Changes

more table setting
Line 38: Line 38:  
For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>f : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</math>
 
For example, consider the case where <math>X = \mathbb{B}.</math>  Then there are exactly four propositions <math>f : \mathbb{B} \to \mathbb{B},</math> and exactly sixteen higher order propositions that are based on this set, all bearing the type <math>m : (\mathbb{B} \to \mathbb{B}) \to \mathbb{B}.</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>
+
Table&nbsp;10 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>
    
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:white; color:black; 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 10.  Higher Order Propositions (''n'' = 1)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>x:</math>
 
| align="right" | <math>x:</math>
Line 119: Line 119:  
|}<br>
 
|}<br>
   −
I am going to put off explaining Table&nbsp;2, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.
+
I am going to put off explaining Table&nbsp;11, that presents a sample of what I call ''interpretive categories'' for higher order propositions, until after we get beyond the 1-dimensional case, since these lower dimensional cases tend to be a bit ''condensed'' or ''degenerate'' in their structures, and a lot of what is going on here will almost automatically become clearer as soon as we get even two logical variables into the mix.
    
{| 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.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''
+
|+ '''Table 11.  Interpretive Categories for Higher Order Propositions (''n'' = 1)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| Measure
 
| Measure
Line 289: Line 289:  
To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.
 
To save a few words in the remainder of this discussion, I will use the terms ''measure'' and ''qualifier'' to refer to all types of higher order propositions and operators.  To describe the present setting in picturesque terms, the propositions of <math>[u, v]\!</math> may be regarded as a gallery of sixteen venn diagrams, while the measures <math>m : (X \to \mathbb{B}) \to \mathbb{B}</math> are analogous to a body of judges or a panel of critical viewers, each of whom evaluates each of the pictures as a whole and reports the ones that find favor or not.  In this way, each judge <math>m_j\!</math> partitions the gallery of pictures into two aesthetic portions, the pictures <math>m_j^{-1}(1)\!</math> that <math>m_j\!</math> likes and the pictures <math>m_j^{-1}(0)\!</math> that <math>m_j\!</math> dislikes.
   −
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;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="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 12.  Higher Order Propositions (''n'' = 2)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
Line 322: Line 322:  
| <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>(u)(v)\!</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>(u) v\!</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>(u)\!</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>u (v)\!</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>(v)\!</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>(u, v)\!</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>(u v)\!</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>u v\!</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>((u, v))\!</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>v\!</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>(u (v))\!</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>u\!</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>((u) v)\!</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>((u)(v))\!</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>
   Line 500: Line 582:     
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 4.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''
+
|+ '''Table 13.  Qualifiers of Implication Ordering:&nbsp; <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| 1100<br>1010
 
| 1100<br>1010
 
| <math>f\!</math>
 
| <math>f\!</math>
 +
| <math>\alpha_0</math>
 +
| <math>\alpha_1</math>
 +
| <math>\alpha_2</math>
 +
| <math>\alpha_3</math>
 +
| <math>\alpha_4</math>
 +
| <math>\alpha_5</math>
 +
| <math>\alpha_6</math>
 +
| <math>\alpha_7</math>
 +
| <math>\alpha_8</math>
 +
| <math>\alpha_9</math>
 +
| <math>\alpha_{10}</math>
 +
| <math>\alpha_{11}</math>
 +
| <math>\alpha_{12}</math>
 +
| <math>\alpha_{13}</math>
 +
| <math>\alpha_{14}</math>
 
| <math>\alpha_{15}</math>
 
| <math>\alpha_{15}</math>
| <math>\alpha_{14}</math>
  −
| <math>\alpha_{13}</math>
  −
| <math>\alpha_{12}</math>
  −
| <math>\alpha_{11}</math>
  −
| <math>\alpha_{10}</math>
  −
| <math>\alpha_9</math>
  −
| <math>\alpha_8</math>
  −
| <math>\alpha_7</math>
  −
| <math>\alpha_6</math>
  −
| <math>\alpha_5</math>
  −
| <math>\alpha_4</math>
  −
| <math>\alpha_3</math>
  −
| <math>\alpha_2</math>
  −
| <math>\alpha_1</math>
  −
| <math>\alpha_0</math>
   
