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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 14:20, 13 December 2008
, 14:20, 13 December 2008redo tables
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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%"
{| 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 8. Simple Qualifiers of Propositions (Version 1)'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="right" | <math>x:</math><br><math>y:</math>
| align="right" | <math>u:</math><br><math>v:</math>
| 1100<br>1010
| 1100<br>1010
| <math>f\!</math>
| <math>f\!</math>
| <math>(\ell_{11})</math><br><math>\text{No } x </math><br><math>\text{is } y </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 } x </math><br><math>\text{is }(y)</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 }(x)</math><br><math>\text{is } y </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 }(x)</math><br><math>\text{is }(y)</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 }(x)</math><br><math>\text{is }(y)</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 }(x)</math><br><math>\text{is } y </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 } x </math><br><math>\text{is }(y)</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 } x </math><br><math>\text{is } y </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>(x)(y)\!</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>(x) y\!</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>(x)\!</math>
| <math>f_3</math>
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 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_4</math> || 0100 || <math>x (y)\!</math>
| <math>f_4</math>
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 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_5</math> || 0101 || <math>(y)\!</math>
| <math>f_5</math>
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
| 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>(x, y)\!</math>
| <math>f_6</math>
| 1 || 0 || 0 || 1 || 0 || 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_7</math> || 0111 || <math>(x y)\!</math>
| <math>f_7</math>
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
| 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>x y\!</math>
| <math>f_8</math>
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
| 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>((x, y))\!</math>
| <math>f_9</math>
| 0 || 1 || 1 || 0 || 1 || 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_{10}</math> || 1010 || <math>y\!</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>(x (y))\!</math>
| <math>f_{11}</math>
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 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_{12}</math> || 1100 || <math>x\!</math>
| <math>f_{12}</math>
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
| 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>((x) y)\!</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>((x)(y))\!</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 9. Simple Qualifiers of Propositions (Version 2)'''
|- 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_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_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_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_{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_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_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_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_{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_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_{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_{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%"
|+ '''Table 10. Relation of Quantifiers to Higher Order Propositions'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>\text{Mnemonic}</math>
| <math>\text{Mnemonic}</math>
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{E}\!</math><br><math>\text{Exclusive}</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{Universal}</math><br><math>\text{Negative}</math>
| <math>\text{All}\ x\ \text{is}\ (y)</math>
| <math>\text{All}\ u\ \text{is}\ (v)</math>
|
|
| <math>\text{No}\ x\ \text{is}\ y </math>
| <math>\text{No}\ u\ \text{is}\ v </math>
| <math>(\ell_{11})</math>
| <math>(\ell_{11})</math>
|-
|-
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{A}\!</math><br><math>\text{Absolute}</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{Universal}</math><br><math>\text{Affirmative}</math>
| <math>\text{All}\ x\ \text{is}\ y </math>
| <math>\text{All}\ u\ \text{is}\ v </math>
|
|
| <math>\text{No}\ x\ \text{is}\ (y)</math>
| <math>\text{No}\ u\ \text{is}\ (v)</math>
| <math>(\ell_{10})</math>
| <math>(\ell_{10})</math>
|-
|-
|
|
|
|
| <math>\text{All}\ y\ \text{is}\ x </math>
| <math>\text{All}\ v\ \text{is}\ u </math>
| <math>\text{No}\ y\ \text{is}\ (x)</math>
| <math>\text{No}\ v\ \text{is}\ (u)</math>
| <math>\text{No}\ (x)\ \text{is}\ y </math>
| <math>\text{No}\ (u)\ \text{is}\ v </math>
| <math>(\ell_{01})</math>
| <math>(\ell_{01})</math>
|-
|-
|
|
|
|
| <math>\text{All}\ (y)\ \text{is}\ x </math>
| <math>\text{All}\ (v)\ \text{is}\ u </math>
| <math>\text{No}\ (y)\ \text{is}\ (x)</math>
| <math>\text{No}\ (v)\ \text{is}\ (u)</math>
| <math>\text{No}\ (x)\ \text{is}\ (y)</math>
| <math>\text{No}\ (u)\ \text{is}\ (v)</math>
| <math>(\ell_{00})</math>
| <math>(\ell_{00})</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (x)\ \text{is}\ (y)</math>
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
|
|
| <math>\text{Some}\ (x)\ \text{is}\ (y)</math>
| <math>\text{Some}\ (u)\ \text{is}\ (v)</math>
| <math>\ell_{00}\!</math>
| <math>\ell_{00}\!</math>
|-
|-
|
|
|
|
| <math>\text{Some}\ (x)\ \text{is}\ y</math>
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ (x)\ \text{is}\ y</math>
| <math>\text{Some}\ (u)\ \text{is}\ v</math>
| <math>\ell_{01}\!</math>
| <math>\ell_{01}\!</math>
|-
|-
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{O}\!</math><br><math>\text{Obtrusive}</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Particular}</math><br><math>\text{Negative}</math>
| <math>\text{Some}\ x\ \text{is}\ (y)</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
|
|
| <math>\text{Some}\ x\ \text{is}\ (y)</math>
| <math>\text{Some}\ u\ \text{is}\ (v)</math>
| <math>\ell_{10}\!</math>
| <math>\ell_{10}\!</math>
|-
|-
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{I}\!</math><br><math>\text{Indefinite}</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Particular}</math><br><math>\text{Affirmative}</math>
| <math>\text{Some}\ x\ \text{is}\ y</math>
| <math>\text{Some}\ u\ \text{is}\ v</math>
|
|
| <math>\text{Some}\ x\ \text{is}\ y</math>
| <math>\text{Some}\ u\ \text{is}\ y</math>
| <math>\ell_{11}\!</math>
| <math>\ell_{11}\!</math>
|}<br>
|}<br>