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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 19:36, 21 November 2008
, 19:36, 21 November 2008restore tables to original order
{| 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. Relation of Quantifiers to Higher Order Propositions'''
|+ '''Table 8. Simple Qualifiers of Propositions (''n'' = 2)'''
|- style="background:ghostwhite"
| align="right" | <math>x:</math><br><math>y:</math>
| 1100<br>1010
| <math>f\!</math>
| <math>(\ell_{11})</math><br>No <math> x </math><br>is <math> y </math>
| <math>(\ell_{10})</math><br>No <math> x </math><br>is <math>(y)</math>
| <math>(\ell_{01})</math><br>No <math>(x)</math><br>is <math> y </math>
| <math>(\ell_{00})</math><br>No <math>(x)</math><br>is <math>(y)</math>
| <math> \ell_{00} </math><br>Some <math>(x)</math><br>is <math>(y)</math>
| <math> \ell_{01} </math><br>Some <math>(x)</math><br>is <math> y </math>
| <math> \ell_{10} </math><br>Some <math> x </math><br>is <math>(y)</math>
| <math> \ell_{11} </math><br>Some <math> x </math><br>is <math> y </math>
|-
| <math>f_0</math> || 0000 || <math>(~)</math>
| 1 || 1 || 1 || 1 || 0 || 0 || 0 || 0
|-
| <math>f_1</math> || 0001 || <math>(x)(y)\!</math>
| 1 || 1 || 1 || 0 || 1 || 0 || 0 || 0
|-
| <math>f_2</math> || 0010 || <math>(x) y\!</math>
| 1 || 1 || 0 || 1 || 0 || 1 || 0 || 0
|-
| <math>f_3</math> || 0011 || <math>(x)\!</math>
| 1 || 1 || 0 || 0 || 1 || 1 || 0 || 0
|-
| <math>f_4</math> || 0100 || <math>x (y)\!</math>
| 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0
|-
| <math>f_5</math> || 0101 || <math>(y)\!</math>
| 1 || 0 || 1 || 0 || 1 || 0 || 1 || 0
|-
| <math>f_6</math> || 0110 || <math>(x, y)\!</math>
| 1 || 0 || 0 || 1 || 0 || 1 || 1 || 0
|-
| <math>f_7</math> || 0111 || <math>(x y)\!</math>
| 1 || 0 || 0 || 0 || 1 || 1 || 1 || 0
|-
| <math>f_8</math> || 1000 || <math>x y\!</math>
| 0 || 1 || 1 || 1 || 0 || 0 || 0 || 1
|-
| <math>f_9</math> || 1001 || <math>((x, y))\!</math>
| 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1
|-
| <math>f_{10}</math> || 1010 || <math>y\!</math>
| 0 || 1 || 0 || 1 || 0 || 1 || 0 || 1
|-
| <math>f_{11}</math> || 1011 || <math>(x (y))\!</math>
| 0 || 1 || 0 || 0 || 1 || 1 || 0 || 1
|-
| <math>f_{12}</math> || 1100 || <math>x\!</math>
| 0 || 0 || 1 || 1 || 0 || 0 || 1 || 1
|-
| <math>f_{13}</math> || 1101 || <math>((x) y)\!</math>
| 0 || 0 || 1 || 0 || 1 || 0 || 1 || 1
|-
| <math>f_{14}</math> || 1110 || <math>((x)(y))\!</math>
| 0 || 0 || 0 || 1 || 0 || 1 || 1 || 1
|-
| <math>f_{15}</math> || 1111 || <math>((~))</math>
| 0 || 0 || 0 || 0 || 1 || 1 || 1 || 1
|}<br>
{| 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'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| Mnemonic
| Mnemonic
| <math>\text{Some}\ x\ \text{is}\ y</math>
| <math>\text{Some}\ x\ \text{is}\ y</math>
| <math>\ell_{11}\!</math>
| <math>\ell_{11}\!</math>
|}<br>
|}<br>