# Toward A Functional Conception Of Quantificational Logic

Jon Awbrey jawbrey at oakland.edu
Mon Dec 24 20:30:40 CET 2001

```¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

Note 128

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

Subj:  Toward A Functional Conception Of Quantificational Logic

I am going to put off explaining Table 11, 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.

| Document History:
|
| Subject:  Inquiry & Analogy
| Contact:  Jon Awbrey <jawbrey at oakland.edu>
| Version:  Draft 3.21
| Created:  01 Jan 1995
| Revised:  24 Dec 2001
| Faculty:  F. Mili & M.A. Zohdy
| Setting:  Oakland University, Rochester, Michigan, USA
| Excerpt:  Section 2.1.2 (Higher Order Propositions & Logical Operators)

2.1.2  Higher Order Propositions & 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 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 collection of critical viewers, each of whom evaluates each
picture 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 type m : (%B%^2 -> %B%) -> %B%.
Table 12 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
will be referred to as the "standard ordering" of 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 12.  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

HO, HO, HO, ...

Jon Awbrey

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

```