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

Revision as of 19:38, 29 May 2008

4,934 bytes added ,  19:38, 29 May 2008
→‎Propositional Forms on Two Variables: update
Line 3,066: Line 3,066:  
|+ '''Table 1.  Propositional Forms on Two Variables'''
 
|+ '''Table 1.  Propositional Forms on Two Variables'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| style="width:16%" | <math>\mathcal{L}_1</math>
+
| <math>\mathcal{L}_1</math>
−
| style="width:16%" | <math>\mathcal{L}_2</math>
+
| <math>\mathcal{L}_2</math>
−
| style="width:16%" | <math>\mathcal{L}_3</math>
+
| <math>\mathcal{L}_3</math>
−
| style="width:16%" | <math>\mathcal{L}_4</math>
+
| <math>\mathcal{L}_4</math>
−
| style="width:16%" | <math>\mathcal{L}_5</math>
+
| <math>\mathcal{L}_5</math>
−
| style="width:16%" | <math>\mathcal{L}_6</math>
+
| <math>\mathcal{L}_6</math>
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| &nbsp;
 
| &nbsp;
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| 0 0 0 0
 
| 0 0 0 0
 
| <math>(~)\!</math>
 
| <math>(~)\!</math>
−
| false
+
| <math>\operatorname{false}</math>
 
| <math>0\!</math>
 
| <math>0\!</math>
 
|-
 
|-
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| 0 0 0 1
 
| 0 0 0 1
 
| <math>(x)(y)\!</math>
 
| <math>(x)(y)\!</math>
−
| neither x nor y
+
| <math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math>
 
| <math>\lnot x \land \lnot y\!</math>
 
| <math>\lnot x \land \lnot y\!</math>
 
|-
 
|-
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| 0 0 1 0
 
| 0 0 1 0
 
| <math>(x)\ y\!</math>
 
| <math>(x)\ y\!</math>
−
| y and not x
+
| <math>y\ \operatorname{without}\ x</math>
 
| <math>\lnot x \land y\!</math>
 
| <math>\lnot x \land y\!</math>
 
|-
 
|-
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| 0 0 1 1
 
| 0 0 1 1
 
| <math>(x)\!</math>
 
| <math>(x)\!</math>
−
| not x
+
| <math>\operatorname{not}\ x</math>
 
| <math>\lnot x\!</math>
 
| <math>\lnot x\!</math>
 
|-
 
|-
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| 0 1 0 0
 
| 0 1 0 0
 
| <math>x\ (y)\!</math>
 
| <math>x\ (y)\!</math>
−
| x and not y
+
| <math>x\ \operatorname{without}\ y</math>
 
| <math>x \land \lnot y\!</math>
 
| <math>x \land \lnot y\!</math>
 
|-
 
|-
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| 0 1 0 1
 
| 0 1 0 1
 
| <math>(y)\!</math>
 
| <math>(y)\!</math>
−
| not y
+
| <math>\operatorname{not}\ y</math>
 
| <math>\lnot y\!</math>
 
| <math>\lnot y\!</math>
 
|-
 
|-
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| 0 1 1 0
 
| 0 1 1 0
 
| <math>(x,\ y)\!</math>
 
| <math>(x,\ y)\!</math>
−
| x not equal to y
+
| <math>x\ \operatorname{not~equal~to}\ y</math>
 
| <math>x \ne y\!</math>
 
| <math>x \ne y\!</math>
 
|-
 
|-
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| 0 1 1 1
 
| 0 1 1 1
 
| <math>(x\ y)\!</math>
 
| <math>(x\ y)\!</math>
−
| not both x and y
+
| <math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math>
 
| <math>\lnot x \lor \lnot y\!</math>
 
| <math>\lnot x \lor \lnot y\!</math>
 
|-
 
|-
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| 1 0 0 0
 
| 1 0 0 0
 
| <math>x\ y\!</math>
 
| <math>x\ y\!</math>
−
| x and y
+
| <math>x\ \operatorname{and}\ y</math>
 
| <math>x \land y\!</math>
 
| <math>x \land y\!</math>
 
|-
 
|-
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| 1 0 0 1
 
| 1 0 0 1
 
| <math>((x,\ y))\!</math>
 
| <math>((x,\ y))\!</math>
−
| x equal to y
+
| <math>x\ \operatorname{equal~to}\ y</math>
 
