Reminder: The thread "Second Enlightenment" (SE) is destined to discuss the rationality of SE as well as to inquire into the sources of the irrational manipulation of masses and to look for remediation.

X1. Scientific Revolution X2. Ontology X3. Ideology X4. Social awareness X5. Establishment

with X=F/S respectively for the first/second enlightenment. We start by the first enlightenment as guidance to the formulation of the second and warning of errors to be avoided.


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The present section deals with the steps S1. Scientific Revolution and S2. Ontology of the second enlightenment.

NOTE: "SPACE" designates the abstract, mathematical construct such as Riemann SPACE, in order to distinguish it from the phenomenal "space" of perception.

THE SCIENTIFIC REVOLUTION

The Paradise

19th Century Physics and Logic were dominated by reaction of Dogmatism against Rationality of the First Enlightenment. Physics was founded in the dogmatic delusion of Aether. Logical systems culminated in Predicate Logic (PL). This noumenal, naive Logic was assumed to be the universal, absolute foundation of transcendental "Reality" and of human knowledge thereof. All areas of Science were assumed to be recursively founded in PL, although Mathematics stayed for some time unfounded. Finally, in 1873 Cantor conceived the naive Set Theory which supplied the missing link and completed the hierarchy of "Cantorian Paradise":

1.(naive) Logic, the universal, absolute, olympian foundation,
2.(naive) Set Theory, founded in (naive) Logic,
3. Mathematics, founded in (naive) Set Theory,
4. Physics, founded in Mathematics and in the mechanistic Dogma, itself founded in (naive) Logic.
5. Other "hard" Sciences, founded in Physics,
6."Soft" Sciences, founded in "hard" Sciences.

This "Paradise" represented the edifice of Science solid and stable as Cheops pyramid, bestowing upon scientists the gratifying conviction that Logic, THEIR Logic, explains and determines the "Real Universe".

Lost Paradise

At the end of the 19th century two breaches appeared in the pyramid setting off the Fin-de-Siecle Crisis of Thought which triggered the Second Scientific Revolution:

1. In Physics the Michelson-Morley experiment showing that the speed of light is invariant with respect to the speed of source and observer and thus falsifying the mechanistic, Dogma.
2. Russell's Paradoxes calling in question the foundations and sense of naive PL and founded in it naive Set Theory.

Question arose, if apparently local breaches in otherwise solid pyramid may be patched locally, or should lead to global reconsideration of the whole structure.

In Physics Einstein has chosen the global approach, revised entire Physics and laid the cornerstone of the Second Enlightenment with its new Rationality.

Logicians did not notice that Russell's Paradoxes did not wreck Logic in general, but only the naive PL and try in vain, till our days, to patch it rather than to scrap the wreck and to construct in its place new rational Logic.

East of Eden

Expelled from Paradise, deprived of reassuring certainties of the divine, olympian "Reality" of Naive View, Science was compelled to count on itself and to engage entirely in the way of "Sapere Aude". As first step it divorced with Naive View and its absolute, noumenal Time and Space. Indeed, invariant C of Electro-Magnetic Field and its implications were totally at odds with Naive View's concepts and representations:

-C invariance itself: Light passing an observer at C whether he moves at 0.99 C in its direction or against it.

-SPACE ceased to be a "container" of Phenomena, but became itself a Phenomenon, not only directly supporting Field and its propagation, but P(henomenally)-Equivalent with Field, both becoming inseparable Aspects or manifestations of Cosmos.

-Time and Space components of SPACE cease to be absolute and invariable, but respectively dilate and contract in a relatively moving Referential in function of its speed.

-Non-Euclidean metric of SPACE whose local curvature is associated with Field density, however neither as cause, nor as effect, but as P-Equivalence.

The demise of Naive View may be illustrated by comparing the propagation of sound and light. An observer moving at V with respect to the source of sound will find himself displaced with respect to the surface of sound's sphere extending in direction of V at sound speed S by S-V. At V=S he will find himself at the surface and at Vs S he will "overtake" the sound and find himself supersonic, outside of the sphere, preceding its advance. The relative speed of sound depends of the speed of observer.

Now, light propagates at C independently of observer's speed so that he will find himself always in the middle of the light sphere, no matter how far he travelled from the source. On the other hand he will find time flowing slower with respect to the sphere than seen from the source. Thus, he will find himself always in the center of the light sphere, which will get smaller in function of his relative speed. At the limit of V=C, time at source will stop flowing for the moving observer and his sphere will contract to a point Field singularity. This basic construction of Extended Relativity is of course totally at odds with the Naive View and "common sens".

