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

Revision as of 05:24, 27 May 2009

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→‎Propositional Forms on Two Variables: table colors
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By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
 
By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions in a number of different languages for zeroth order logic.
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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<br>
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 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 1.  Propositional Forms on Two Variables'''
 
|+ '''Table 1.  Propositional Forms on Two Variables'''
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_2</math>
 
| <math>\mathcal{L}_2</math>
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| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_6</math>
 
| <math>\mathcal{L}_6</math>
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|- style="background:ghostwhite; height:48px"
+
|- style="background:#e6e6ff; height:48px"
 
| &nbsp;
 
| &nbsp;
 
|
 
|
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{| align="right" style="background:ghostwhite; text-align:right"
+
{| align="right" style="background:#e6e6ff; text-align:right"
 
|-
 
|-
 
| <math>x\!</math> :
 
| <math>x\!</math> :
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|}
 
|}
 
|
 
|
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{| align="center" style="background:ghostwhite"
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{| align="center" style="background:#e6e6ff"
 
|-
 
|-
 
| 1 1 0 0
 
| 1 1 0 0
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|}
 
|}
 
|}
 
|}
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<br>
 
<br>
    
Table&nbsp;2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes.
 
Table&nbsp;2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes.
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{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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<br>
 +
 
 +
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 2.  Propositional Forms on Two Variables'''
 
|+ '''Table 2.  Propositional Forms on Two Variables'''
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_1</math>
 
| <math>\mathcal{L}_2</math>
 
| <math>\mathcal{L}_2</math>
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| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_5</math>
 
| <math>\mathcal{L}_6</math>
 
| <math>\mathcal{L}_6</math>
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|- style="background:ghostwhite; height:48px"
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|- style="background:#e6e6ff; height:48px"
 
| &nbsp;
 
| &nbsp;
 
|
 
|
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{| align="right" style="background:ghostwhite; text-align:right"
+
{| align="right" style="background:#e6e6ff; text-align:right"
 
|-
 
|-
 
| <math>x\!</math> :
 
| <math>x\!</math> :
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|}
 
|}
 
|
 
|
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{| align="center" style="background:ghostwhite"
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{| align="center" style="background:#e6e6ff"
 
|-
 
|-
 
| 1 1 0 0
 
| 1 1 0 0
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| <p><math>1\!</math></p>
 
| <p><math>1\!</math></p>
 
|}
 
|}
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<br>
 
<br>
    
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
 
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
   −
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
<br>
 +
 
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
 
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
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|- style="background:ghostwhite; height:48px"
+
|- style="background:#e6e6ff; height:48px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
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| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| colspan="2" | Fixed Point Total :
 
| colspan="2" | Fixed Point Total :
 
| <math>4\!</math>
 
| <math>4\!</math>
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| <math>16\!</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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
 
|+ '''Table 4.  <math>\operatorname{D}f</math> Expanded Over Differential Features <math>\{ \operatorname{d}x, \operatorname{d}y \}</math>'''
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
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| <math>(~)\!</math>
 
| <math>(~)\!</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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|+ '''Table 5.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
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| <math>((~))\!</math>
 
| <math>((~))\!</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="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|+ '''Table 6.  <math>\operatorname{D}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
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|- style="background:ghostwhite; height:36px"
+
|- style="background:#e6e6ff; height:36px"
 
| &nbsp;
 
| &nbsp;
 
| <math>f\!</math>
 
| <math>f\!</math>
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| <math>(~)\!</math>
 
| <math>(~)\!</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
Jon Awbrey
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