Version 23
1.
Introduction
In 1985, Adi developed a theory of the Arabic
language by looking at an old Arabic book and asking a single question
thousands of times: "Do the many contexts in which a word is used point to
an abstract parallel between the word structure and the structure of the
physical context to which the word refers?"
In Arabic, there is one-to-one correspondence
between letters and sounds. Arabic has
28 letters and four vowels. Most Arabic word stems are verbs and most of them
consist of three consonants. Vowels are
added in the course of creating words out of stems and short vowels are usually
not written. Short vowels are not even
considered letters. Even long
(stretched) vowels are sometimes not written.
Vowels do not change the basic meaning.
First, Adi examined the contextual usage of
prefixes, suffixes, prepositions, conjunctives, pronouns and other
morphological and syntactic units that consisted of a single consonant and a
vowel. Then he turned to stems that had
two soft consonants (ya, hamza, waw, ha). Soft consonants are related to vowels
and are known to add much less meaning to stems than other consonants. Then, he examined stems with one soft
consonant, and finally stems without soft consonants. Adi started by examining stems that denote simple processes that
are easy to visualize.
The Adi Ontology is the result of these analyses.
Management Information Technologies, Inc. has patented the structures and
procedures of the Adi Ontology and implemented them in its Readware
tecnology. Readware text analysis
products have excelled in major international competitions such as TREC 7 and
TREC 8.
This work is an attempt by Adi to re-express his
ontology in abstract mathematical formalisms with the help of Paul Prueitt
who--among other things--masters abstract math. Prueitt has made countless comments and suggestions that Adi used
to re-express his ontological observations and insights. Prueitt did not contribute to the Adi
Ontology itself and was careful not to suggest any changes to it.
Once this work is finished, Adi and Prueitt plan to
have a discussion of how this ontology relates to stratified theory and Karl
Pribram's scientific work.
2. Notions
Observed in Contextual Usage of Arabic Vowels and Consonants Indicate Abstract
Aspects
In the course of attempting to identify an abstract
aspect that represents real things, one encounters intermediate notions that we
call indicators. We say notion
X is an indicator of aspect Y or notion
X indicates aspect Y.
Adi observed that each Arabic vowel or consonant
plays different roles as indicator of multiple abstract aspects. In a refinement process, he grouped similar indicators under a single
abstract aspect. Then, he grouped
aspects that seemed related into aspect sets.
Elements of aspect sets are the contents of
frames at three layers of
abstraction:
1. Substructural ontology (indicated by vowels and consonants)
2. Functional ontology (indicated by word stems)
3. Cognitive ontology (implementation of functions)
We will discuss functional ontology in Part II and
cognitive ontology in Part III.
In every single context, each vowel or consonant
points to notions that always indicate the same pair from the cross product of
the two aspect sets of boundary
{open, closed} and engagement {self,
engaged}. Adi referred to these ordered
pairs as polarities (closed, self),
(open, self), (closed, engaged), (open, engaged), and labeled them for short inward, outward, engaged and separate, respectively.
The correspondence between sets of consonants and a
single polarity created an observational basis for the discoveries that
followed.
A high degree of regularity is seen in the observed
correspondence between polarities and the short form of vowels. The long form of vowels (so-called stretched
vowels) repeats and extends the correspondence.
In addition to a unique polarity, each consonant
also points to notions that indicate up to three abstract processes. They form
the aspect set of processes {assignment, manifestation, containment}.
Remember that most Arabic stems are verbs, and that
Adi focused on stems that express simple processes that could be easily
visualized.
But Adi also observed notions that indicated a
special aspect associated with processes that he called the static view. This is like the way people look at something as if it had no
internal or relative motion or change.
Our hearts are beating and the earth is turning and traveling fast, but
we see ourselves standing still. Adi
called the static view of processes objects,
and labeled them element, domain and order, respectively.
