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Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory (view source)
Revision as of 21:58, 18 November 2008
, 21:58, 18 November 2008HTML → TeX
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
With this interpretation in mind we note the following correspondences between classical quantifications and higher order indicator functions:
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
|+ '''Table 7. Syllogistic Premisses as Higher Order Indicator Functions'''
|+ '''Table 7. Syllogistic Premisses as Higher Order Indicator Functions'''
| <math>\mathrm{A}\!</math>
| <math>\mathrm{A}\!</math>
Tables 8 and 9 develop these ideas in more detail.
Tables 8 and 9 develop these ideas in more detail.
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
|+ '''Table 8. Relation of Quantifiers to Higher Order Propositions'''
|+ '''Table 8. Relation of Quantifiers to Higher Order Propositions'''
|- style="background:paleturquoise"
|- style="background:ghostwhite"
|Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
|Mnemonic||Category||Classical Form||Alternate Form||Symmetric Form||Operator
|-
|-
| E<br>Exclusive
| <math>\mathrm{E}\!</math><br>Exclusive
| Universal<br>Negative
| Universal<br>Negative
| align=left | All x is (y)
| align=left | All x is (y)
| align=left |
| align=left |
| align=left | No x is y
| align=left | No x is y
| (''L''<sub>11</sub>)
| <math>(\!| L_{11} |\!)</math>
|-
|-
| A<br>Absolute
| <math>\mathrm{A}\!</math><br>Absolute
| Universal<br>Affirmative
| Universal<br>Affirmative
| align=left | All x is y
| align=left | All x is y
| align=left |
| align=left |
| align=left | No x is (y)
| align=left | No x is (y)
| (''L''<sub>10</sub>)
| <math>(\!| L_{10} |\!)</math>
|-
|-
|
|
| align=left | No y is (x)
| align=left | No y is (x)
| align=left | No (x) is y
| align=left | No (x) is y
| (''L''<sub>01</sub>)
| <math>(\!| L_{01} |\!)</math>
|-
|-
|
|
| align=left | No (y) is (x)
| align=left | No (y) is (x)
| align=left | No (x) is (y)
| align=left | No (x) is (y)
| (''L''<sub>00</sub>)
| <math>(\!| L_{00} |\!)</math>
|-
|-
|
|
| align=left |
| align=left |
| align=left | Some (x) is (y)
| align=left | Some (x) is (y)
| ''L''<sub>00</sub>
| <math>L_{00}\!</math>
|-
|-
|
|
| align=left |
| align=left |
| align=left | Some (x) is y
| align=left | Some (x) is y
| ''L''<sub>01</sub>
| <math>L_{01}\!</math>
|-
|-
| O<br>Obtrusive
| <math>\mathrm{O}\!</math><br>Obtrusive
| Particular<br>Negative
| Particular<br>Negative
| align=left | Some x is (y)
| align=left | Some x is (y)
| align=left |
| align=left |
| align=left | Some x is (y)
| align=left | Some x is (y)
| ''L''<sub>10</sub>
| <math>L_{10}\!</math>
|-
|-
| I<br>Indefinite
| <math>\mathrm{I}\!</math><br>Indefinite
| Particular<br>Affirmative
| Particular<br>Affirmative
| align=left | Some x is y
| align=left | Some x is y
| align=left |
| align=left |
| align=left | Some x is y
| align=left | Some x is y
| ''L''<sub>11</sub>
| <math>L_{11}\!</math>
|}<br>
|}<br>