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Jon Awbrey


Registered: Feb 2004
Posts: 557

Introduction to Inquiry Driven Systems

INTRO. Note 31

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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Jon Awbrey


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Posts: 557

Introduction to Inquiry Driven Systems

INTRO. Note 32

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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Jon Awbrey


Registered: Feb 2004
Posts: 557

Introduction to Inquiry Driven Systems

INTRO. Note 33

3.2.2. Umpire Operators

| Nota Bene
|
| For this production, the part of the upper-case
| Greek character Upsilon will be played by "!Y!".
|
| In addition, I am going to experiment with marking
| Cactus Language or Ref Log expressions by means of
| single underscore marks at their beginning and end.

In order to get a handle on the space of higher order propositions and
eventually to carry out a functional approach to quantification theory,
it serves to construct some specialized tools. Specifically, I define
a higher order operator !Y!, called the "umpire operator", which takes
up to three propositions as arguments and returns a single truth value
as the result. Formally, this so-called "multi-grade" property of !Y!
can be expressed as a union of function types, in the following manner:


    !Y! : |_|^(m = 1, 2, 3) ((B^k -> B)^m -> B).

In contexts of application the intended sense can be discerned by
the number of arguments that actually appear in the argument list.
Often, the first and last arguments appear as indices, the one in
the middle being treated as the main argument while the other two
arguments serve to modify the sense of the operation in question.
Thus, we have the following forms:

    !Y!_p^r q = !Y!(p, q, r)

    !Y!_p^r : (B^k -> B) -> B

The intention of this operator is that we evaluate the proposition q
on each model of the proposition p and combine the results according
to the method indicated by the connective parameter r. In principle,
the index r might specify any connective on as many as 2^k arguments,
but usually we have in mind a much simpler form of combination, most
often either collective products or collective sums. By convention,
each of the accessory indices p, r is assigned a default value that
is understood to be in force when the corresponding argument place
is left blank, specifically, the constant proposition 1 : B^k -> B
for the lower index p, and the continued conjunction or continued
product operation ]¯[ for the upper index r. Taking the upper
default value gives license to the following readings:

    1. !Y!_p q = !Y!(p, q) = !Y!(p, q, product).

    2. !Y!_p = !Y!(p, -, product) : (B^k -> B) -> B.

This means that !Y!_p q = 1 if and only if q holds for all models of p.
In propositional terms, this is tantamount to the assertion that p => q,
or that _(p (q))_ = 1.

Throwing in the lower default value permits the following abbreviations:

    3. !Y!q = !Y!(q) = !Y!_1 q = !Y!(1, q, product).

    4. !Y! = !Y!(1, -, product) : (B^k -> B) -> B.

This means that !Y!q = 1 if and only if q holds for the whole
universe of discourse in question, that is, if and only q is the
constantly true proposition 1 : B^k -> B. The ambiguities of this
usage are not a problem so long as we distinguish the context of
definition from the context of application and restrict all
shorthand notations to the latter.

Jon Awbrey

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Jon Awbrey


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Posts: 557

Introduction to Inquiry Driven Systems

INTRO. Note 34

3.2.3. Measure for Measure

An acquaintance with the functions of the umpire operator can
be gained from Tables 10 and 11, where the 2-dimensional case
is worked out in full.

The auxiliary notations:


  • !a!_i f = !Y!(f_i, f),

  • !b!_i f = !Y!(f, f_i),

define two series of measures:

  • !a!_i, !b!_i : (B^2 -> B) -> B,

incidentally providing compact names for
the column headings of the next two Tables.

Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| `| x | 1100 | ` `f` ` `|a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
| `| y | 1010 | ` ` ` ` `|1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \ `| ` ` `| ` ` ` ` `|5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_0 `| 0000 | ` `() ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_1 `| 0001 | `(x)(y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_2 `| 0010 | `(x) y` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_3 `| 0011 | `(x)` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_4 `| 0100 | ` x (y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_5 `| 0101 | ` ` (y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1 `1 ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_6 `| 0110 | `(x, y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 1 ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_7 `| 0111 | `(x `y) `| ` ` ` ` ` ` ` ` ` ` ` `1` 1 `1` 1 `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_8 `| 1000 | ` x `y` `| ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_9 `| 1001 | ((x, y))`| ` ` ` ` ` ` ` ` `1` 1 ` ` ` ` ` ` ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_10`| 1010 | ` ` `y` `| ` ` ` ` ` ` ` 1 ` ` 1 ` ` ` ` ` ` ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_11`| 1011 | `(x (y))`| ` ` ` ` ` `1` 1 `1` 1 ` ` ` ` ` ` `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_12`| 1100 | ` x ` ` `| ` ` ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_13`| 1101 | ((x) y) `| ` ` `1` 1 ` ` ` `1` 1 ` ` ` `1` 1 ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_14`| 1110 | ((x)(y))`| ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_15`| 1111 | ` (())` `|1` 1 `1` 1 `1` 1 `1` 1 `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

Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
| `| x | 1100 | ` `f` ` `|b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
| `| 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
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_0 `| 0000 | ` `() ` `|1` 1 `1` 1 `1` 1 `1` 1 `1` 1 `1` 1 `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_1 `| 0001 | `(x)(y) `| ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_2 `| 0010 | `(x) y` `| ` ` `1` 1 ` ` ` `1` 1 ` ` ` `1` 1 ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_3 `| 0011 | `(x)` ` `| ` ` ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_4 `| 0100 | ` x (y) `| ` ` ` ` ` `1` 1 `1` 1 ` ` ` ` ` ` `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_5 `| 0101 | ` ` (y) `| ` ` ` ` ` ` ` 1 ` ` 1 ` ` ` ` ` ` ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_6 `| 0110 | `(x, y) `| ` ` ` ` ` ` ` ` `1` 1 ` ` ` ` ` ` ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_7 `| 0111 | `(x `y) `| ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_8 `| 1000 | ` x `y` `| ` ` ` ` ` ` ` ` ` ` ` `1` 1 `1` 1 `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_9 `| 1001 | ((x, y))`| ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 1 ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_10`| 1010 | ` ` `y` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1` 1 ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_11`| 1011 | `(x (y))`| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_12`| 1100 | ` x ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1` 1 `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_13`| 1101 | ((x) y) `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_14`| 1110 | ((x)(y))`| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_15`| 1111 | ` (())` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 |
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

Applied to a given proposition f, the qualifiers !a!_i and !b!_i tell whether
f rests "above f_i" or "below f_i", respectively, in the implication ordering.
By way of example, let us trace the effects of several such measures, namely,
those that occupy the limiting positions of the Tables.

  • !a!_00 f = 1 iff f_00 => f, iff 0 => f, hence !a!_00 f = 1 for all f.

  • !a!_15 f = 1 iff f_15 => f, iff 1 => f, hence !a!_15 f = 1 iff f = 1.

  • !b!_00 f = 1 iff f => f_00, iff f => 0, hence !b!_00 f = 1 iff f = 0.

  • !b!_15 f = 1 iff f => f_15, iff f => 1, hence !b!_15 f = 1 for all f.

Thus, !a!_0 = !b!_15 is a totally indiscriminate measure,
one that accepts all propositions f : B^2 -> B, whereas
!a!_15 and !b!_0 are measures that value the constant
propositions 1 : B^2 -> B and 0 : B^2 -> %B%,
respectively, above all others.

Finally, in conformity with the use of the fiber notation to
indicate sets of models, it is natural to use notations like:

  • [| !a!_i |] = (!a!_i)^(-1)(1),

  • [| !b!_i |] = (!b!_i)^(-1)(1),

  • [| !Y!_p |] = (!Y!_p)^(-1)(1),

to denote sets of propositions that satisfy the umpires in question.

Jon Awbrey

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Introduction to Inquiry Driven Systems

INTRO. Note 35

3.2.4. Extending the Existential Interpretation to Quantificational Logic

Previously I introduced a calculus for propositional logic, fixing its meaning
according to what C.S. Peirce called the "existential interpretation". As far
as it concerns propositional calculus this interpretation settles the meanings
that are associated with merely the most basic symbols and logical connectives.
Now we must extend and refine the existential interpretation to comprehend the
analysis of "quantifications", that is, quantified propositions. In doing so
we recognize two additional aspects of logic that need to be developed, over
and above the material of propositional logic. At the formal extreme there
is the aspect of higher order functional types, into which we have already
ventured a little above. At the level of the fundamental content of the
available propositions we have to introduce a different interpretation
for what we may call "elemental" or "singular" propositions.

Let us return to the 2-dimensional case X° = [x, y].
In order to provide a bridge between propositions and
quantifications it serves to define a set of qualifiers
L_uv : (B^2 -> B) -> B that have the following characters:



  • L_00 f

    = L_"(x)(y)" f

    = !a!_1 f

    = !Y!_"(x)(y)" f

    = !Y! "(x)(y) => f"

    = "f likes (x)(y)"


  • L_01 f

    = L_"(x) y " f

    = !a!_2 f

    = !Y!_"(x) y " f

    = !Y! "(x) y => f"

    = "f likes (x) y "


  • L_10 f = L_" x (y)" f

    = !a!_4 f

    = !Y!_"x (y)" f

    = !Y! "x (y) => f"

    = "f likes x (y)"


  • L_11 f

    = L_"x y" f

    = !a!_8 f

    = !Y!_"x y" f

    = !Y! "x y => f"

    = "f likes x y"