ONTOLOGY

Introduction

In Kant's wake, the ontology of the Second Enlightenment (SE) intends to be axiomatic and derived from the second scientific revolution, to wit from the Extended Relativity.

Derivation of ontology from physics consists in identifying underlying it ontological principles of the "physical reality". Axiomatic ontology, like any axiomatic model, has to guarantee the falsifiability of its axioms. This request if automatically satisfied for axioms underlying the Extended Relativity by reduction to the latter. However, ontology is more general than foundation of physics and has to found additionally a rational, axiomatic logic. We shal briefly outline these two aspects in "Physical Reality" and "Logic".

Physical Reality

SE ontology extends over the total human reality. The term "Physical Reality" stresses its derivation from physics and its compatibility with its first outline - Einstein's "Physics and Reality".

We shall present in this section only the basic axioms and definitions without detailed justification, which may be found in links time awareness and events structures of mind natural model of http://findgeorges.com/

Time and clocks

Introspection tells me that I intuit time as continuous, uninterrupted setting of foreground events, whether percepts, emotions, images or intellections, as their perpetual and universal, but unattainable and inscrutable background. Foreground events discretize the continuous background flow of time. Particular, periodically recurring events may function as clocks, the frequency of their occurrences or "ticks" indicating the speed of time flow.

Dichotomy Continuum/Discreteness (CD)

Time and clocks appear as Aspects of the CD Dichotomy . Clocks integrate time into discrete periods, bounded by the ticks and conserving in between the continuous fabric. We shall call the continuous time aspect "Intuitive Time" and the discrete - "Eventtual Time".

Basic fuzziness of the reality

Values of time are necessarily affected by fuzziness determined by tick intervals of involved clocks. As time is the universal background of all events, its fuzziness extends over the totality of human reality.

The continuous, universal, but inscrutable intuitive time, which sets discrete events appears indissociable from the intuition of the thematic awareness:

FUNDAMENTAL EQUIVALENCE PRINCIPLE:

INTUITIVE(CONTINUOUS) ASPECT OF TIME IS EQUIVALENT WITH AWARENESS.

intuition stays unaffected by scientific, philosophical and other models, so that considering awareness or time as seen by different models we really examine their views of events and clocks. Considering awareness or time "as such", else then via their eventtual complements would be an empty speculation.

All events, all attainable eventtual elements of human Universe point to the unique background, the intuitive awareness and time:

POSTULATE OF RELATIVITY

ALL ELEMENTS OF HUMAN UNIVERSE ARE RELATIVE, EXCEPTING AWARENESS, ITS ABSOLUTE FOUNDATION.

And the corollary:

THE DICHOTOMY CONTINUUM/DISCRETENESS IS THE BASIC STRUCTURE OF ALL ELEMENTS OF HUMAN REALITY WITH THE FOUNDATIONAL PREPONDERANCE OF THE CONTINUOUS ASPECT.

MIND

PERCEPTION

Focusing on the apple I perceive it steady and fixed in its
3D sphericity and other sense impressions, but when I center on the garden behind, strange things happen. The compact spheroid splits into two flat disks hazy and transparent, letting the garden to be seen through with clarity. Is the apple still there, or it decayed into two blurred shadows? When I focus on one of them, the other jumps to join it and there it is back again, the green, opaque, spherical apple, mysteriously knocked together from two blurred shadows. But what happened to the garden when I focused on the apple? All trees and bushes behind the green, opaque spheroid split into couples of blurred shadows, some closer-, some farther apart, muddled up in confused, undistinguishable mess. So, the garden, our world out there, would also consist of flat blurs, every now and then covering one another and making snapshots of opaque 3D objects, scrambling the rest to chaotic mess?

Making a mountain out of a molehill - one might say - Simple stereoscopy. Two slightly different views of the apple on each retina fuse in the brain to the 3D object. Agreed, but then the apple I perceive would be a construct of my brain, not the apple out there. Projection, representation of it, perhaps, but construct of my brain anyway. But even that view - though compatible with superficial ophthalmology - would not satisfy a neurologist, who would say: - All I see in brain and retinas are EM waves, which are deemed covariant, but certainly not identical with certain sense impressions. And we clearly experience sense impressions, not waves, so we necessarily need some sort of mind to support their covariance with waves.- So, finally, the sense impressions I perceive and the apple they make up are constructs of my mind.

IMAGERY

Mind's faculty supporting perception will be called "Imagery" to keep it compatible with another essential function - Imagination. Elements of imagery will be called Images, of which percepts are a subclass.