| <math>x = y\!</math>
 
| <math>x = y\!</math>
 
|-
 
|-
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| 1 0 1 0
 
| 1 0 1 0
 
| <math>y\!</math>
 
| <math>y\!</math>
−
| y
+
| <math>y\!</math>
 
| <math>y\!</math>
 
| <math>y\!</math>
 
|-
 
|-
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| 1 0 1 1
 
| 1 0 1 1
 
| <math>(x\ (y))\!</math>
 
| <math>(x\ (y))\!</math>
−
| not x without y
+
| <math>\operatorname{not}\ x\ \operatorname{without}\ y</math>
 
| <math>x \Rightarrow y\!</math>
 
| <math>x \Rightarrow y\!</math>
 
|-
 
|-
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| 1 1 0 0
 
| 1 1 0 0
 
| <math>x\!</math>
 
| <math>x\!</math>
−
| x
+
| <math>x\!</math>
 
| <math>x\!</math>
 
| <math>x\!</math>
 
|-
 
|-
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| 1 1 0 1
 
| 1 1 0 1
 
| <math>((x)\ y)\!</math>
 
| <math>((x)\ y)\!</math>
−
| not y without x
+
| <math>\operatorname{not}\ y\ \operatorname{without}\ x</math>
 
| <math>x \Leftarrow y\!</math>
 
| <math>x \Leftarrow y\!</math>
 
|-
 
|-
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| 1 1 1 0
 
| 1 1 1 0
 
| <math>((x)(y))\!</math>
 
| <math>((x)(y))\!</math>
−
| x or y
+
| <math>x\ \operatorname{or}\ y</math>
 
| <math>x \lor y\!</math>
 
| <math>x \lor y\!</math>
 
|-
 
|-
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| 1 1 1 1
 
| 1 1 1 1
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
−
| true
+
| <math>\operatorname{true}</math>
 
| <math>1\!</math>
 
| <math>1\!</math>
−
|}<br>
+
|}
 +
<br>
 +
 
 +
Table&nbsp;2 exhibits the same information in a different order, grouping the sixteen functions into seven natural classes.
 +
 