Although notions that indicate polarities of
processes are different from notions that indicate polarities of objects, the
notion difference is due to the static view of the process that carries the
polarity. A judgment was made that
polarities themselves do not carry a dynamic versus static view. The introduction of a separate aspect such
as a "static view" of polarity was not justified. It would only be justified if some nuance
related to, say, a static view of polarity versus a static view of process was
useful.
To illustrate and motivate substructural ontology, a
collection of descriptions is organized in a grouped correspondence to a banding of an 8x4 matrix holding the
set of 32 vowels and consonants of the Arabic language. The banding treats the first row as the top
band. The second, third and fourth rows
are treated as the second band of the matrix.
The fifth, sixth and seventh rows are treated as the third band. The eighth and last row of the matrix is
treated as the fourth band of the matrix.
The banding follows a decomposition of the number 8
(rows) into 1 + 3 + 3 + 1 (rows), which are the numbers of subsets of size 0,
1, 2, or 3 of a universal set of size 3.
The bands partition the cells of the 8x4 matrix into groups having 4,
12, 12, 4 elements in each group. Each
group has a categorical difference of some importance.
The formal presentation of the substructural
ontology follows, where we will establish a one-to-one correspondence between
this 8x4 matrix and an 8x4 matrix representing what we conjecture to be a
reasonable way to organize categories of abstractions over the set of natural
kinds.
In the examples below, aspects are in bold type.
1) The Arabic vowel group (vowel_i, vowel_a, vowel_u
and sukoon) points to notions that indicate the polarities inward, outward, engaged and separate.
Short vowels at the end of Arabic nouns and verbs mark grammatical aspects.
Vowel_i at the end of nouns
means--among other things--that someone owns or is receiving something, notions
that indicate inward polarity.
Vowel_a means that something
is dual or is being acted upon:
indicators of outward
polarity.
Vowel_u means activeness for
nouns and the pending tense (mudari') for verbs: indicators of engaged
polarity.
Sukoon (silence, the null
vowel) means a special condition for verbs: an indicator of separate polarity.
2) The soft consonant group (ya, hamza, waw,
ha)--called soft because thay are relatives of vowels (vowel_i, vowel_a,
vowel_u, sukoon, respectively)--points to notions that indicate the polarities inward, outward, engaged and separate of the assignment
process or its static view, the element object.
The address article
"ya" (O, spelled ya and stretched vowel_a) is an article used to
address someone, an indication of inward
in using the assignment process.
The "hamza" is a
question article, an indicator of the outward
(open) polarity of the assignment
process. As a verb prefix, hamza offers two indicators of outward
assignment: removal and granting.
As conjunctive
"waw" offers different indicators to the engaged polarity of the assignment
process. It means "and"
(connects two things), "while" (as in "eating while
talking", engagement into something) or "I swear by" (a
commitment).
"Ha" is at the
beginning of all third person pronouns "hoowa" (he),
"heeya" (she), "hum" (they), etc. The third person indicates the separate polarity of the element object, the static view of the assignment process.
3) The consonant group (meem, fa, dal, thal) points
to notions that indicate the polarities
inward, outward, engaged and separate of the aspect of the manifestation process or its static view, the domain object.
As prefix, "meem"
is used profusely in Arabic morphology to define
a number of things: a person, an activity, a time, a place, a method or an
instrument. These are indicators of the
inward polarity (closed self, defined) of the manifestation process or its static
view, the domain object.
In "foo" (mouth,
spelled fa waw) and "fee" (in, into, spelled faa ya), consonant
"fa" offers indicators of the outward
polarity of the domain object. Outward manifestation is indicated by the notion of outward action in the sense "into" of "fee" and
also in the conjunctive "fa" which means "then."
"Dal" offers
indicators of engaged domain or engaged manifestation in the notions
carried by the following words where "dal" is the only non-soft
consonant: "wadi" (valley,
engaged domain), "diyah" (compensation--engaged manifestation--for
involuntary manslaughter) and "adaa" (paying back debt, engaged
manifestation).
