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

Revision as of 18:46, 13 December 2008

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→‎Higher Order Propositional Expressions: redo table
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Table&nbsp;1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>f : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>f_i.\!</math>
 
Table&nbsp;1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion:  Columns&nbsp;1 and 2 form a truth table for the four <math>f : \mathbb{B} \to \mathbb{B},</math> turned on its side from the way that one is most likely accustomed to see truth tables, with the row leaders in Column&nbsp;1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column&nbsp;2 give the values of each function for the argument values that are listed in the corresponding column head.  Column&nbsp;3 displays one of the more usual expressions for the proposition in question.  The last sixteen columns are topped by a collection of conventional names for the higher order propositions, also known as the ''measures'' <math>m_j,\!</math> for <math>j\!</math> = 0 to 15, where the entries in the body of the Table record the values that each <math>m_j\!</math> assigns to each <math>f_i.\!</math>
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{| 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:white; color:black; font-weight:bold; text-align:center; width:96%"
 
|+ '''Table 1.  Higher Order Propositions (''n'' = 1)'''
 
|+ '''Table 1.  Higher Order Propositions (''n'' = 1)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| align="right" | <math>x</math>:
+
| align="right" | <math>x:</math>
 
| 1 0
 
| 1 0
−
| <math>f</math>
+
| <math>f\!</math>
 
| <math>m_0</math>
 
| <math>m_0</math>
 
| <math>m_1</math>
 
| <math>m_1</math>
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| 0 0
 
| 0 0
 
| <math>0\!</math>
 
| <math>0\!</math>
−
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1 || 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
 
|-
 
|-
 
| <math>f_1</math>
 
| <math>f_1</math>
 
| 0 1
 
| 0 1
 
| <math>(x)\!</math>
 
| <math>(x)\!</math>
−
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 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
 
|-
 
|-
 
| <math>f_2</math>
 
| <math>f_2</math>
 
| 1 0
 
| 1 0
 
| <math>x\!</math>
 
| <math>x\!</math>
−
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 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
 
|-
 
|-
 
| <math>f_3</math>
 
| <math>f_3</math>
 
| 1 1
 
| 1 1
 
| <math>1\!</math>
 
| <math>1\!</math>
−
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
+
| 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
 
|}<br>
 
|}<br>
  
Jon Awbrey
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