Changes

MyWikiBiz, Author Your Legacy — Sunday December 01, 2024
Jump to navigationJump to search

Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy (view source)

Revision as of 12:14, 19 November 2009

461 bytes added ,  12:14, 19 November 2009
→‎Higher Order Propositions and Logical Operators (''n'' = 2): reformat table
Line 313: Line 313:  
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;12 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;12 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="background:white; color:black; font-weight:bold; text-align:center; width:96%"
+
{| align="center" style="background:white; color:black; text-align:center; width:90%"
 
|+ '''Table 12.  Higher Order Propositions (''n'' = 2)'''
 
|+ '''Table 12.  Higher Order Propositions (''n'' = 2)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| align="right" | <math>u:</math><br><math>v:</math>
+
| align="right" | <math>\begin{matrix}u\!:\\v\!:\end{matrix}</math>
−
| 1100<br>1010
+
| <math>\begin{matrix}1100\\1010\end{matrix}</math>
 
| <math>f\!</math>
 
| <math>f\!</math>
−
| <math>m_0</math>
+
| <math>\underset{0}{m}</math>
−
| <math>m_1</math>
+
| <math>\underset{1}{m}</math>
−
| <math>m_2</math>
+
| <math>\underset{2}{m}</math>
−
| <math>m_3</math>
+
| <math>\underset{3}{m}</math>
−
| <math>m_4</math>
+
| <math>\underset{4}{m}</math>
−
| <math>m_5</math>
+
| <math>\underset{5}{m}</math>
−
| <math>m_6</math>
+
| <math>\underset{6}{m}</math>
−
| <math>m_7</math>
+
| <math>\underset{7}{m}</math>
−
| <math>m_8</math>
+
| <math>\underset{8}{m}</math>
−
| <math>m_9</math>
+
| <math>\underset{9}{m}</math>
−
| <math>m_{10}</math>
+
| <math>\underset{10}{m}</math>
−
| <math>m_{11}</math>
+
| <math>\underset{11}{m}</math>
−
| <math>m_{12}</math>
+
| <math>\underset{12}{m}</math>
−
| <math>m_{13}</math>
+
| <math>\underset{13}{m}</math>
−
| <math>m_{14}</math>
+
| <math>\underset{14}{m}</math>
−
| <math>m_{15}</math>
+
| <math>\underset{15}{m}</math>
−
| <math>m_{16}</math>
+
| <math>\underset{16}{m}</math>
−
| <math>m_{17}</math>
+
| <math>\underset{17}{m}</math>
−
| <math>m_{18}</math>
+
| <math>\underset{18}{m}</math>
−
| <math>m_{19}</math>
+
| <math>\underset{19}{m}</math>
−
| <math>m_{20}</math>
+
| <math>\underset{20}{m}</math>
−
| <math>m_{21}</math>
+
| <math>\underset{21}{m}</math>
−
| <math>m_{22}</math>
+
| <math>\underset{22}{m}</math>
−
| <math>m_{23}</math>
+
| <math>\underset{23}{m}</math>
 
|-
 
|-
 
| <math>f_0</math>
 
| <math>f_0</math>
−
| 0000
+
| <math>0000</math>
 
| <math>\texttt{(~)}</math>
 
| <math>\texttt{(~)}</math>
 
| 0 || style="background:black; color:white" | 1
 
| 0 || style="background:black; color:white" | 1
Line 361: Line 361:  
|-
 
|-
 
| <math>f_1</math>
 
| <math>f_1</math>
−
| 0001
+
| <math>0001</math>
 
| <math>\texttt{(u)(v)}</math>
 
| <math>\texttt{(u)(v)}</math>
 
| 0 || 0
 
| 0 || 0
Line 383: Line 383:  
|-
 
|-
 
| <math>f_2</math>
 
| <math>f_2</math>
−
| 0010
+
| <math>0010</math>
 
| <math>\texttt{(u) v}</math>
 
| <math>\texttt{(u) v}</math>
 
| 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0
Line 402: Line 402:  
|-
 
|-
 
| <math>f_3</math>
 
| <math>f_3</math>
−
| 0011
+
| <math>0011</math>
 
| <math>\texttt{(u)}</math>
 
| <math>\texttt{(u)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 416: Line 416:  
|-
 
|-
 
| <math>f_4</math>
 
| <math>f_4</math>
−
| 0100
+
| <math>0100</math>
 
| <math>\texttt{u (v)}</math>
 
| <math>\texttt{u (v)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 430: Line 430:  
|-
 
|-
 
| <math>f_5</math>
 
| <math>f_5</math>
−
| 0101
+
| <math>0101</math>
 
| <math>\texttt{(v)}</math>
 
| <math>\texttt{(v)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 437: Line 437:  
|-
 
|-
 
| <math>f_6</math>
 
| <math>f_6</math>
−
| 0110
+
| <math>0110</math>
 
| <math>\texttt{(u, v)}</math>
 
| <math>\texttt{(u, v)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 444: Line 444:  
|-
 
|-
 
| <math>f_7</math>
 
| <math>f_7</math>
−
| 0111
+
| <math>0111</math>
 
| <math>\texttt{(u v)}</math>
 
| <math>\texttt{(u v)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 451: Line 451:  
|-
 
|-
 
| <math>f_8</math>
 
| <math>f_8</math>
−
| 1000
+
| <math>1000</math>
 
| <math>\texttt{u v}</math>
 
| <math>\texttt{u v}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 458: Line 458:  
|-
 
|-
 
| <math>f_9</math>
 
| <math>f_9</math>
−
| 1001
+
| <math>1001</math>
 
| <math>\texttt{((u, v))}</math>
 
| <math>\texttt{((u, v))}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 465: Line 465:  
|-
 
|-
 
| <math>f_{10}</math>
 
| <math>f_{10}</math>
−
| 1010
+
| <math>1010</math>
 
| <math>\texttt{v}</math>
 
| <math>\texttt{v}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 472: Line 472:  
|-
 
|-
 
| <math>f_{11}</math>
 
| <math>f_{11}</math>
−
| 1011
+
| <math>1011</math>
 
| <math>\texttt{(u (v))}</math>
 
| <math>\texttt{(u (v))}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 479: Line 479:  
|-
 
|-
 
| <math>f_{12}</math>
 
| <math>f_{12}</math>
−
| 1100
+
| <math>1100</math>
 
| <math>\texttt{u}</math>
 
| <math>\texttt{u}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 486: Line 486:  
|-
 
|-
 
| <math>f_{13}</math>
 
| <math>f_{13}</math>
−
| 1101
+
| <math>1101</math>
 
| <math>\texttt{((u) v)}</math>
 
| <math>\texttt{((u) v)}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 493: Line 493:  
|-
 
|-
 
| <math>f_{14}</math>
 
| <math>f_{14}</math>
−
| 1110
+
| <math>1110</math>
 
| <math>\texttt{((u)(v))}</math>
 
| <math>\texttt{((u)(v))}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
Line 500: Line 500:  
|-
 
|-
 
| <math>f_{15}</math>
 
| <math>f_{15}</math>
−
| 1111
+
| <math>1111</math>
 
| <math>\texttt{((~))}</math>
 
| <math>\texttt{((~))}</math>
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
 
| 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>
    
===Umpire Operators===
 
===Umpire Operators===
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/98674"

Navigation menu