|-
 
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0000
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(~)</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0001
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0010
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_3</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0011
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1    || 1    || 1
+
| <math>(u)\!</math>
 +
| 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_4</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0100
|&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_5</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0101
|&nbsp;||&nbsp;|| 1    || 1    ||&nbsp;||&nbsp;|| 1    || 1
+
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_6</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0110
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_7</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 0111
| 1    || 1    || 1    || 1    || 1    || 1    || 1    || 1
+
| <math>(u v)\!</math>
 +
| 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
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_8</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1000
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_9</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| 1001
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    || 1
+
| <math>((u, v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1010
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_{11}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1    || 1    || 1
+
| 1011
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1    || 1    || 1
+
| <math>(u (v))\!</math>
 +
| 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{12}</math>
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 1100
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>u\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
|&nbsp;||&nbsp;|| 1   || 1    ||&nbsp;||&nbsp;|| 1   || 1
+
| 1101
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>((u) v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
+
| <math>f_{14}</math>
|&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 1110
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>((u)(v))\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{15}</math> || 1111 || <math>((~))</math>
+
| <math>f_{15}</math>
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 1111
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>((~))</math>
 +
| 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
 +
| 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>
    
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 5.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''
+
|+ '''Table 14.  Qualifiers of Implication Ordering:&nbsp; <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
Line 610: Line 948:  
| <math>\beta_{15}</math>
 
| <math>\beta_{15}</math>
 
|-
 
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| 0000
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>(~)</math>
 +
| 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
 +
| 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_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| 0001
|&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>(u)(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| 0010
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u) v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_3</math>
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0011
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_4</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| 0100
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u (v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_5</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| 0101
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>(v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_6</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| 0110
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>(u, v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_7</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| 0111
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_8</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1000
| 1   || 1   || 1   || 1   || 1   || 1   || 1   || 1
+
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_9</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1001
|&nbsp;|| 1   ||&nbsp;|| 1   ||&nbsp;|| 1    ||&nbsp;|| 1
+
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1010
|&nbsp;||&nbsp;|| 1   || 1   ||&nbsp;||&nbsp;|| 1   || 1
+
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_{11}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1011
|&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;||&nbsp;||&nbsp;|| 1
+
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{12}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1100
|&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1   || 1   || 1
+
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1101
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   ||&nbsp;|| 1
+
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
+
| <math>f_{14}</math>
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;
+
| 1110
|&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 1   || 1
+
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <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;|| 1
+
| <math>((~))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|}<br>
 
|}<br>
   Line 757: Line 1,351:  
\end{array}</math></center>
 
\end{array}</math></center>
   −
Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math>  Figure&nbsp;6 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
+
Intuitively, the <math>\ell_{ij}\!</math> operators may be thought of as qualifying propositions according to the elements of the universe of discourse that each proposition positively values.  Taken together, these measures provide us with the means to express many useful observations about the propositions in <math>X^\circ = [u, v],</math> and so they mediate a subtext <math>[\ell_{00}, \ell_{01}, \ell_{10}, \ell_{11}]\!</math> that takes place within the higher order universe of discourse <math>X^{\circ 2} = [X^\circ] = [[u, v]].\!</math>  Figure&nbsp;15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>
    
<pre>
 
<pre>
Line 805: Line 1,399:  
|                                                          |
 
|                                                          |
 
o-----------------------------------------------------------o
 
o-----------------------------------------------------------o
Figure 6.  Higher Order Universe of Discourse [L_ij] c [[u, v]]
+
Figure 15.  Higher Order Universe of Discourse [L_ij] c [[u, v]]
 
</pre>
 
</pre>
   Line 817: Line 1,411:     
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; width:96%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; width:96%"
|+ '''Table 7.  Syllogistic Premisses as Higher Order Indicator Functions'''
+
|+ '''Table 16.  Syllogistic Premisses as Higher Order Indicator Functions'''
 
|
 
|
 
<math>\begin{array}{clcl}
 
<math>\begin{array}{clcl}
Line 839: Line 1,433:  
|}<br>
 
|}<br>
   −
Tables&nbsp;8 and 9 develop these ideas in more detail.
+
The following Tables develop these ideas in more detail.
 +
 
