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Note: Neologies and ambiguities clarified in Glossary are marked "[G]".
CCA.COGNITIVE NETWORK
A.Foreword
This chapter presents the principles of the phenomenological
Logical System called "Cognitive Network" (CN). It may be
considered as epistemological description, or as introduction
to practical developing and using concrete CN applications.
We have seen in BAA.STRUCTURE OF MIND
that imaginary Event Patterns of "Imagery" or "Phenomenal
#Space"[G] (PS) map into symbolic Entity structures of
"Symbolics" or "Abstract #Space"[G] (AS). In particular,
Patterns of Events related by CAUSALITY
map into Network structures of Entities related by
Implication, which we shall call "ER" Networks.
We shall define "Logic" as Mind's dynamic construct, a system
involving symbolic ER Networks processed by the Faculty of
Inference.
Thus defined Logic designates a subconscious Mind's system
used instinctively as support of daily behavior. This
"instinctive" Logic is irreplaceable as pilot of simple
activities, but it may be misleading due to the vagueness of
Causality and it is incapable to handle complex cases due to
Mind's limited working memory and to its incapacity to
concentrate simultaneously on numerous issues.
Facing shortcomings of their natural faculties, humans
usually produced compensating tools: hammer to assist
striking and extrinsic "Logical Systems" to assist intrinsic
Mind's reasoning, Our "Cognitive Network" (CN), as any
Logical System, may be justified exclusively by its capacity
to support Mind's intrinsic, ER based Logic. Its applications
have so far stood this test. It replaces consistently and
simply both, the ill-founded Set Theory and its underlying
noumenal, naive Predicate (pseudo-)Logic.
( CA2.INTRODUCTION TO PREDICATE LOGIC )
CN is an extrinsic Entity-Relation (ER) structure. Its
vertices or nodes are Entities valued by Certainty, a
continuous variable ranging from 0 to 1, replacing in Fuzzy
System the binary 1/0 (true/false) variable of Exact Systems.
Its edges are Relations. From the point of view of Fuzzy
Logical Calculus, Entities are Operands and Relations -
Operators. Operators evaluate the Certainty of an Entity
inductively, in function of its lower neighbors related into
expressions. Structure, dimensionality of nodes and types
of Operators are similar to those described
in CAB.ND EXACT PROPOSITIONAL CALCULUS
with binary logical variable (true/false, or 1/0) replaced
by the continuous fuzzy Certainty. Expressions of Entities
related with fuzzy Operators involve quite complex algorithms
(see APPENDIX).
For conciseness' sake we shall call lower level neighbors of
a node its "parts" and higher level ones - its "aggregates".
B.Inference
Philosophical and scientific inquiry reposes essentially on
Inference exerted explicitly or implicitly over a CN-like
network structure.
Inference, a "two-way" procedure encompasses Deduction and
Induction scanning CN respectively top-down and bottom-up.
Deduction scans the CN top-down creating a concrete instance
of the hypothetical Theory and setting fuzzy Operatore in
Relations according to Theory definition. Top nodes having
no aggregates (deductive premises), thus nothing to be
deduced from, are Axioms, granted as certain, but subject to
inductive, factual verification 'see below). Middle nodes,
having both, aggregates and parts are Theorems.
Induction scans the CN bottom-up setting fuzzy Certainty of
nodes in function of their parts (inductive premises) and
connecting Relations (fuzzy Operators), It starts with bottom
nodes which, having no parts, thus nothing to be induced from,
are set by extralogical Observations, or Facts. Inductive
scan verifies or falsifies CN's Theorems and Axioms in the
light of Facts.
C.Diagnostics, an example.
This example presents printouts of a particular CN program
written for a Company which used it for identifying
misfunctions in space crafts. It's defined in terms used in
medical diagnostics, more familiar to the average reader.
The form (indented printouts, etc) is not inherent in CN,
but pertains to the particular program.
