[Inquiry] Re: Introduction to Inquiry Driven Systems

[Jon Awbrey]

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INTRO.  Note 32

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3.2.1.2.  Higher Order Propositions and Logical Operators (n = 2)

By way of reviewing notation and preparing to extend it to
higher order universes of discourse, let us first consider
the universe of discourse X° = [$X$] = [x_1, x_2] = [x, y],
based on two logical features or boolean variables x and y.

1.  The points of X° are collected in the space:

    X   =   <<x, y>>   =   {<x, y>}   ~=~   B^2.

    In other words, written out in full:

    X   =   {<"(x)", "(y)">,
             <"(x)", " y ">,
             <" x ", "(y)">,
             <" x ", " y ">}

    X  ~=~  {<0, 0>,
             <0, 1>,
             <1, 0>,
             <1, 1>}

2.  The propositions of X° make up the space:

    ^X^  =  (X -> %B%)  =  {f : X -> %B%}  ~=~  (%B%^2 -> %B%).

As always, it is frequently convenient to omit a few of the
finer markings of distinctions among isomorphic structures,
so long as one is aware of their presence and knows when
it is crucial to call upon them again.

The next higher order universe of discourse that is built on X° is
X°2 = [X°] = [[x, y]], which may be developed in the following way.
The propositions of X° become the points of X°2, and the mappings
of the type m : (X -> B) -> B become the propositions of X°2.
In addition, it is convenient to equip the discussion with
a selected set of higher order operators on propositions,
all of which have the form w : (B^2 -> B)^k -> B.

To save a few words in the remainder of this discussion, I will
use the terms "measure" and "qualifier" to refer to all types of
"higher order" (HO) propositions and operators.  To describe the
present setting in picturesque terms, the propositions of [x, y]
may be regarded as a gallery of sixteen venn diagrams, while the
measures m : (X -> B) -> B are analogous to a body of judges or
a panel of critical viewers, each of whom evaluates each of the
pictures as a whole and reports the ones that find favor or not.
In this way, each judge m_j partitions the gallery of pictures
into two aesthetic portions, the pictures (m_j)^(-1)(1) that
m_j likes and the pictures (m_j)^(-1)(0) that m_j dislikes.

There are 2^16 = 65536 measures of the type m : (B^2 -> B) -> B.
Table 9 introduces the first 16 of these measures in the fashion
of the HO truth table that I used before.  The column headed "m_j"
shows the values of the measure m_j on each of the propositions
f_i : B^2 -> B, for i = 0 to 15, with blank entries in the Table
being optional for values of zero.  The arrangement of measures
that continues according to the plan indicated here is referred
to as the "standard ordering" of these measures.  In this scheme
of things, the index j of the measure m_j is the decimal equivalent
of the bit string that is associated with m_j's functional values,
which can be obtained in turn by reading the j^th column of binary
digits in the Table as the corresponding range of boolean values,
taking them up in the order from bottom to top.

Table 9.  Higher Order Propositions (n = 2)
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| `| x | 1100 | ` `f` ` `|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.|
| `| y | 1010 | ` ` ` ` `|0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.|
| f \ `| ` ` `| ` ` ` ` `|0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.|
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_0 `| 0000 | ` `() ` `|0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_1 `| 0001 | `(x)(y) `| ` `1 1 0 0 1 1 0 0 1 1 0 0 1 1` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_2 `| 0010 | `(x) y` `| ` ` ` `1 1 1 1 0 0 0 0 1 1 1 1` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_3 `| 0011 | `(x)` ` `| ` ` ` ` ` ` ` `1 1 1 1 1 1 1 1` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_4 `| 0100 | ` x (y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_5 `| 0101 | ` ` (y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_6 `| 0110 | `(x, y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_7 `| 0111 | `(x `y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_8 `| 1000 | ` x `y` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_9 `| 1001 | ((x, y)) | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_10`| 1010 | ` ` `y` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_11`| 1011 | `(x (y))`| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_12`| 1100 | ` x ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_13`| 1101 | ((x) y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_14`| 1110 | ((x)(y))`| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_15`| 1111 | ` (())` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o------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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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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