"Image" is a strict synonym of "Event" as defined in "TIME, AWARENESS AND EVENTS". Due to common usages, it's easier to talk about events in the most general context of ontology and about images in the much more specific one, that of mind and its structures.

SENSORIUM

Images are brought about amongst others by sensations. We have seen that a tactile sensation induces fully fledged green, spheroid, compact image of apple. Sensations' output provides input of Imagery setting off perceptual images as maps of a hypothetical territory assumed as input of sensations, but unattainable and incognizable as such, which we shall call "Transcendency".

On the other hand, I may chose to pick the apple and this event of volition will output into kinesthetic sensations, closing the perceptual-motor coupling.

Calling "Sensorium" the mind's faculty handling sensations, we may consider Imagery as a map of a hypothetical, unattainable, transcendental territory, projected by Sensorium.

SYMBOLISM

We have seen that events function as clocks integrating the fundamental continuity of time/awareness into periods bounded by their ticks and conserving in between the continuous fabric. It holds of course for images, synonymous with events and due to the universality of the CD dichotomy we naturally seek a discrete counterpart of the intrinsic continuity of images. We find it in Symbols. Images may be mapped to symbols which act psychologically as mnemonic pointers and physically as fixed point attractors.

We shall call "Symbolism" the mind's faculty handling abstract symbols and consider it as the discrete complement of continuous images.

Images are seldom reduced to single percepts, like the apple of our example. Usually they represent compound patterns or "situations" in the sense Sartre gave this term, which are mapped to Symbolism as complex abstract structures e. g. linguistic, logical and mathematical expressions.

We have seen that sensations provide perceptual input to Imagery, which is intuited as "real". Concepts represent another input source of "conceptual" images.

RECOLLECTION ET (RE)COGNITION

Images are memorized. Previously memorized ones may be recalled to Awareness. A newly perceived Image may be recognized as homomorphic with a recalled one, in which case it will be considered as known.

Recognition underlies Cognition: an Image is known when it's recognized.

REFLECTION AND MEANING

"Reflection" will denote the faculty of reciprocal transfers between Imagery and Symbolism, involving:
1."Symbolizing" or Mapping Images into symbolic or abstract structures.
2."Understanding", or regressing symbolic Structures to Images, which embody their "Meaning".

ABSTRACTIONS POSTULATE

In the light of above defined reflection and meaning we may specify the Abstractions Postulate (AP) which underlies science and epistemology since the Second Scientific Revolution of the Extended Relativity:

** Abstract, symbolic constructs may be justified solely by their capacity to coordinate events which represent their unique meaning and justification, where coordination of events implies considering them in their context, i. e. upon their background of continuum.
**

PHYSICAL BODIES

We believe that the first step in the setting of the physical reality is the formation of images of "objects", or, to comply with Einstein's terms - "physical bodies". Out of the multitude of his event-images one takes, mentally and arbitrarily, certain repeatedly occurring patterns of sensations, and collects them into a secondary image under a particular symbolic name. One attributes to this name its originating secondary image as meaning, thus constructing a meaningful concept of a particular "body".

It's crucial to emphasize that one intuits such secondary constructs as unshakably certain and "real", unlike their originating percepts affected by doubt and uncertainty and that finally, one conceives the "physical world" as populated with them.

MYSELF, SIGNS AND OTHERS

Associating a body with "reflexive" events stemming from somatic and kinesthetic sensations, I construct the particularly important "(my)own body" symbolized originally by some primitive name, which, with the apprenticeship of natural languages becomes "I", "me" or "myself".

In the wake of kinesthetic sensations I perceive particular bodies which get associated with "myself" as its "signs", "expressions" or "manifestations".

Out of the multitude of my event-images I take some, which, while lacking somatic and kinesthetic sensations, are otherwise homomorphic with my "own body" and similarly associated with signs. I collect them under "others" and considers their signs as expressions of their inaccessible but alleged sensations, images and symbols.

Besides direct sensations, signs provide additional input of the mentally constructed "reality", enlarging my own experience, by that of the others.

SPACE

There are "rigid" bodies to which one can ascribe, within a certain sphere of perception, no alteration of state, but only alterations of position

Mind's faculty of putting every body situated in any arbitrary manner into contact with the quasi rigid continuation of a chosen body of relation B0 is the basis of our intuition of space. In pre-scientific thinking, the solid earth's crust plays the role of B0. The very name geometry indicates that the idea of space is mentally connected with the earth considered as the relation body.