CA.Step 1
Fig. 1 is a graphic representation of Step 1 of defining the
AI Application, the general, static, deductive Class
Structure:
"diagnostics" implies
("diagnosis" implies
("syndrome" implies
("symptom")))
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Aggregates and Parts of a node are respectively its deductive
and inductive premises. Top nodes lacking Aggregates are
Axioms, granted as certain, but subject to inductive check.
Middle nodes having both, Parts and Aggregates, are Theorems.
Bottom nodes (Symptoms) are Factual Predictions, which lack
Parts and can be set only by corresponding Facts observed in
the Phenomenal #Space[G] (PS). Symptoms' Certainty, spreading
bottom-up throughout the Network verifies or falsifies
Theorems and Axioms.
At the Step 1, Axioms and Theorems marked in red represent
purely deductive domain of the Theory restricted to the
Abstract #Space[G] (AS). Potential Facts marked in green,
represent the intersection AS/PS, AS Predictions open to
verifying PS Observations.
NOTE: -#Space denotes abstract topological concept such as
Riemann #Space, synonymous with Domain" or "Class" and
sharply distinct from its unfortunate homonym, the "space"
of direct perception. PS and AS are defined in
BAA.STRUCTURE OF MIND as
observing and symbolizing domains of Mind.
CB.Step 2
Fig. 2 illustrates Step 2, the dynamic, inductive Set
Structure:
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CBA.Particularities of Step 2:
CBAA.Concretisation.
Deductive Symptoms Class of Step 1, is concretised by PS
Observations into Factual Set of Symptoms. Following the
concretising rule, Set-Parts concretise their deductive
Class Relations into concrete inductive Set Relations.
Set Relations concretise in turn involved Aggregates from
Classes into Sets. The process is recursive, so that all
Entities and Relations of our simple Hierarchy become
concrete Sets and Set Relations.
In the general case of a Network structure not all Aggregates
would be concretised, but only those implying recursively the
Facts.
(see "Fig.2, Symbolic Structure 1, inductive" of
REFLECTION SPIRAL )
CBAB.Inductive Inference Rules.
Concrete Set Relations are associated with Fuzzy Operators.
which compose a rather complex domain outlined in Appendix.
Here we recall from ND EXACT PROPOSITIONAL CALCULUS
that dimensionality is local and for each Aggregate is
determined by the number of its Parts. This unlimited
dimensionality requests Operators capable to deal with any
number of inductive premises (Parts). Some Operators map one
to one from the 2D to the nD Calculus, but other fork to more
than one in nD. Such is the case of "orr" (exclusive or)
which maps to nD as n distinct Operators, in 3D to "one-of",
"two-of" and "not-all".
Our example uses two Fuzzy Operators:
-"AND", as a syndrome implies all symptoms and a diagnosis
implies all syndromes,
-"OOF" (One Of), as only one diagnosis may be chosen as base
of subsequent therapy.
Inference operates not on general Sets, but on particular Set
Instances. In Step 2 we define general inference Rules (in
form of Operators associated with Relations), which will be
applied to Set Instances in the subsequent Step 3.
CC.Step 3 Inductive Inference Run
Fig. 3 illustrates Step 3, the inductive Inference Instance
Structure, showing in graphic form the Instances of
"diagnostics" Application interrelated and ready for imputing
the observed Certainty of symptoms and for subsequent
inductive Inference.
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Application's printout of the Instances Structure in indented
form is shown in Fig. 4. It represents "Explosion" or
"Extension" of "1 diagnostics", i.e. a recursive structured
enumeration of its parts. As it shows recursively the whole
structure, we call it "Deep Explosion".
Fig. 4. Deep Explosion of "diagnostics".
