12,080
edits
Changes
Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy (view source)
Revision as of 15:23, 16 November 2008
, 15:23, 16 November 2008Import Raw Text (IRT)
{{DISPLAYTITLE:Functional Logic : Inquiry and Analogy}}
{{DISPLAYTITLE:Functional Logic : Inquiry and Analogy}}
<pre>
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
IDS -- Fumctional Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 1
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Inquiry and Analogy
| Version: Draft 3.25
| Created: 01 Jan 1995
| Revised: 24 Dec 2001
| Revised: 12 Mar 2004
Abstract
This report discusses C.S. Peirce's treatment of analogy,
placing it in relation to his overall theory of inquiry.
The first order of business is to introduce the three
fundamental types of reasoning that Peirce adopted
from classical logic. In Peirce's analysis both
inquiry and analogy are complex programs of
reasoning which develop through stages of
these three types, although normally in
different orders.
1. Three Types of Reasoning
1.1. Types of Reasoning in Aristotle
1.2. Types of Reasoning in C.S. Peirce
1.3. Comparison of the Analyses
1.4. Aristotle's "Apagogy": Abductive Reasoning as Problem Reduction
1.5. Aristotle's "Paradigm": Reasoning by Analogy or Example
1.6. Peirce's Formulation of Analogy
1.7. Dewey's "Sign of Rain": An Example of Inquiry
2. Functional Conception of Quantification Theory
Up till now quantification theory has been based on the assumption of
individual variables ranging over universal collections of perfectly
determined elements. Merely to write down quantified notations like
"(For All)_(x in X) F(x)" and "(For Some)_(x in X) F(x)" involves a
subscription to such notions, as shown by the membership relations
invoked in their indices. Reflected on pragmatic and constructive
principles, these ideas begin to appear as problematic hypotheses
whose warrants to be granted are not beyond question, as projects
of exhaustive determination that overreach the powers of finite
information and control to manage. Consequently, it is worth
considering how we might shift the medium of quantification
theory closer to familiar ground, toward the predicates
themselves that represent our continuing acquaintance
with phenomena.
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.
2.1.1. Higher Order Propositions and Logical Operators (n = 1)
A "higher order" 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 higher order propositions
that are based on this set, all bearing the type m : (B -> B) -> B.
Table 10 lists the sixteen higher order 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, turned on its side from the
way that one is most likely accustomed to see truth tables, with the
row leaders in Column 1 displaying the names of the functions F_i,
for 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
corresponding column head. Column 3 displays one of the
more usual expressions for the proposition in question.
The last sixteen columns are topped by a collection of
conventional names for the higher order 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 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
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.
Table 11. 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
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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 type m : (X -> B) -> B become the propositions of the second
order universe of discourse 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" propositions
and operators. To describe the present arrangement 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 sixteen of these measures in the
fashion of the higher order truth table that I set out 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
a blank entry in the Table being optional for a value of zero.
The arrangement of measures that continues in accord with this
plan 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
in 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
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
2.1.3. Umpire Operators
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 modulate the sense of the operator in question.
Accordingly, the employment of umpires may be
contingent on the following application 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 apply when the corresponding place is left blank,
specifically, the constant proposition 1 : B^k -> B for the lower
index p, and the continued conjunction or continued product ]^[
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 licences 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 true over the entire
universe of discourse in question, in other words, if and only if q
is the constant true proposition 1 : B^k -> B. The ambiguities of
this trial usage will not be a problem so long as we distinguish
the context of definition from the context of application and
restrict all shorthand notations to the latter.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 4
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
2.1.4. Measure for Measure
An acquaintance with the functions of the umpire operator can be gained
from Tables 13 & 14, 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 these two Tables.
Table 13. 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 14. 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.
In short, !a!_0 = !b!_15 is a totally indiscriminate measure,
one that accepts all of the propositions f : B^2 -> B, while
!a!_15 and !b!_0 are measures that appreciate 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
the following to denote classes of propositions that satisfy
the umpires in question:
[| !a!_i |] = (!a!_i)^(-1)(1)
[| !b!_i |] = (!b!_i)^(-1)(1)
[| !Y!_p |] = (!Y!_p)^(-1)(1)
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 5
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
2.1.5. 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 universe X% = [x, y].
In order to construct a bridge between propositions and
quantifications it serves to create a set of qualifiers
L_uv : (B^2 -> B) -> B that take on the following forms:
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 15 summarizes the action of the L_uv on the f_i in 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 15. Higher Order Universe [L_uv] c [[x, y]]
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 6
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
2.1.6. 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 establish a type of "existence-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 16. Syllogistic Premisses as Higher Order Indicator Functions
o---o---------------------o-----------------o----------------------------o
| | | | |
| A | Universal Affrmtve | All x is y | Indicator of < x (y)> = 0 |
| | | | |
| E | Universal Negative | All x is (y) | Indicator of < x y > = 0 |
| | | | |
| I | Particular Affrmtve | 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 17 and 18 develop these ideas in greater detail.
Table 17. 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 18. 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 |
| | | | |
| f_1 | 0001 | (x)(y) | 1 1 1 1 |
| | | | |
| f_2 | 0010 | (x) y | 1 1 1 1 |
| | | | |
| f_3 | 0011 | (x) | 1 1 1 1 |
| | | | |
| f_4 | 0100 | x (y) | 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 1 1 |
| | | | |
| f_8 | 1000 | x y | 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 1 1 |
| | | | |
| f_12 | 1100 | x | 1 1 1 1 |
| | | | |
| f_13 | 1101 | ((x) y) | 1 1 1 1 |
| | | | |
| f_14 | 1110 | ((x)(y)) | 1 1 1 1 |
| | | | |
| f_15 | 1111 | (()) | 1 1 1 1 |
| | | | |
o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 7
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Note 0
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
To: NKS
I will need to focus on the DATA thread while I have
the concentration to do so, but an exagoric inquiry
from a reader of that thread prompts me to revisit
for a moment a topic where I previously noticed
some fractal shades lurking about my tables.
I am reorganizing this material on a couple
of other discussion lists, starting here:
FL. http://suo.ieee.org/ontology/thrd1.html#05480
FL. http://stderr.org/pipermail/inquiry/2004-March/thread.html#1256
See especially FL Note 4, at either of these web loci:
FL 4. http://suo.ieee.org/ontology/msg05483.html
FL 4. http://stderr.org/pipermail/inquiry/2004-March/001259.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
FL. Functional Logic
Ontology List
01. http://suo.ieee.org/ontology/msg05480.html
02. http://suo.ieee.org/ontology/msg05481.html
03. http://suo.ieee.org/ontology/msg05482.html
04. http://suo.ieee.org/ontology/msg05483.html
05. http://suo.ieee.org/ontology/msg05484.html
06. http://suo.ieee.org/ontology/msg05485.html
07.
Inquiry List
01. http://stderr.org/pipermail/inquiry/2004-March/001256.html
02. http://stderr.org/pipermail/inquiry/2004-March/001257.html
03. http://stderr.org/pipermail/inquiry/2004-March/001258.html
04. http://stderr.org/pipermail/inquiry/2004-March/001259.html
05. http://stderr.org/pipermail/inquiry/2004-March/001260.html
06. http://stderr.org/pipermail/inquiry/2004-March/001261.html
07.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
</pre>