"Thal" (pronounced
like "the") is combined with vowels to express notions that indicate
a separate manifestation or domain:
"thoo" (special attribute), "tha" (this, that),
"allthee" (the one who, that which) etc. "Itha" (when)
indicates specific time (separate domain object).
4) The consonant group ('ain, noon, qaf, ghain)
points to notions that indicate the polarities
inward, outward, engaged and separate of the aspect of the containment process or its static view, the order object.
In "wi'a"
(container, waw 'ain stretched
vowel_a hamza), "'ain" is the only non-soft consonant. The container notion obviously indicates inward containment.
"Ayna" (where,
hamza ya noon) indicates an outward (undefined) containment.
The stem "waw qaf ya" means shielding, indicating
engaged containment. Consider the
stems "qaf dal ra"
(quantity) and "qaf waw
meem" (straight) that contain "qaf." They indicate engaged
order.
The stem "ghain ya ba" means to hide, an
indicator of separate containment. "Ghain waw ya" means "to
stray," an indicator of separate order.
So far, we have encountered three substructural
aspect sets:
substructural
boundary:
closed, open
substructural
engagement:
self, engaged
substructural processes: assignment,
manifestation, containment
and a special abstract aspect that we called the static view of a substructural process.
The four polarities are formed from a cross product
of boundary and engagement aspect sets.
Into these polarities are combined the referential indicators associated
with vowels. The four polarities also
create a partition into four categories of the referential indicators
associated with the consonants discussed so far. These indicators also indicate
processes and the static view. We thus
find places within an 8x4 framework for the three sets of four consonants
discussed above.
In the following, each consonant is associated with
notions that each simultaneously indicate two substructural processes. These two processes appear to interact and
we notice a forth abstract aspect, substructural
precedence, that is indicated by notions regarding which process is active and which process is passive. The process that has substructural precedence is active and the
other is passive.
In the following, substructural precedence is
written in italics when it is spelled
out by prepositions such as "assignment to a domain"
(assignment is active and domain is passive).
Substructural precedence may also be implied. For example, "manifestation assignment" implies that
assignment is active and manifestation is passive.
Sometimes, the passive process is likely to be
viewed as static, i.e. as object.
5) The consonant group (ra, lam, ba, ta) points to
notions that indicate the polarities
inward, outward, engaged and separate of a permutation of the aspect
of assignment process or its static
view (element object) with the
aspect of manifestation process or
its static view (domain object).
"Waraa," where
"ra" is the only non-soft consonant, means "behind,"
indicating an inward (backward) assignment in domain.
"Lam" is a
preposition that means "belonging to," indicating an outward assignment to a domain.
"Ba" is a
preposition that expresses causality, an indicator of engaged assignment of a manifestation.
"Ta" is an oath preposition, an indicator of separate (special)
assignment to a manifestation.
6) The consonant group (seen, zay, sad, tha) points
to notions that indicate the polarities
inward, outward, engaged and separate of a permutation of the aspect of assignment process or
its static view (element object) with the aspect of containment process or its static view (order object).
"Seen waw ya" means "equal" or "level,"
indicating assignment to
inward (defined) order. "Seen
ya lam" (flood) indicates assignment
to an inward containment.
"Zay waw ra" means "perjury," "veer" and
"visit," an indication of assignment
to outward (undefined) order
or containment (visit indicates
outward containment).
"Wa sad ya" means "a will" or
"commandment," indicators of engaged
assignment of order.
"Tha wa ya" means "to lodge," an indicator of assignment of separate containment.
7) The consonant group (kaf, ddad, tta, kha) points
to notions that indicate the polarities
inward, outward, engaged and separate of a permutation of the aspect
of manifestation process or its static view (domain object)
with the aspect of containment process
or its static view (order object).