Intuitively, the L_uv operators may be thought of as qualifying propositions
according to the elements of the universe of discourse that each proposition
positively values. Taken together, these measures provide us with the means
to express many useful observations about the propositions in X° = [x, y],
and so they mediate a subtext [L_00, L_01, L_10, L_11] that takes place
within the higher order universe of discourse X°2 = [X°] = [[x, y]].
Figure 12 summarizes the action of the L_uv on the f_i within X°2.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `/ \` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/x ` y\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` / o---o \ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` `\ /` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` / \ ` o ` / \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/` `\` | `/` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` \ @ / ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/`x` `y`\`/`x` `y`\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o `o---o` o `o---o` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/ \` \ ` `/ \` ` / `/ \` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` \ `@` / ` \ `@` / ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` ` `\` `/` ` `\` `/` ` `\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` / ` y ` \ / ` ` ` \ / ` y ` \ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o` ` @ ` `o` ` @ ` `o` ` o ` `o` ` ` ` ` ` ` |
| ` ` ` ` ` ` / \ ` ` ` / \ ` ` ` / \ ` | ` / \ ` ` ` ` ` ` |
| ` ` ` ` ` `/` `\` ` `/` `\` ` `/` `\` @ `/` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` \ ` /x` `y\ ` / ` ` \ ` / ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` x y `\`/`o` `o`\`/` x y `\`/`x` `y`\` ` ` ` ` |
| ` ` ` ` o ` `@` ` o ` \ / ` o ` `o` ` o `o` `o` o ` ` ` ` |
| ` ` ` ` |\` ` ` `/`\` `o` `/`\` `|` `/`\` \ / `/| ` ` ` ` |
| ` ` ` ` | \ ` ` / ` \ `|` / ` \ `@` / ` \ `@` / | ` ` ` ` |
| ` ` ` ` | `\` `/` ` `\`@`/` ` `\` `/` ` `\` `/` | ` ` ` ` |
| ` ` ` ` | ` \ / ` x ` \ / x ` y \ / ` x ` \ / ` | ` ` ` ` |
| ` ` ` ` | ` `o` ` @ ` `o` o---o `o` ` o ` `o` ` | ` ` ` ` |
| ` ` ` ` | ` `|\ ` ` ` / \ `\ /` / \ ` | ` /|` ` | ` ` ` ` |
| ` ` ` ` | ` `|`\` ` `/` `\` @ `/` `\` @ `/`|` ` | ` ` ` ` |
| ` ` ` ` | ` `|` \ ` / ` ` \ ` / ` ` \ ` / `|` ` | ` ` ` ` |
| ` ` ` ` |L_11|` `\`/` `o y \`/`x o ` \`/` `|L_00| ` ` ` ` |
| ` ` ` ` o---------o ` `| ` `o` ` | ` `o---------o ` ` ` ` |
| ` ` ` ` ` ` `|` ` `\`x`@` `/`\` `@`y`/` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` \ ` ` / ` \ ` ` / ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` `\` `/` ` `\` `/` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|L_10` ` \ / ` o ` \ / ` `L_01|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o---------o` ` | ` `o---------o` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` @ ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 12. Higher Order Universe of Discourse [L_uv] c [[x, y]]

Jon Awbrey

---------------------------------------------------------------------------------

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Introduction to Inquiry Driven Systems

INTRO. Note 36

3.2.5. Application of Higher Order Propositions to Quantification Theory

Our excursion into the vastening landscape of higher order propositions
has finally come round to the stage where we can bring its returns to
bear on opening up new perspectives for quantificational logic.

There is a question arising next that is still experimental in my mind.
Whether it makes much difference from a purely formal point of view is
not a question I can answer yet, but it does seem to aid the intuition
to invent a slightly different interpretation for the two-valued space
that we use as the target of our basic indicator functions. Therefore,
let us declare a type of "existential-valued" functions f : B^k -> !E!,
where !E! = {-e-, +e+} = {"empty", "exist"} is a couple of values that
we interpret as indicating whether of not anything exists in the cells
of the underlying universe of discourse, venn diagram, or other domain.
As usual, let us not be too strict about the coding of these functions,
reverting to binary codes whenever the interpretation is clear enough.

With this interpretation in mind we note the following correspondences
between classical quantifications and higher order indicator functions:

Table 13. Syllogistic Premisses as Higher Order Indicator Functions
o---o------------------------o-----------------o---------------------------o
| ` | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A | Universal Affirmative `| All ` x `is `y` | Indicator of " x`(y)" = 0 |
| ` | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| E | Universal Negative` ` `| All ` x `is (y) | Indicator of " x` y " = 0 |
| ` | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| I | Particular Affirmative`| Some` x `is `y` | Indicator of " x` y " = 1 |
| ` | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| O | Particular Negative ` `| Some` x `is (y) | Indicator of " x`(y)" = 1 |
| ` | ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---o------------------------o-----------------o---------------------------o

Tables 14 and 15 develop these ideas in more detail.

