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| | | |
| ====Tables==== | | ====Tables==== |
| + | |
| + | The auxiliary notations: |
| + | |
| + | : <math>\alpha_i f = \Upsilon (f_i, f),\!</math> |
| + | |
| + | : <math>\beta_i f = \Upsilon (f, f_i),\!</math> |
| + | |
| + | define two series of measures: |
| + | |
| + | : <math>\alpha_i, \beta_i : (\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B},</math> |
| + | |
| + | incidentally providing compact names for the column headings of the next two Tables. |
| + | |
| + | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 1. Qualifiers of Implication Ordering: <math>\alpha_i f = \Upsilon (f_i \Rightarrow f)</math>''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>p:</math><br><math>q:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</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> |
| + | | || || || || || || || |
| + | | || || || || || || || 1 |
| + | |- |
| + | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | <math>f_2</math> || 0010 || <math>(p) q\!</math> |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | <math>f_3</math> || 0011 || <math>(p)\!</math> |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_4</math> || 0100 || <math>p (q)\!</math> |
| + | | || || || || || || || |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | <math>f_5</math> || 0101 || <math>(q)\!</math> |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | <math>f_7</math> || 0111 || <math>(p q)\!</math> |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_8</math> || 1000 || <math>p q\!</math> |
| + | | || || || || || || || 1 |
| + | | || || || || || || || 1 |
| + | |- |
| + | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | <math>f_{10}</math> || 1010 || <math>q\!</math> |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_{12}</math> || 1100 || <math>p\!</math> |
| + | | || || || 1 || || || || 1 |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | <math>f_{14}</math> || 1110 || <math>((p)(q))\!</math> |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | <math>f_{15}</math> || 1111 || <math>((~))</math> |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |}<br> |
| + | |
| + | {| align="center" border="1" cellpadding="1" cellspacing="0" style="font-weight:bold; text-align:center; width:96%" |
| + | |+ '''Table 2. Qualifiers of Implication Ordering: <math>\beta_i f = \Upsilon (f \Rightarrow f_i)</math>''' |
| + | |- style="background:ghostwhite" |
| + | | align="right" | <math>p:</math><br><math>q:</math> |
| + | | 1100<br>1010 |
| + | | <math>f\!</math> |
| + | | <math>\beta_0</math> |
| + | | <math>\beta_1</math> |
| + | | <math>\beta_2</math> |
| + | | <math>\beta_3</math> |
| + | | <math>\beta_4</math> |
| + | | <math>\beta_5</math> |
| + | | <math>\beta_6</math> |
| + | | <math>\beta_7</math> |
| + | | <math>\beta_8</math> |
| + | | <math>\beta_9</math> |
| + | | <math>\beta_{10}</math> |
| + | | <math>\beta_{11}</math> |
| + | | <math>\beta_{12}</math> |
| + | | <math>\beta_{13}</math> |
| + | | <math>\beta_{14}</math> |
| + | | <math>\beta_{15}</math> |
| + | |- |
| + | | <math>f_0</math> || 0000 || <math>(~)</math> |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_1</math> || 0001 || <math>(p)(q)\!</math> |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | <math>f_2</math> || 0010 || <math>(p) q\!</math> |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | <math>f_3</math> || 0011 || <math>(p)\!</math> |
| + | | || || || 1 || || || || 1 |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | <math>f_4</math> || 0100 || <math>p (q)\!</math> |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_5</math> || 0101 || <math>(q)\!</math> |
| + | | || || || || || 1 || || 1 |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | <math>f_6</math> || 0110 || <math>(p, q)\!</math> |
| + | | || || || || || || 1 || 1 |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | <math>f_7</math> || 0111 || <math>(p q)\!</math> |
| + | | || || || || || || || 1 |
| + | | || || || || || || || 1 |
| + | |- |
| + | | <math>f_8</math> || 1000 || <math>p q\!</math> |
| + | | || || || || || || || |
| + | | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_9</math> || 1001 || <math>((p, q))\!</math> |
| + | | || || || || || || || |
| + | | || 1 || || 1 || || 1 || || 1 |
| + | |- |
| + | | <math>f_{10}</math> || 1010 || <math>q\!</math> |
| + | | || || || || || || || |
| + | | || || 1 || 1 || || || 1 || 1 |
| + | |- |
| + | | <math>f_{11}</math> || 1011 || <math>(p (q))\!</math> |
| + | | || || || || || || || |
| + | | || || || 1 || || || || 1 |
| + | |- |
| + | | <math>f_{12}</math> || 1100 || <math>p\!</math> |
| + | | || || || || || || || |
| + | | || || || || 1 || 1 || 1 || 1 |
| + | |- |
| + | | <math>f_{13}</math> || 1101 || <math>((p) q)\!</math> |
| + | | || || || || || || || |
| + | | || || || || || 1 || || 1 |
| + | |- |
| + | | <math>f_{14}</math> || 1110 || <math>((p)(y))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || 1 || 1 |
| + | |- |
| + | | <math>f_{15}</math> || 1111 || <math>((~))\!</math> |
| + | | || || || || || || || |
| + | | || || || || || || || 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 1. Simple Qualifiers of Propositions (''n'' = 2)''' | + | |+ '''Table 3. Simple Qualifiers of Propositions (''n'' = 2)''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
| | align="right" | <math>p:</math><br><math>q:</math> | | | align="right" | <math>p:</math><br><math>q:</math> |
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| {| 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 2. Relation of Quantifiers to Higher Order Propositions''' | + | |+ '''Table 4. Relation of Quantifiers to Higher Order Propositions''' |
| |- style="background:ghostwhite" | | |- style="background:ghostwhite" |
| | <math>\text{Mnemonic}</math> | | | <math>\text{Mnemonic}</math> |