Preposition "kaf"
means "similar to," indicating a manifestation
of inward (defined) order.
"Ddad lam lam" means "stray," indicating manifestation of outward order (disorder).
"Tta waw ya" means "fold," indicating a manifestation of engaged containment.
"Kha waw ya" means "empty," indicating a manifestation of separate containment
(removed content).
We notice that assignment has process precdence over
manifestation and containment, and that manifestation has process precdence
over containment. This establishes the descending substructural precedence
sequence: assignment, manifestation, containment.
In the last group, each consonant is associated with
all substructural processes.
8) The consonant group (hha, sheen, geem, zza) points
to notions that indicate the polarites
inward, outward, engaged and separate of a permutation of the aspect
of assignment process or its static
view (element object) with the
aspect of manifestation process or
its static view (domain object) and with the aspect of containment process or its static view (order object). Because of
substructural precedence, we have the compound process of assignment of
manifestion and containment. These consonants are also used with general
purpose notions.
"Hha waw ya" means "slither," indicating inward assignment of manifestation and containment.
"Sheen ya hamza" means "want" or "thing"
indicating outward assignment of manifestation
and containment.
"Geem ya hamza" means "to come," indicating an
engaged assignment of manifestation and containment.
"Zza lam lam" means "to remain" or "shade,"
indicating separate assignment of manifestation
and containment.
Adi uses the cross product of boundary and
engagement (polarities) in one dimension, and all possible combinations of
processes in another dimension, to create the 4x8 abstract matrix Q that is
described in the following section. The
order of the second to fourth rows corresponds to a descending order of
substructural precedence that governs the interaction of processes in the
following rows. The static view of a process also plays a role in this matrix.
3. Complex
Abstract Frameworks
The notion of a framework is central to at least
three scholars’ work: Zachman, Sowa and Ballard [1].
Let S1, S2, … , Sn
be n abstract aspect sets. An abstract
framework F is the cross product
F = S1 x S2
x … x Sn
The framework F
is said to have n dimensions and a size equal to the product of the sizes of
all framework aspects sets.
For example, let A and B
A = {aspect1, aspect2}
B = {aspect3, aspect4, aspect5}
be two aspect sets of the framework F.
Then
F =
A x B
size(F) = size(A) * size(B) = 2 * 3 = 6
F may be represented as a
matrix where the cells contain the elements of the cross product. In this case we have a 2x3 matrix
F = [ f(i, j) | i = 1, 2 and j = 1, 2, 3 ]
The cell f(1,1) contains the ordered pair (aspect1,
aspect3), and so forth. The cell is
called a frame. The framework consists of single frames that
are each ordered tuples, members of the cross product of the aspect sets..
The aspect sets of a framework are constructed by different people in different ways and for different purposes. But aspects are descriptive names of abstract things and they should be selected to be as independent as possible from each other.
Adi derived his aspects by induction over a complete survey of all the notions associated with all Arabic vowels and consonants and around a thousand Arabic word stems in an old Arabic book. In other words, Adi recorded all the notions in the around five hundred pages of the book, associated with every single text spot where a vowel, a consonant, or one of the around thousand word stems is used.
The notions he observed indicated several abstract
aspects. In a refinement process, he
grouped similar indicators under a
single abstract aspect. Then, he grouped aspects that seemed related into separate aspect sets.
The presence of abstract precedence rules was also is indicated by notions observed by Adi.
The nature of these aspect sets and their relationships requires the definition of a complex abstract framework that goes beyond a simple framework.
Let S = {S(i) | i = 1 , 2, ..., n} be a set of abstract aspect sets with precedence, i.e. there is at least one central aspect set S(k) where an abstract precedence operator exists that determines which element of any subset of S(k) has precedence over the remaining elements of the subset. A complex abstract framework is an operator defined from operations over S to operations over S.