Table 14. Relation of Quantifiers to Higher Order Propositions
o------------o------------o-----------o-----------o-----------o-----------o
| Mnemonic` `| Category` `| Classical | Alternate | Symmetric | Operator` |
| ` ` ` ` ` `| ` ` ` ` ` `| ` Form` ` | ` Form` ` | ` Form` ` | ` ` ` ` ` |
o============o============o===========o===========o===========o===========o
| ` ` E ` ` `| Universal `| `All` `x` | ` ` ` ` ` | ` No` `x` | `(L_11) ` |
| Exclusive `| `Negative `| ` is` (y) | ` ` ` ` ` | ` is` `y` | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` A ` ` `| Universal `| `All` `x` | ` ` ` ` ` | ` No` `x` | `(L_10) ` |
| Absolute` `| `Affrmtve `| ` is` `y` | ` ` ` ` ` | ` is` (y) | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` ` ` ` `| ` ` ` ` ` `| `All` `y` | ` No` `y` | ` No` (x) | `(L_01) ` |
| ` ` ` ` ` `| ` ` ` ` ` `| ` is` `x` | ` is` (x) | ` is` `y` | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` ` ` ` `| ` ` ` ` ` `| `All` (y) | ` No` (y) | ` No` (x) | `(L_00) ` |
| ` ` ` ` ` `| ` ` ` ` ` `| ` is` `x` | ` is` (x) | ` is` (y) | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` ` ` ` `| ` ` ` ` ` `| Some` (x) | ` ` ` ` ` | Some` (x) | ` L_00` ` |
| ` ` ` ` ` `| ` ` ` ` ` `| ` is` (y) | ` ` ` ` ` | ` is` (y) | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` ` ` ` `| ` ` ` ` ` `| Some` (x) | ` ` ` ` ` | Some` (x) | ` L_01` ` |
| ` ` ` ` ` `| ` ` ` ` ` `| ` is` `y` | ` ` ` ` ` | ` is` `y` | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` O ` ` `| Particular`| Some` `x` | ` ` ` ` ` | Some` `x` | ` L_10` ` |
| Obtrusive `| `Negative `| ` is` (y) | ` ` ` ` ` | ` is` (y) | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o
| ` ` I ` ` `| Particular`| Some` `x` | ` ` ` ` ` | Some` `x` | ` L_11` ` |
| Indefinite`| `Affrmtve `| ` is` `y` | ` ` ` ` ` | ` is` `y` | ` ` ` ` ` |
o------------o------------o-----------o-----------o-----------o-----------o

Table 15. Simple Qualifiers of Propositions (n = 2)
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| `| x | 1100 | ` `f` ` `|(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 |
| `| y | 1010 | ` ` ` ` `|no `x|no `x|no ~x|no ~x|sm ~x|sm ~x|sm `x|sm `x|
| f \ `| ` ` `| ` ` ` ` `|is `y|is ~y|is `y|is ~y|is ~y|is `y|is ~y|is `y|
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_0 `| 0000 | ` `() ` `| `1 ` ` 1 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` ` 0 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_1 `| 0001 | `(x)(y) `| `1 ` ` 1 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 0 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_2 `| 0010 | `(x) y` `| `1 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_3 `| 0011 | `(x)` ` `| `1 ` ` 1 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` ` 0 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_4 `| 0100 | ` x (y) `| `1 ` ` 0 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` ` 1 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_5 `| 0101 | ` ` (y) `| `1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_6 `| 0110 | `(x, y) `| `1 ` ` 0 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 1 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_7 `| 0111 | `(x `y) `| `1 ` ` 0 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` ` 1 ` ` 0 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_8 `| 1000 | ` x `y` `| `0 ` ` 1 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` ` 0 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_9 `| 1001 | ((x, y)) | `0 ` ` 1 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 0 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_10`| 1010 | ` ` `y` `| `0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_11`| 1011 | `(x (y)) | `0 ` ` 1 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` ` 0 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_12`| 1100 | ` x ` ` `| `0 ` ` 0 ` ` 1 ` ` 1 ` ` 0 ` ` 0 ` ` 1 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_13`| 1101 | ((x) y) `| `0 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_14`| 1110 | ((x)(y)) | `0 ` ` 0 ` ` 0 ` ` 1 ` ` 0 ` ` 1 ` ` 1 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| f_15`| 1111 | ` (())` `| `0 ` ` 0 ` ` 0 ` ` 0 ` ` 1 ` ` 1 ` ` 1 ` ` 1 `|
| ` ` `| ` ` `| ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o

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Introduction to Inquiry Driven Systems

Note (November 2013). The current version of this material can be found here:

Introduction to Inquiry Driven Systems

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