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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)

Revision as of 17:44, 13 December 2008

877 bytes added ,  17:44, 13 December 2008
→‎Higher Order Propositions and Logical Operators (''n'' = 2): redo table in fractal checkerboard style
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There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;3 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
 
There are <math>2^{16} = 65536\!</math> measures of the type <math>m : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>  Table&nbsp;3 introduces the first 24 of these measures in the fashion of the higher order truth table that I used before.  The column headed <math>m_j\!</math> shows the values of the measure <math>m_j\!</math> on each of the propositions <math>f_i : \mathbb{B}^2 \to \mathbb{B},</math> for <math>i\!</math> = 0 to 23, with blank entries in the Table being optional for values of zero.  The arrangement of measures that continues according to the plan indicated here is referred to as the ''standard ordering'' of these measures.  In this scheme of things, the index <math>j\!</math> of the measure <math>m_j\!</math> is the decimal equivalent of the bit string that is associated with <math>m_j\!</math>'s functional values, which can be obtained in turn by reading the <math>j^\mathrm{th}\!</math> column of binary digits in the Table as the corresponding range of boolean values, taking them up in the order from bottom to top.
   −
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
+
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:white; color:black; font-weight:bold; text-align:center; width:96%"
 
|+ '''Table 3.  Higher Order Propositions (''n'' = 2)'''
 
|+ '''Table 3.  Higher Order Propositions (''n'' = 2)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| align="right" | <math>x:</math><br><math>y:</math>
+
| align="right" | <math>u:</math><br><math>v:</math>
 
| 1100<br>1010
 
| 1100<br>1010
 
| <math>f\!</math>
 
| <math>f\!</math>
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| <math>m_{23}</math>
 
| <math>m_{23}</math>
 
|-
 
|-
−
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
−
| 0   || 1   || 0    || 1   || 0    || 1   || 0    || 1
+
| 0000
−
| 0   || 1   || 0    || 1   || 0    || 1   || 0    || 1
+
| <math>(~)</math>
−
| 0   || 1   || 0    || 1   || 0    || 1   || 0    || 1
+
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 +
| 0 || style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_1</math> || 0001 || <math>(x)(y)\!</math>
+
| <math>f_1</math>
−
|&nbsp;||&nbsp;|| 1   || 1   || 0   || 0   || 1   || 1
+
| 0001
−
| 0   || 0   || 1   || 1   || 0   || 0   || 1   || 1
+
| <math>(u)(v)\!</math>
−
| 0   || 0   || 1   || 1   || 0   || 0   || 1   || 1
+
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_2</math> || 0010 || <math>(x) y\!</math>
+
| <math>f_2</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| 0010
−
| 0   || 0   || 0   || 0   || 1   || 1   || 1   || 1
+
| <math>(u) v\!</math>
−
| 0   || 0   || 0   || 0   || 1   || 1   || 1   || 1
+
| 0 || 0 || 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0 || 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0 || 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_3</math> || 0011 || <math>(x)\!</math>
+
| <math>f_3</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0011
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>(u)\!</math>
−
| 0   || 0   || 0   || 0   || 0   || 0   || 0   || 0
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_4</math> || 0100 || <math>x (y)\!</math>
+
| <math>f_4</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>u (v)\!</math>
−
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
−
| <math>f_5</math> || 0101 || <math>(y)\!</math>
+
| <math>f_5</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(v)\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_6</math> || 0110 || <math>(x, y)\!</math>
+
| <math>f_6</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u, v)\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_7</math> || 0111 || <math>(x y)\!</math>
+
| <math>f_7</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u v)\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_8</math> || 1000 || <math>x y\!</math>
+
| <math>f_8</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1000
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>u v\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_9</math> || 1001 || <math>((x, y))\!</math>
+
| <math>f_9</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((u, v))\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{10}</math> || 1010 || <math>y\!</math>
+
| <math>f_{10}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>v\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{11}</math> || 1011 || <math>(x (y))\!</math>
+
| <math>f_{11}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>(u (v))\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{12}</math> || 1100 || <math>x\!</math>
+
| <math>f_{12}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1100
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>u\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{13}</math> || 1101 || <math>((x) y)\!</math>
+
| <math>f_{13}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1101
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((u) v)\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math>
+
| <math>f_{14}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1110
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((u)(v))\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|-
 
|-
−
| <math>f_{15}</math> || 1111 || <math>((~))\!</math>
+
| <math>f_{15}</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1111
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| <math>((~))\!</math>
−
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 +
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
|}<br>
 
|}<br>
  
Jon Awbrey
12,080

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