ONT Re: Zeroth Order Theories (ZOT's)
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I continue with the previous Example, that I bring forward and sum up here:
| boy male girl female
| o---o child o---o child
| male female \ / \ / child human
| o---o o o o--o
| \ / | | |
| @ @ @ @
|
| (male , female) ((boy , male child))((girl , female child))(child (human))
For my master's piece in Quantitative Psychology (Michigan State, 1989),
I wrote a program, "Theme One" (TO) by name, that among its other duties
operates to process the expressions of the cactus language in many of the
most pressing ways that we need in order to be able to use it effectively
as a propositional calculus. The operational component of TO where one
does the work of this logical modeling is called "Study", and the core
of the logical calculator deep in the heart of this Study section is
a suite of computational functions that evolve a particular species
of "normal form", analogous to a "disjunctive normal form" (DNF),
from whatever expression they are prebendered as their input.
This "canonical", "normal", or "stable" form of logical expression --
I'll refine the distinctions among these subforms all in good time --
permits succinct depiction as an "arboreal boolean expansion" (ABE).
Once again, the graphic limitations of this space prevail against
any disposition that I might have to lay out a really substantial
case before you, of the brand that might have a chance to impress
you with the aptitude of this ilk of ABE in rooting out the truth
of many complexly obscurely subtly adamant patterning of evidence.
So let me just illustrate the way of it with one conjunct of our Example.
What follows will be a sequence of expressions, each one after the first
being logically equal to the one that precedes it:
Step 1
| g fc
| o---o
| \ /
| o
| |
| @
Step 2
| o
| fc | fc
| o---o o---o
| \ / \ /
| o o
| | |
| g o-------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
Step 3
| f c
| o
| | f c
| o o
| | |
| g o-------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
Step 4
| o
| |
| c o o c o
| | | |
| o o c o o c
| | | | |
| f o---o--o f f o---o--o f
| \ / \ /
| g o-------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
Step 5
| o c o
| c | |
| f o---o--o f f o---o--o f
| \ / \ /
| g o-------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
Step 6
| o
| |
| o o o
| | | |
| c o---o--o c o c o---o--o c
| \ / | \ /
| f o-------------o--o f f o-------------o--o f
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| g o---------------------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
Step 7
| o o
| | |
| c o---o--o c o c o---o--o c
| \ / | \ /
| f o-------------o--o f f o-------------o--o f
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| \ / \ /
| g o---------------------------o--o g
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| \ /
| @
This last expression is the ABE of the input expression.
It can be transcribed into ordinary logical language as:
| either girl and
| either female and
| either child and true
| or not child and false
| or not female and false
| or not girl and
| either female and
| either child and false
| or not child and true
| or not female and true
The expression "((girl , female child))" is sufficiently evaluated
by considering its logical values on the coordinate tuples of %B%^3,
or its indications on the cells of the associated venn diagram that
depicts the universe of discourse, namely, on these eight arguments:
<1, 1, 1> = girl female child ,
<1, 1, 0> = girl female (child),
<1, 0, 1> = girl (female) child ,
<1, 0, 0> = girl (female)(child),
<0, 1, 1> = (girl) female child ,
<0, 1, 0> = (girl) female (child),
<0, 0, 1> = (girl)(female) child ,
<0, 0, 0> = (girl)(female)(child).
The ABE output expression tells us the logical values of
the input expression on each of these arguments, doing so
by attaching the values to the leaves of a tree, and acting
as an "efficient" or "lazy" evaluator in the sense that the
process that generates the tree follows each path only up to
the point in the tree where it can determine the values on the
entire subtree beyond that point. Thus, the ABE tree tells us:
girl female child -> 1
girl female (child) -> 0
girl (female) -> 0
(girl) female child -> 0
(girl) female (child) -> 1
(girl)(female) -> 1
Picking out the interpretations that yield the truth of the expression,
and expanding the corresponding partial argument tuples, we arrive at
the following interpretations that satisfy the input expression:
girl female child -> 1
(girl) female (child) -> 1
(girl)(female) child -> 1
(girl)(female)(child) -> 1
In sum, if it's a female and a child, then it's a girl,
and if it's either not a female or not a child or both,
then it's not a girl.
Enough for now ...
Jon Awbrey
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