Changes

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

Revision as of 22:32, 13 December 2008

32,011 bytes added , 22:32, 13 December 2008

more table setting

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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 1 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 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 1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 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 10 lists the sixteen higher order propositions about propositions on one boolean variable, organized in the following fashion: Columns 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 1 displaying the names of the functions <math>f_i,\!</math> for <math>i\!</math> = 1 to 4, while the entries in Column 2 give the values of each function for the argument values that are listed in the corresponding column head. Column 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%"

−

|+ '''Table 1. Higher Order Propositions (''n'' = 1)'''

+

|+ '''Table 10. Higher Order Propositions (''n'' = 1)'''

|- style="background:ghostwhite"

| align="right" | <math>x:</math>

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|}

−

I am going to put off explaining Table 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 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%"

−

|+ '''Table 2. Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

+

|+ '''Table 11. Interpretive Categories for Higher Order Propositions (''n'' = 1)'''

|- style="background:ghostwhite"

| Measure

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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 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 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"

| align="right" | <math>u:</math> <math>v:</math>

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| <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~~;|| 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~~;|| 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~~;

+

| 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>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 0100

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~&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>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 0101

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>(v)\!</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

|-

−

| <math>f_6</math> || 0110 || <math>(u, v)\!</math>

+

| <math>f_6</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 0110

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>(u, v)\!</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

|-

−

| <math>f_7</math> || 0111 || <math>(u v)\!</math>

+

| <math>f_7</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 0111

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>(u v)\!</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

|-

−

| <math>f_8</math> || 1000 || <math>u v\!</math>

+

| <math>f_8</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1000

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>u v\!</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

|-

−

| <math>f_9</math> || 1001 || <math>((u, v))\!</math>

+

| <math>f_9</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1001

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>((u, v))\!</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

|-

−

| <math>f_{10}</math> || 1010 || <math>v\!</math>

+

| <math>f_{10}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1010

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>v\!</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

|-

−

| <math>f_{11}</math> || 1011 || <math>(u (v))\!</math>

+

| <math>f_{11}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1011

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>(u (v))\!</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

|-

−

| <math>f_{12}</math> || 1100 || <math>u\!</math>

+

| <math>f_{12}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1100

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>u\!</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

|-

−

| <math>f_{13}</math> || 1101 || <math>((u) v)\!</math>

+

| <math>f_{13}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1101

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>((u) v)\!</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

|-

−

| <math>f_{14}</math> || 1110 || <math>((u)(v))\!</math>

+

| <math>f_{14}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1110

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>((u)(v))\!</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

|-

−

| <math>f_{15}</math> || 1111 || <math>((~))\!</math>

+

| <math>f_{15}</math>

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| 1111

−

|~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~||~~ ~~

+

| <math>((~))\!</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

|}

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{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ '''Table 4. Qualifiers of Implication Ordering:  <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>'''

+

|+ '''Table 13. Qualifiers of Implication Ordering:  <math>\alpha_i f = \Upsilon \langle f_i, f \rangle = \Upsilon \langle f_i \Rightarrow f \rangle</math>'''

|- style="background:ghostwhite"

| align="right" | <math>u:</math> <math>v:</math>

| 1100 1010

| <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_{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

|}

{| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"

−

|+ '''Table 5. Qualifiers of Implication Ordering:  <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>'''

+

|+ '''Table 14. Qualifiers of Implication Ordering:  <math>\beta_i f = \Upsilon \langle f, f_i \rangle = \Upsilon \langle f \Rightarrow f_i \rangle</math>'''

|- style="background:ghostwhite"

| align="right" | <math>u:</math> <math>v:</math>

Line 610: Line 948:

| <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

|}

Line 757: Line 1,351:

\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 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 15 summarizes the action of the <math>\ell_{ij}\!</math> operators on the <math>f_i\!</math> within <math>X^{\circ 2}.\!</math>

<pre>

Line 805: Line 1,399:

| |

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>

Line 817: Line 1,411:

{| 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}

Line 839: Line 1,433:

|}

−

Tables~~ 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> <math>v:</math>

+

| 1100 1010

+

| <math>f\!</math>

+

| <math>(\ell_{11})</math> <math>\text{No } u </math> <math>\text{is } v </math>

+

| <math>(\ell_{10})</math> <math>\text{No } u </math> <math>\text{is }(v)</math>

+

| <math>(\ell_{01})</math> <math>\text{No }(u)</math> <math>\text{is } v </math>

+

| <math>(\ell_{00})</math> <math>\text{No }(u)</math> <math>\text{is }(v)</math>

+

| <math> \ell_{00} </math> <math>\text{Some }(u)</math> <math>\text{is }(v)</math>

+

| <math> \ell_{01} </math> <math>\text{Some }(u)</math> <math>\text{is } v </math>

+

| <math> \ell_{10} </math> <math>\text{Some } u </math> <math>\text{is }(v)</math>

+

| <math> \ell_{11} </math> <math>\text{Some } u </math> <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

+

|}

{| 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"

| align="right" | <math>u:</math> <math>v:</math>

Line 856: Line 1,658:

| <math> \ell_{11} </math> <math>\text{Some } u </math> <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

|}

{| 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"

| <math>\text{Mnemonic}</math>

Jon Awbrey

12,080

edits