To facilitate his presentations, Adi makes use of shorthand notation. For example, instead of an ordered pair Adi may use a single term. Adi did this with polarities in the previous section. In the following section Adi will also use bold type to indicate an ordered pair.
4. First Layer
of Abstraction--Substructural Ontology
For notational convenience and completeness, we introduce the following operations defined on sets, frames and lists.
Define a list
as a comma-separated sequence of any kind of elements, sets, structures,
functions or operators, so
a, {b}, (c, d), f, D is a list
where f is a function and D
is an operator
Define a
frame as a unity, indicated by "(-)", filled with a list -, so
(a(i) | i = 1 to n) is a frame, where a(i) can be anything.
Define frame(-) = (-), where "-" is a list, so frame(a, b, c) = (a, b, c).
Let
T = {closed, open}
G = {self, engaged}
be the boundary
aspect set and the engagement aspect
set, respectively.
Let
R =
T x G = { r(i) | i = 1 to 4 }
= { (closed, self),
(open, self), (closed, engaged), (open, engaged) }
or using a shorthand
notation
= {inward, outward, engaged, separate}
be called the set of substructural polarities.
Let
P =
{ p(i) | i = 1, 2, 3 } = {assignment,
manifestation, containment}
be the aspect set of substructural processes. For convenience we may write p1 for
p(1), p2 for p(2) and p3 for p(3).
We enumerate the power set of P
P*
= {s(i) | i = 1 to 8}
= { {}, {p1}, {p2}, {p3}, {p1, p2}, {p1, p3}, {p2, p3}, {p1,
p2, p3} }
Note that the size of P* is 8.
In the subsets of P* that contain more than one process, the processes interact in
certain ways. One process will be active
while the other process(es) will be passive. We have seen process interaction in the
example sets (5) to (8) in Section 2 above.
Let
Z = { active, passive }
be the aspect set of process states. Define the substructural precedence operator C
from P* to the cross product P* x Z that assigns a status out of Z to the elements of each multi-process subset of P* according to the descending substructural precedence sequence: assignment, manifestation,
containment. We enumerate C
C (
s(i) ) = s(i) i = 1, 2, 3, 4
C (
s(5) ) = { (assignment, active),
(manifestation, passive) }
C (
s(6) ) = { (assignment, active),
(containment, passive) }
C (
s(7) ) = { (manifestation, active),
(containment, passive) }
C (
s(8) ) = { (assignment, active),
(manifestation, passive), (containment, passive) }
Let
W = { static }
be the static
view aspect set that is associated with each substructural process. The static view of a substructural process
is called an object. In Section 2, we called the static views of
the substructural processes assignment, manifestation and containment by the
object names element, domain and order, respectively.
Objects are not separate aspects of the ontology,
but the result of applying a single aspect, the static view, to processes. In other words, the set of substructural
objects B is the cross product W x P
B = W x P = { b(i) | i = 1, 2, 3 }
= { {static, assignment}, {static, manifestation}, {static,
containment} }
= { element, domain, order }
Define the operator V from P to B
V(p(i))
= b(i) where i = 1, 2, 3
that generates the static view of a
process, the corresponding object.
The substructural
ontology is a complex abstract
framework defined by the operator q
over the set of aspect sets H = {T, G,
P, W, Z}
q(i,
j) = frame( r(j), C (s(i)) , V ) where i = 1 to 8 and j = 1 to 4
where r(j) is an element of T x G
and s(i) is a member of P*
and substructural precedence
operator C is defined from P* to P* x Z
and V is the static view operator from P to W x P
The complex abstract framework operator q generates the 4x8 matrix Q whose elements we call substructural frames.