1 diagnostics axiom
2 diagnosis_1_3 oof thrm
3 syndrome_1_acf and thrm
4 symptom_a and fact
4 symptom_c and fact
4 symptom_f and fact
3 syndrome_3_bgh and thrm
4 symptom_b and fact
4 symptom_g and fact
4 symptom_h and fact
2 diagnosis_2_4 oof thrm
3 syndrome_2_bcd and thrm
4 symptom_b and fact
4 symptom_c and fact
4 symptom_d and fact
3 syndrome_4_aeg and thrm
4 symptom_a and fact
4 symptom_g and fact
4 symptom_e and fact
2 diagnosis_3_4 oof thrm
3 syndrome_3_bgh and thrm repetition
3 syndrome_4_aeg and thrm repetition
Legend:
1.Numbers starting the lines are "levels" of the Structure.
Entity of level N implies directly Enities of level N+1:
"1 diagnostics" implies "2 diagnosis_1_3", "2 diagnosis_2_4",
"2 diagnosis_3_4".
"3 syndrome_2_bcd" implies "4 symptom_b", "4 symptom_c",
"4 symptom_d".
2."oof", "and" are Operators associated with relation between
Enities of level N+1 and N: 1 syndrome_1_acf = and 2 symptom_a
and 2 symptom_c and 2 symptom_f, or in Polish Notation:
1 syndrome_1_acf = and (2 symptom_a, 2 symptom_c, 2 symptom_f)
In Polish Notation:
1 diagnostics = oof(2 diagnosis_1_3, 2 diagnosis_2_4,
2 diagnosis_3_4) (is one of them).
"axiom" denotes the top Node which has Parts, but no
Aggregates. In terms of the Model it is Axiom, i.e. Node
set arbitrarily to "true".
"thrm" denotes middle Nodes having both Parts and Aggregates.
In terms of the Model they are Theorems.
"fact" denotes bottom Nodes which have no Parts and are
set by extralogical Observations. In terms of the Model they
are Facts, or Observations of the Phenomenal #Space.
"repetition" denotes an Entity whose Explosion appeares
above in the indented display and is not repeated for
conciseness' sake.
Discussing the Fig. 1 above we have indicated that bottom
Entities of CN structure represent Facts or Observations of
the Phenomenon #Space. In our example, bottom Entities are
"symptoms", marked accordingly as "fact". In the case of our
"diagnostics", the Factual "symptoms" may be set by sensors
of some technological device such as a space craft, or by a
physician who observes symptoms of a patient, which are more
or less typical, strong or in one word "certain". Fig. 5
shows an example of a distribution of symptoms' certainties
inputted to CN and printed out.
Fig. 5. Input of symptoms' Certainties.
NOTE: All Certainties are expressed in percents.
symptom_a and 98 fact
symptom_b and 95 fact
symptom_c and 96 fact
symptom_d and 25 fact
symptom_e and 18 fact
symptom_f and 97 fact
symptom_g and 98 fact
symptom_h and 94 fact
Starting from those premises CN executes the inductive,
bottom up Inference scan results are shown in Fig. 6.
It's the same structure as that of Fig. 4 with additional,
inductively evaluated Certainties.
Fig. 6. Deep Explosion of evaluated Instances.
1 diagnostics 77 axiom
2 diagnosis_1_3 oof 80 thrm
3 syndrome_1_acf and 93 thrm
4 symptom_a and 98 fact
4 symptom_c and 96 fact
4 symptom_f and 97 fact
3 syndrome_3_bgh and 89 thrm
4 symptom_b and 95 fact
4 symptom_g and 98 fact
4 symptom_h and 94 fact
2 diagnosis_2_4 oof 1 thrm
3 syndrome_2_bcd and 18 thrm
4 symptom_b and 95 fact
4 symptom_c and 96 fact
4 symptom_d and 25 fact
3 syndrome_4_aeg and 12 thrm
4 symptom_a and 98 fact
4 symptom_g and 98 fact
4 symptom_e and 18 fact
2 diagnosis_3_4 oof 6 thrm
3 syndrome_3_bgh and 89 thrm repetition
3 syndrome_4_aeg and 12 thrm repetition
Deep Explosion of the top Entity is in practical cases too
long and too complex to be grasped at a glance. Even our
very small example may be found not quite limpid. It is
utile to navigate through the structure with help of one
level or "flat" explosion, as shown in Fig. 7-9.
