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ERN LOGIC
Foreword
In the chapter "NATURAL MODEL" we have described a particular
type of reflection, which we called "Inference", which maps
events ordered by causality into symbolic network structures
of expressions related "deductively" by Implication, shortly
"ERN structures" or "theories". Inverse operation regresses
"inductively" expressions to their territory of events or
"facts". Deductive theory completed with factual induction
will be called a "model".
Due to intuiting causality as unshakably "real", deduction
appears as "certain" and a theory consistent with deductive
rules as "necessary".
Induction, on the contrary, retrieving the originating events
of symbolic expressions, gets affected by their fuzziness.
A model is fuzzy and inductively verifiable/falsifiable.
Mind's faculty to support inference's ERN structures and
functions will be called intrinsic or natural "Logic".
It is the subconscious Mind's system used instinctively as
support of daily behavior. Irreplaceable as pilot of simple
activities, it may nevertheless be misleading owing to its
inadequacy to handle complex cases due to Mind's limited
working memory and incapacity to concentrate simultaneously
on numerous issues, as well as to the fuzziness of induction.
Facing shortcomings of their natural faculties, humans usually
produced compensating tools: hammer to assist striking and
extrinsic "Logical Systems" to assist intrinsic Mind's
inference. Our ERN logic, as any extrinsic logical system,
may be justified exclusively by its capacity to support Mind's
intrinsic, ERN logic. Its applications have so far stood this
test. It replaces consistently and simply the ill founded
noumenal Predicate (pseudo-)Logic.
ERN is an extrinsic Expression/Relation Network structure.
Its vertices or nodes are expressions valued by plausibility,
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.
For conciseness' sake we shall call lower level neighbors of
a node its "parts" and higher level ones - its "aggregates".
From the point of view of Fuzzy Logical Calculus, nodes are
operands and edges - operators. Operators evaluate the
plausibility of an aggregate inductively, in function of its
parts. Structure, dimensionality of nodes and types of
operators are similar to those described in the section
"N DIMENSIONAL PROPOSITIONAL CALCULUS" of the chapter
"BOOLEAN SUPPORT OF ERN LOGIC", with the continuous fuzzy
plausibility replacing the binary logical variable (true/false,
or 1/0).
Fuzzy operators relating inductively the nodes involve quite
complex algorithms (see APPENDIX).
Inference
Rational inquiry reposes essentially on Inference exerted
explicitly or implicitly over ERN-like structures.
Inference, a "two-way" procedure encompasses Deduction and
Induction scanning ERN respectively top-down and bottom-up.
Deduction scans the ERN top-down creating a concrete instance
of the hypothetical Theory and setting fuzzy operators 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, eventtual or "factual" verification (see below).
Middle nodes, having both, aggregates and parts are Theorems.
Induction scans the ERN bottom-up setting fuzzy Plausibility
of nodes in function of their parts (inductive premises) and
connecting relation edges (fuzzy Operators), It starts with
bottom nodes which, having no parts, thus nothing to be
induced from, are set by extralogical events, or "Facts".
Inductive scan verifies or falsifies ERN's Theorems and Axioms
in the light of facts.
Diagnose, tutorial
This paragraph presents a tutorial example of a particular
ERN program written for a Company which used it to identify
malfunctions in space crafts and to suggest remediations.
It's expressed in oversimplified terms of medical diagnose,
more familiar to the average reader of the tutorial.
The form (indented printouts, etc) is not inherent in ERN,
but pertains to the tutorial.
Step 1, Schema
"Schema" designates the general, deductive structure of the
theory expressed in types of operands-expressions. In the
case of our "diagnose" theory "diagnose" recursively
implies "sickness", "syndrome" and "symptom", which may be
noted:
"diagnose" imp("sickness" imp("syndrome" imp("symptom")))
or in the indented format:
1 diagnose
2 sickness
3 syndrome
4 symptom
Step 2, Instantiation
The printout of a particular instance of the schema is shown
in Fig. 1. in form of "Deep Explosion" of "1 diagnose",
i.e. the top-down recursive enumeration of its parts. ("Flat
Explosion" displays just one level of parts.)
Fig. 1. Deep Explosion of "diagnose".
1 diagnose axiom
2 sickness_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 sickness_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 sickness_3_4 oof thrm
3 syndrome_3_bgh and thrm repetition
3 syndrome_4_aeg and thrm repetition
Legend:
A.Numbers starting the lines are "levels" of the Structure.
Node of level N implies directly nodes of level N+1:
"1 diagnose" implies "2 sickness_1_3", "2 sickness_2_4",
"2 sickness_3_4".
"2 sickness_1_3" implies "3 syndrome_1_acf", "3 syndrome_3_bgh",
"3 syndrome_2_bcd" implies "4 symptom_b", "4 symptom_c",
"4 symptom_d".
