ONT Re: Toward A Functional Conception Of Quantificational Logic
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Note 127
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A "higher order" (HO) proposition is, very roughly speaking,
a proposition about propositions. If the original order of
propositions is a class of indicator functions F : X -> %B%,
then the next higher order of propositions consists of maps
of the type m : (X -> %B%) -> %B%.
For example, consider the case where X = %B%. Then there are
exactly four propositions F : %B% -> %B%, and exactly sixteen
HO propositions, all bearing the type m : (%B% -> %B%) -> %B%.
Table 10 lists the sixteen HO propositions about propositions
on one boolean variable, organized in the following fashion:
Columns 1 & 2 form a truth table for the four F : %B% -> %B%,
perhaps turned on its side from the way one is accustomed to
see truth tables, with the row leaders in Column 1 displaying
the names of the functions F_i, i = 1 to 4, while the entries
in Column 2 give the values of each function for the argument
values that are listed in its column head. Column 3 displays
one of the usual expressions for the proposition in question.
The last sixteen columns are topped by a set of conventional
names for the HO propositions, also known as the "measures"
m_j, for j = 0 to 15, with the entries in the body of the
Table giving the values of each m_j on each F_i.
Table 10. Higher Order Propositions (n = 1)
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| \ x | 1 0 | F | m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |
| F \ | | | 00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15|
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| | | | |
| F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| | | | |
| F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| | | | |
| F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| | | | |
| F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| | | | |
o------o-----o-----o---o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Jon Awbrey
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