 +
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 +
|+ '''Table 17.  Simple Qualifiers of Propositions (Version 1)'''
 +
|- style="background:ghostwhite"
 +
| align="right" | <math>u:</math><br><math>v:</math>
 +
| 1100<br>1010
 +
| <math>f\!</math>
 +
| <math>(\ell_{11})</math><br><math>\text{No } u </math><br><math>\text{is } v </math>
 +
| <math>(\ell_{10})</math><br><math>\text{No } u </math><br><math>\text{is }(v)</math>
 +
| <math>(\ell_{01})</math><br><math>\text{No }(u)</math><br><math>\text{is } v </math>
 +
| <math>(\ell_{00})</math><br><math>\text{No }(u)</math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{00} </math><br><math>\text{Some }(u)</math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{01} </math><br><math>\text{Some }(u)</math><br><math>\text{is } v </math>
 +
| <math> \ell_{10} </math><br><math>\text{Some } u </math><br><math>\text{is }(v)</math>
 +
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 +
|-
 +
| <math>f_0</math>
 +
| 0000
 +
| <math>(~)</math>
 +
| 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_1</math>
 +
| 0001
 +
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_2</math>
 +
| 0010
 +
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_3</math>
 +
| 0011
 +
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_4</math>
 +
| 0100
 +
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_5</math>
 +
| 0101
 +
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_6</math>
 +
| 0110
 +
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_7</math>
 +
| 0111
 +
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
|-
 +
| <math>f_8</math>
 +
| 1000
 +
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_9</math>
 +
| 1001
 +
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{10}</math>
 +
| 1010
 +
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{11}</math>
 +
| 1011
 +
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{12}</math>
 +
| 1100
 +
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{13}</math>
 +
| 1101
 +
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{14}</math>
 +
| 1110
 +
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
|-
 +
| <math>f_{15}</math>
 +
| 1111
 +
| <math>((~))</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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
 +
|}<br>
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 8.  Simple Qualifiers of Propositions (''n'' = 2)'''
+
|+ '''Table 18.  Simple Qualifiers of Propositions (Version 2)'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| align="right" | <math>u:</math><br><math>v:</math>
 
| align="right" | <math>u:</math><br><math>v:</math>
Line 856: Line 1,658:  
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 
| <math> \ell_{11} </math><br><math>\text{Some } u </math><br><math>\text{is } v </math>
 
|-
 
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
+
| <math>f_0</math>
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
+
| 0000
 +
| <math>(~)</math>
 +
| 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:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_1</math> || 0001 || <math>(u)(v)\!</math>
+
| <math>f_1</math>
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
+
| 0001
 +
| <math>(u)(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_2</math> || 0010 || <math>(u) v\!</math>
+
| <math>f_2</math>
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
+
| 0010
 +
| <math>(u) v\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_3</math> || 0011 || <math>(u)\!</math>
+
| <math>f_4</math>
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
+
| 0100
 +
| <math>u (v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_4</math> || 0100 || <math>u (v)\!</math>
+
| <math>f_8</math>
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
+
| 1000
 +
| <math>u v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_5</math> || 0101 || <math>(v)\!</math>
+
| <math>f_3</math>
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
+
| 0011
 +
| <math>(u)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_6</math> || 0110 || <math>(u, v)\!</math>
+
| <math>f_{12}</math>
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
+
| 1100
 +
| <math>u\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_7</math> || 0111 || <math>(u v)\!</math>
+
| <math>f_6</math>
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
+
| 0110
 +
| <math>(u, v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_8</math> || 1000 || <math>u v\!</math>
+
| <math>f_9</math>
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
+
| 1001
 +
| <math>((u, v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_9</math> || 1001 || <math>((u, v))\!</math>
+
| <math>f_5</math>
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
+
| 0101
 +
| <math>(v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{10}</math> || 1010 || <math>v\!</math>
+
| <math>f_{10}</math>
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
+
| 1010
 +
| <math>v\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>
+
| <math>f_7</math>
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
+
| 0111
 +
| <math>(u v)\!</math>
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 
|-
 
|-
| <math>f_{12}</math> || 1100 || <math>u\!</math>
+
| <math>f_{11}</math>
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
+
| 1011
 +
| <math>(u (v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>
+
| <math>f_{13}</math>
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
+
| 1101
 +
| <math>((u) v)\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>
+
| <math>f_{14}</math>
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
+
| 1110
 +
| <math>((u)(v))\!</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:white; color:black" | 0
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 +
| style="background:black; color:white" | 1
 
|-
 
|-
| <math>f_{15}</math> || 1111 || <math>((~))</math>
+
| <math>f_{15}</math>
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
+
| 1111
 +
| <math>((~))</math>
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 0
 +
| style="background:white; color:black" | 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
 
|}<br>
 
|}<br>
    
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 9.  Relation of Quantifiers to Higher Order Propositions'''
+
|+ '''Table 19.  Relation of Quantifiers to Higher Order Propositions'''
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
 
| <math>\text{Mnemonic}</math>
 
| <math>\text{Mnemonic}</math>
12,080

edits