For the fifth to eighth rows of the matrix Q, we introduce a shorthand
notation. A process X in bold type is
short for the ordered pair (X, active) and a process X in plain type is short
for the ordered pair (X, passive). For
example
C (
s(5) ) = { (assignment, active),
(manifestation, passive) }
={ assignment,
manifestation }
Q = [q(i, j) = frame( r(j), C (s(i)) , V ) | i = 1
to 8 and j = 1 to 4 ]
|
|
(inward, {}, V) |
(outward, {},
V) |
(engaged, {}, V) |
(separate, {}, V) |
|
|
|
(inward, {assignment}, V) |
(outward, {assignment}, V) |
(engaged, {assignment}, V) |
(separate, {assignment}, V) |
|
|
Q = { |
(inward, {manifestation}, V) |
(outward, {manifestation}, V) |
(engaged, {manifestation}, V) |
(separate, {manifestation}, V) |
} |
|
|
(inward, {containment}, V) |
(outward, {containment}, V) |
(engaged, {containment}, V) |
(separate, {containment}, V) |
|
|
|
(inward, {assignment, manifestation}, V) |
(outward, {assignment, manifestation}, V) |
(engaged, {assignment, manifestation}, V) |
(separate, {assignment, manifestation}, V) |
|
|
|
(inward, {assignment, containment}, V) |
(outward, {assignment, containment}, V) |
(engaged, {assignment, containment}, V) |
(separate, {assignment, containment}, V) |
|
|
|
(inward, {manifestation, containment}, V) |
(outward, {manifestation, containment}, V) |
(engaged, {manifestation, containment}, V) |
(separate, {manifestation, containment}, V) |
|
|
|
(inward, {assignment, manifestation, containment}, V) |
(outward, {assignment, manifestation, containment}, V) |
(engaged, {assignment, manifestation, containment}, V) |
(separate, {assignment, manifestation, containment}, V) |
|
The substructural static view operator V is available as a discretionary ontological operator to
be applied or not (at the discretion of the ontology user) to every element of
each subset of P* that is contained
in each substructural frame. V means that each substructural process
in each substructural frame may be viewed (measured, observed) in a static way
as a substructural object, as something that is not associated with motion or
change. This is like a snapshot, a
freeze frame, of a process.
Based in intuitive notions, the substructural
ontology Q appears to contain all
possible substructural frames. A sense
of completeness appears to exist. As
required, each of the 32 elements of Q
has an independent status, and each appears to have a legitimate basis for
being included.
We find that we can arrange the 4 vowels and 28
consonants of the Arabic language in a certain order in a 4x8 matrix A
A = [ a(i, j) | i = 1 to 8
and j = 1 to 4 ]
|
|
vowel_i |
vowel_a |
vowel_u |
sukoon |
|
|
|
ya |
hamza |
waw |
ha |
|
|
A = [ |
meem |
fa |
dal |
thal |
] |
|
|
'ain |
noon |
qaf |
ghain |
|
|
|
ra |
lam |
ba |
ta |
|
|
|
seen |
zay |
ssad |
tha |
|
|
|
kaf |
ddad |
tta |
kha |
|
|
|
hha |
sheen |
geem |
zza |
|
There is a one-to-one correspondence between the
vowels and consonants of the Arabic language and the substructural frames of
our ontology.
Adi derived Q
from A by induction over a
preponderance of evidence from thousands of notions that he observed in the
contextual usage of Arabic in an old Arabic book. This induction can be verified by scholars of Arabic.
Adi verified this induction by connecting to Q
notions associated with English letters in the contextual usage of thousands of
English words. The same was done for
many words from twenty different languages.
This verification created an 8x4 A'
matrix for each language examined.
However, Adi was not able to fill all the cells of any A'.
Adi conjectures
that Q is an abstract representation
of all elementary processes in the real world.
In Part II, we will discuss our second layer of abstraction that is
indicated by Arabic word stems, the functional ontology. In Part III, we will discuss cognitive
ontology, i.e. the ways functional ontology is used in cognition and knowledge
representation.
[1] See http://www.ontologystream.com/beads/enumeration/gFfoundations.htm for a review by Paul Prueitt of these frameworks and a discussion of the framework concept.