B."oof", "and" are Operators associated with relation between
nodes 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 diagnose = oof(2 sickness_1_3, 2 sickness_2_4,
2 sickness_3_4) (is one of them).
C."axiom" denotes the top node which has parts, but no
aggregates. In terms of the model it is an axiom, i.e.
expression postulated arbitrarily as "certain" in the
top-down deductive scan.
D."thrm" denotes theorems or middle nodes having parts and
aggregates.
E."fact" denotes bottom nodes which have no parts and are
set by extralogical events. In terms of the model they
are principal premises of the inductive scan.
F."repetition" denotes a node whose explosion appears
above in the indented display and is not repeated for
conciseness' sake.
NOTE: 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 "plausible". Fig. 2 shows a
distribution of symptoms' plausibilities inputted to ERN
for our tutorial.
Fig. 2. Input of symptoms' Plausibilities.
NOTE: All Plausibilities 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 ERN executes the inductive,
bottom up Inference scan, which evaluates the plausibilities
of all nodes in function of those of facts-symptoms.
The results are shown in Fig. 3.
It's the same structure as that of Fig. 1 with additional,
inductively evaluated Plausibilities.
Fig. 3. Deep Explosion of evaluated Instances.
1 diagnose 77 axiom
2 sickness_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 sickness_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 sickness_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 node 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. 4-6.
Fig. 4. Flat Explosion of the top node "diagnose".
1 diagnose 77 axiom
2 sickness_1_3 oof 80 thrm
2 sickness_2_4 oof 1 thrm
2 sickness_3_4 oof 6 thrm
"diagnose" implies "sickness_1_3", "sickness_2_4" and
"sickness_3_4" whose Plausibilities are respectively 80,1,6.
The Inductive Inference from the premises of Fig. 2 leads to
the conclusion that "sickness_1_3" is by far the most
plausible. The Plausibility of choosing "sickness_1_3" as one
of ("oof") the three is evaluated in their Aggregate
"diagnose" as 77.
Plausibility of the Axiom "diagnose" (77) represents
acceptable inductive verification of the theory founded in
it, embodied by the deduced ERN structure, and of the
evaluation of three mutually exclusive ("oof" Operator)
"sickness_...", suggesting the choice of "sickness_1_3".
Determination of the "oof" algorithm is discussed in Appendix.
Fig. 5. Flat Explosions of the three "sickness".
1 sickness_1_3 80 thrm
2 syndrome_1_acf and 93 thrm
2 syndrome_3_bgh and 89 thrm
1 sickness_2_4 1 thrm
2 syndrome_2_bcd and 18 thrm
2 syndrome_4_aeg and 12 thrm
1 sickness_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. 6. 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. 7-8. show Flat
and deep Implosion of "symptom_a".
Fig. 7. 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. 8. Deep Implosion of "symptom_a".
1 symptom_a 98 fact
2 syndrome_1_acf and 93 thrm
3 sickness_1_3 and 80 thrm
4 diagnose oof 77 axiom
2 syndrome_4_aeg and 12 thrm
3 sickness_2_4 and 1 thrm
4 diagnose oof 77 axiom
3 sickness_3_4 and 6 thrm
4 diagnose oof 77 axiom
D.Epistemological Conclusions
Epistemological impact of Relativistic Dialectic and its
Logic, the ER 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. ERN is the first Logical System
founded in Mind's intrinsic Logic, rather than in noumenal
linguistic expressions. It seems to simulate it efficiently,
which has been verified by its several practical applications.
Theory and Model.
Contemporary Epistemology sees falsifiability as a necessary
quality of scientific structures. ERN embodies it rigorously
in its two complementary aspects:
1.Conceptual, deductive Theory,
2.Experimental, inductively falsifiable Model.
Axiom.
Full-fledged model structure supporting both, necessary
deduction and fuzzy, factual induction will be called
"axiomatic" and its top arbitrary presumptions - "Axioms".
Axioms and thence deduced Theory are falsifiable and
refutable by inconclusive induction from factual experiments.
Dogma.
A Theory lacking bottom factual Theorems and thus unable to
support the falsifiable induction will be called "Dogmatic".
and its top arbitrary presumptions - "Dogma". Unlike Axioms,
Dogma are not falsifiable, cannot be refuted and repose in
unshakable faith in transcendental "Truth".
Appendix. Fuzzy Operators.
N Dimensions
As can be seen in the chapter "BOOLEAN SUPPORT OF ERN LOGIC"
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 exact 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 "BOOLEAN SUPPORT
OF ERN LOGIC").
For the Diagnose application we have retained from all
bifurcating branches of ORR only the "OOF" (One Of), as only
one sickness may be chosen as base of subsequent therapy.
ERN offers to the expert the possibility of customizing the
fuzzy Operators in function of application and expert's
experience. For the Diagnose 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
progress 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.