 +
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 +
|+ '''Table 2.  Propositional Forms on Two Variables'''
 +
|- style="background:ghostwhite"
 +
| <math>\mathcal{L}_1</math>
 +
| <math>\mathcal{L}_2</math>
 +
| <math>\mathcal{L}_3</math>
 +
| <math>\mathcal{L}_4</math>
 +
| <math>\mathcal{L}_5</math>
 +
| <math>\mathcal{L}_6</math>
 +
|- style="background:ghostwhite"
 +
| <p>&nbsp;</p>
 +
| align="right" | <p><math>x\!</math> :</p>
 +
| <p>1 1 0 0</p>
 +
| <p>&nbsp;</p>
 +
| <p>&nbsp;</p>
 +
| <p>&nbsp;</p>
 +
|- style="background:ghostwhite"
 +
| <p>&nbsp;</p>
 +
| align="right" | <p><math>y\!</math> :</p>
 +
| <p>1 0 1 0</p>
 +
| <p>&nbsp;</p>
 +
| <p>&nbsp;</p>
 +
| <p>&nbsp;</p>
 +
|-
 +
| <p><math>f_{0}\!</math></p>
 +
| <p><math>f_{0000}\!</math></p>
 +
| <p>0 0 0 0</p>
 +
| <p><math>(~)\!</math></p>
 +
| <p><math>\operatorname{false}</math></p>
 +
| <p><math>1\!</math></p>
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{1}\!</math></p>
 +
<p><math>f_{2}\!</math></p>
 +
<p><math>f_{4}\!</math></p>
 +
<p><math>f_{8}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{0001}\!</math></p>
 +
<p><math>f_{0010}\!</math></p>
 +
<p><math>f_{0100}\!</math></p>
 +
<p><math>f_{1000}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 0 0 1</p>
 +
<p>0 0 1 0</p>
 +
<p>0 1 0 0</p>
 +
<p>1 0 0 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>(x)(y)\!</math></p>
 +
<p><math>(x)\ y\!</math></p>
 +
<p><math>x\ (y)\!</math></p>
 +
<p><math>x\ y\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\operatorname{neither}\ x\ \operatorname{nor}\ y</math></p>
 +
<p><math>y\ \operatorname{without}\ x</math></p>
 +
<p><math>x\ \operatorname{without}\ y</math></p>
 +
<p><math>x\ \operatorname{and}\ y</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\lnot x \land \lnot y</math></p>
 +
<p><math>\lnot x \land y</math></p>
 +
<p><math>x \land \lnot y</math></p>
 +
<p><math>x \land y</math></p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{3}\!</math></p>
 +
<p><math>f_{12}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{0011}\!</math></p>
 +
<p><math>f_{1100}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 0 1 1</p>
 +
<p>1 1 0 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>(x)\!</math></p>
 +
<p><math>x\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\operatorname{not}\ x</math></p>
 +
<p><math>x\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\lnot x</math></p>
 +
<p><math>x\!</math></p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{6}\!</math></p>
 +
<p><math>f_{9}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{0110}\!</math></p>
 +
<p><math>f_{1001}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 1 0</p>
 +
<p>1 0 0 1</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>(x,\ y)\!</math></p>
 +
<p><math>((x,\ y))\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>x\ \operatorname{not~equal~to}\ y</math></p>
 +
<p><math>x\ \operatorname{equal~to}\ y</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>x \ne y</math></p>
 +
<p><math>x = y\!</math></p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{5}\!</math></p>
 +
<p><math>f_{10}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{0101}\!</math></p>
 +
<p><math>f_{1010}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 0 1</p>
 +
<p>1 0 1 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>(y)\!</math></p>
 +
<p><math>y\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\operatorname{not}\ y</math></p>
 +
<p><math>y\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\lnot y</math></p>
 +
<p><math>y\!</math></p>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{7}\!</math></p>
 +
<p><math>f_{11}\!</math></p>
 +
<p><math>f_{13}\!</math></p>
 +
<p><math>f_{14}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>f_{0111}\!</math></p>
 +
<p><math>f_{1011}\!</math></p>
 +
<p><math>f_{1101}\!</math></p>
 +
<p><math>f_{1110}\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p>0 1 1 1</p>
 +
<p>1 0 1 1</p>
 +
<p>1 1 0 1</p>
 +
<p>1 1 1 0</p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>(x\ y)\!</math></p>
 +
<p><math>(x\ (y))\!</math></p>
 +
<p><math>((x)\ y)\!</math></p>
 +
<p><math>((x)(y))\!</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\operatorname{not~both}\ x\ \operatorname{and}\ y</math></p>
 +
<p><math>\operatorname{not}\ x\ \operatorname{without}\ y</math></p>
 +
<p><math>\operatorname{not}\ y\ \operatorname{without}\ x</math></p>
 +
<p><math>x\ \operatorname{or}\ y</math></p>
 +
|}
 +
|
 +
{| align="center"
 +
|
 +
<p><math>\lnot x \lor \lnot y</math></p>
 +
<p><math>x \Rightarrow y</math></p>
 +
<p><math>x \Leftarrow y</math></p>
 +
<p><math>x \lor y</math></p>
 +
|}
 +
|-
 +
| <p><math>f_{15}\!</math></p>
 +
| <p><math>f_{1111}\!</math></p>
 +
| <p>1 1 1 1</p>
 +
| <p><math>((~))\!</math></p>
 +
| <p><math>\operatorname{true}</math></p>
 +
| <p><math>1\!</math></p>
 +
|}
 +
<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%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
−
|+ '''Table 2.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
+
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| style="width:16%" | &nbsp;
 
| style="width:16%" | &nbsp;
Line 3,326: Line 3,578:     
<pre>
 
<pre>
−
Table 3.  Df Expanded Over Ordinary Features {x, y}
+
Table 4.  Df Expanded Over Ordinary Features {x, y}
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
Line 3,380: Line 3,632:  
</pre>
 
</pre>
 
<pre>
 
<pre>
−
Table 4.  Ef Expanded Over Differential Features {dx, dy}
+
Table 5.  Ef Expanded Over Differential Features {dx, dy}
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
Line 3,440: Line 3,692:  
</pre>
 
</pre>
 
<pre>
 
<pre>
−
Table 5.  Df Expanded Over Differential Features {dx, dy}
+
Table 6.  Df Expanded Over Differential Features {dx, dy}
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
Line 3,500: Line 3,752:  
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 4.
+
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.
    
<pre>
 
<pre>
−
Table 4.  Ef Expanded Over Differential Features {dx, dy}
+
Table 5.  Ef Expanded Over Differential Features {dx, dy}
 
o------o------------o------------o------------o------------o------------o
 
o------o------------o------------o------------o------------o------------o
 
|      |            |            |            |            |            |
 
|      |            |            |            |            |            |
Jon Awbrey
12,080

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