Fig. 7. Flat Explosion of the top Entity "diagnostics".
1 diagnostics 74 axiom
2 diagnosis_1_3 oof 80 thrm
2 diagnosis_2_4 oof 1 thrm
2 diagnosis_3_4 oof 6 thrm
"diagnostics" implies "diagnosis_1_3", "diagnosis_2_4" and
"diagnosis_3_4" whose Certainties are respectively 80,1,6.
The Inductive Inference from the premises of Fig. 5 leads to
the conclusion that "diagnosis_1_3" is by far the most
certain. The Certainty of choosing "diagnosis_1_3" as one
of ("oof") the three is evaluated in their
Aggregate "diagnostics" as 74.
Certainty of the Axiom "diagnostics" (74) represents
acceptable inductive verification of the theory founded in
it, embodied by the deduced CN structure, and of the
evaluation of three mutually exclusive ("oof" Operator)
"diagnosis_...", suggesting the choice of "diagnosis_1_3".
Determination of the rather complex "oof" algorithm is
discussed in Appendix.
Fig. 8. Flat Explosions of the three "diagnosis".
1 diagnosis_1_3 80 thrm
2 syndrome_1_acf and 93 thrm
2 syndrome_3_bgh and 89 thrm
1 diagnosis_2_4 1 thrm
2 syndrome_2_bcd and 18 thrm
2 syndrome_4_aeg and 12 thrm
1 diagnosis_3_4 6 thrm
2 syndrome_3_bgh and 89 thrm
2 syndrome_4_aeg and 12 thrm
We note that and(93,89) = 80; and(18,12) = 1; and(89,12) = 6;
Details of "and" algorithm is discussed in Appendix.
Fig. 9. Flat Explosions of "syndromes".
1 syndrome_1_acf 93 thrm
2 symptom_a and 98 fact
2 symptom_c and 96 fact
2 symptom_f and 97 fact
1 syndrome_2_bcd 18 thrm
2 symptom_b and 95 fact
2 symptom_c and 96 fact
2 symptom_d and 25 fact
1 syndrome_3_bgh 89 thrm
2 symptom_b and 95 fact
2 symptom_g and 98 fact
2 symptom_h and 94 fact
1 syndrome_4_aeg 12 thrm
2 symptom_a and 98 fact
2 symptom_g and 98 fact
2 symptom_e and 18 fact
Explosion structures show for an Aggregate the Parts it
implies, either directly (Flat Explosion) or recursively,
till the bottom of structure (Deep Explosion).
One may be on the other hand interested for a Part by which
Aggregates it's implied (to which inductive Conclusions it
contributes), directly (Flat Implosion), or recursively till
the top of structure (Deep Implosion). Fig. 10-11. show Flat
and deep Implosion of "symptom_a".
Fig. 10. Flat Implosion of "symptom_a".
1 symptom_a 98 fact
2 syndrome_1_acf and 93 thrm
2 syndrome_4_aeg and 12 thrm
Fig. 11. Deep Implosion of "symptom_a".
1 symptom_a 98 fact
2 syndrome_1_acf and 93 thrm
3 diagnosis_1_3 and 80 thrm
4 diagnostics oof 74 axiom
2 syndrome_4_aeg and 12 thrm
3 diagnosis_2_4 and 1 thrm
4 diagnostics oof 74 axiom
3 diagnosis_3_4 and 6 thrm
4 diagnostics oof 74 axiom
D.Epistemological Conclusions
Epistemological impact of Relativistic Dialectic and its
Logic, the Cognitive Network, concerns mainly
-foundations,
-definitions and distinction of "Theory" and "Model",
-definitions and distinction of "Axiom" and "Dogma".
Foundations.
