[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INTRO.  Note 31

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3.2.1.  Higher Order Propositional Expressions

By way of equipping this inquiry with a bit of concrete material, I begin
with a consideration of "higher order propositional expressions" (HOPE's),
in particular, those that stem from the propositions on 1 and 2 variables.

3.2.1.1.  Higher Order Propositions and Logical Operators (n = 1)

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, where, as usual, B = {0, 1}.

For example, consider the case where X = B^1 = B.  Then there
are exactly four propositions F : B -> B, and exactly sixteen
higher order propositions, all of the type m : (B -> B) -> B.
Table 7 lists the sixteen HO propositions about propositions
on one boolean variable, organized in the following fashion:
Columns 1 and 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 the 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, where the entries in the body of the
Table record the values that each m_j assigns to each F_i.

Table 7.  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

I am going to put off explaining Table 8, that presents a sample of
what I call "Interpretive Categories for Higher Order Propositions",
until after we get beyond the 1-dimensional case, since these lower
dimensional cases tend to be a bit "condensed" or "degenerate" in
their structures, and a lot of what is going on here will almost
automatically become clearer as soon as we get even two logical
variables into the mix.

Table 8.  Interpretive Categories for Higher Order Propositions (n = 1)
o-------o----------o------------o------------o----------o----------o-----------o
|Measure| Happening| Exactness `| Existence `| Linearity|Uniformity|Information|
o-------o----------o------------o------------o----------o----------o-----------o
| m_0 ` | nothing `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | happens `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_1 ` | ` ` ` ` `| ` ` ` ` ` `| nothing ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just false`| exists` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_2 ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just not x`| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_3 ` | ` ` ` ` `| ` ` ` ` ` `| nothing ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| is x` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_4 ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just x` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_5 ` | ` ` ` ` `| ` ` ` ` ` `| everything`| F is ` ` | ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| is x` ` ` `| linear ` | ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_6 ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| F is not`| F is ` ` `|
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| uniform `| informed `|
o-------o----------o------------o------------o----------o----------o-----------o
| m_7 ` | ` ` ` ` `| not ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just true `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_8 ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just true `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_9 ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| F is` ` `| F is not` |
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| uniform `| informed` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_10` | ` ` ` ` `| ` ` ` ` ` `| something `| F is not`| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| is not x` `| linear` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_11` | ` ` ` ` `| not ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just x` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_12` | ` ` ` ` `| ` ` ` ` ` `| something `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| ` ` ` ` ` `| is x` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_13` | ` ` ` ` `| not ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just not x`| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_14` | ` ` ` ` `| not ` ` ` `| something `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | ` ` ` ` `| just false`| exists` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o
| m_15` | anything`| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
| ` ` ` | happens `| ` ` ` ` ` `| ` ` ` ` ` `| ` ` ` ` `| ` ` ` ` `| ` ` ` ` ` |
o-------o----------o------------o------------o----------o----------o-----------o

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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