: Line 4,661:  
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|}
 
|}
 
|
 
|
 
{| align="center"
 
{| align="center"
 
|-
 
|-
−
| height="36px" | <math>((~))\!</math>
+
| height="36px" | <math>\operatorname{d}x</math>
 
|-
 
|-
−
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Line 4,855: Line 4,855:  
So let us do just that.
 
So let us do just that.
   −
I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way.  The data for that account is contained in Table&nbsp;5.
+
I will first rationalize the novel grouping of propositional forms in the last set of Tables, as that will extend a gentle invitation to the mathematical subject of group theory, and demonstrate its relevance to differential logic in a strikingly apt and useful way.  The data for that account is contained in Table&nbsp;3.
    
The shift operator ''E'' can be understood as enacting a substitution operation on the proposition that is given as its argument.  In our immediate example, we have the following data and definition:
 
The shift operator ''E'' can be understood as enacting a substitution operation on the proposition that is given as its argument.  In our immediate example, we have the following data and definition:
Jon Awbrey
12,089

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