We have postulated that Logical Systems may be evaluated and
justified exclusively by their capacity to simulate Mind's
intrinsic, ER based Logic. CN is the first Logical System
founded in Mind's intrinsic Logic, rather than in naive
linguistic expressions. It seems to simulate it efficiently,
as has been verified by its several practical applications.
Theory and Model.
Contemporary Epistemology sees falsifiability as a necessary
quality of scientific structures. CN embodies it rigorously
in its two consecutive phases:
1.Conceptual, deductive Theory,
2.Experimental, inductively falsifiable Model.
Axiom and Dogma.
Full-fledged structure encompassing both, Theory and Model
will be" called "axiomatic" and its top arbitrary presumption
- an "Axiom". Axiom and thence deduced Theory are falsifiable
and refutable by inconclusive experiments of the Model phase.
A Theory lacking bottom factual Theorems and thus unable to
pass to the falsifiable Model phase will be called "Dogmatic"
and its top arbitrary presumption - a "Dogma". Unlike Axiom,
a Dogma is not falsifiable, cannot be refuted and reposes in
naive unshakable faith in aprioristic "Truth".
Appendix. Fuzzy Operators.
N Dimensions
As can be seen in ND EXACT PROPOSITIONAL CALCULUS
For N dimensions the Number of operators (2^(2^N) increases
very fast with N. For N=2 we had 16 operators which may be
learnt by heart, like the multiplication table, so that with
a bit of practice one can execute and program all operations
of the 2D Calculus from memory. However, For N=4 we have
2^(2^4)=65536 and for n=5 2^(2^5)=2^32=4294967296 operators.
And 5 is small for practical applications. We may have 20
symptoms of a disease or 100 "symptoms" of some breakdown in
a spacecraft. The respective diagnostic systems would extend
over 2^(2^20) and 2^(2^100) operators. A bit to much to
learn by heart, to describe in a textbook, or, for that
matter, in the whole Congress Library.
It's clear that for higher N's only a few operators can be
chosen from endless lists in function of their utility for
a particular problem. The user has to tailor his logic to
his problem by choosing pertinent operators and designing
their evaluation algorithms.
OOF (One Of) Operator
Some Operators like "OR", or "AND" map from 2D to ND as one
to one, but for instance the 2D Operator ORR ("exclusive or",
"either-or") forks for ND to N distinct operators from
"One Of" to "(N-1) Of" and "Not All".
(see ND EXACT PROPOSITIONAL CALCULUS )
For the Diagnostics application we have retained only the
"OOF" (One Of), as only one diagnosis may be chosen as base
of subsequent therapy.
CN offers to the expert the possibility of customizing the
fuzzy Operators in function of application and expert's
experience. For the Diagnostics application we established
the OOF algorithm as follows:
Meaningful cases encompass any number of Operands
N greater than 1.
The values (in %) of concerned Operands are split into "MAX"
(the maximum value, or first of equal greatest values in
Operands' vector) and the rest.
SIG: sum of all N Operands.
The average of all but MAX: NOMAX = (SIG - MAX) / (N-1)
and
OOF = (MAX * (100-NOMAX) / 100
For the ideal distribution (MAX=100,NOMAX=0) OOF = 100.
With decreasing MAX and increasing NOMAX OOF decreases.
AND Operator
MIN: smallest value or first of equal smallest values in
Operands' vector.
MED: Average of all concerned Operands.
AND = (MIN * MED) / 100
The apparently simple AND Operator has raised more
discussions than any other one. The first approach is to
treat it with the probability Multiplication Rule. For two
rather certain Operands of 90% it seems reasonable to say
that AND(90,90) = 81. Yet, the experts argued that if the
progres of science discovers other 5 symptoms all confirmed
in our instance at 90%, it should make the syndrome more
certain, or at least equal and not disqualify it at 47% as
the Multiplication Rule would do. Finally they accepted the
above algorithm which maintains the syndrome at 81% for any
number of 90% symptoms.