We shall find the intuition of space founded by rigid bodies at the base of Einstein's Covering Principle which requires physical distance to be measured, also in mental experiments, with physical rods complying with physical rules, such as the Lorentz Contraction. The Covering Principle underlies directly or indirectly the entire Extended Relativity. For the first time in the history of science an ontological principle is directly incorporated in mathematical formulation of physical theorems.

In mathematical terms geometry symbolizes the basic intuitive image of continuous space and arithmetic - its discrete covering measurements. The ultimate foundation of mathematics is the continuum. Discrete concepts starting with that of "number" are founded in continuum and symbolize its covering measurements. Shortly, arithmetic is founded in geometry.

SPACE-TIME and GENERALIZED COVERING PRINCIPLE

We have introduced above the eventtual time as indications or ticks of a "clock," i. e. as recurrences of a periodical event.

Clocks disretize intuitive time as "physical rods" discretize intuitive space, which allows to formulate the Generalized Covering Principle:

EVENTS DISCRETIZE INTUITIVE TIME AND SPACE INTO EVENTTUAL PERIODS AND DISTANCES.

Associating each point of the intuitive space with a clock we construct intuitive Space-Time continuum accessible via rod/clock discretized eventtual bulks. We shall call this continuum "SPACE", capitalized in order to distinguish it from its "space" components.

INTUITIVE CAUSALITY AND LOGIC

Similarly to constructing "bodies", one selects some pattern of event-images apparently always followed by another one and under names respectively "cause"/"effect" collects them into the secondary image under the symbolic name "causality". Like in the case of "physical body", one attributes to this secondary construct "causality" the unshakable "physical reality" and certainty, unlike to its "cause/effect" instances. And, having constructed the "physical world" by populating it with "physical bodies", one orders it with the principal orderer "causality". A particular type of Reflection, which we shall call "Inference", maps events ordered by causality into symbolic structures of expressions related "deductively" by Implication, shortly "ER structures". Inverse operation regresses "inductively" expressions to their territory of events.

Due to intuiting causality as unshakably "real", deduction appears as "necessary" or "certain".

Induction, on the contrary, retrieving the originating events of symbolic expressions, gets affected by their uncertainty and fuzziness (see "FUZZINESS" below).

Common sense taking the illusory "reality" at its face value, considers the intuitive causality/logic as an evident rule of practical behavior, applying it instinctively to the day-to-day practice. One knows intuitively that hitting a nail with a hammer will drive it into the wall and that standing in the way of the oncoming bus will cause not a little misery. In the social praxis people tending to stand in the way of oncoming buses are isolated in lunatic asylums with others who cast similar doubts on causality and logic.

However, extended over intellection, the common sense unveils the prejudices it has gathered through the daily practice, and rigorous rationality starts by overcoming it. The rigorous view of causality presented in the chapter "CAUSALITY AND IMPLICATION" of http://findgeorges.com/ exceeds the scope of the present essay.

The natural faculty of inferring, also called "Logical Reflection" or "Logic" is rather inefficient, due to mind's restricted capacity of simultaneously recalling numerous expressions and executing numerous operations. Humans attempt to enhance their limited natural capacities by creating tools. Thus hammer enhances the striking capacity of human hand and extrinsic logic enhances the limited capacity of human intrinsic, natural inference. Being an enhancement of the natural faculty, extrinsic logical systems may only be justified by extending and simulating the ER structures of the natural Logic, by accounting for the illimited number of logical operators as well as for the fuzziness of the induction. Extrinsic purely deductive logical systems will be called "Theories" and complete, deductive-inductive ones - "Models" Extrinsic ERN (Expression-Relation-Network) Logic rigorously extends the ER structures of the natural Logic. The concise description of the ERN Logic is presented in POSTSCRIPT.

ESTHETICS AND ETHICS

Some "reflexive" events underlying the construction of "(my)own body" or "myself" are perceived as (dis)satisfactory. They may be arranged in esthetic ER structures similar to the logical ones with fuzzy "satisfaction" replacing "plausibility". Axioms represent "taste", intermediary theorems symbolize reflexive events and bottom ones - the current factual percepts. Current percepts verify/falsify the taste estimations. Falsification calls the posited taste in question claiming its adjustment and refinement.

Ethics is founded in Esthetics. The difference consists in Ethics considering others, rather than myself. Ethical ER structures are populated not with reflexive "my own" events, but with posited others' events assumed as homomorphic with my own. Metalanguage term "taste" may be replaced with "code" and "satisfaction" with "duty".

It does not mean, of course, that each time I exceed the speed limit I have to run the ERN procedure. However, for establishing the highway code, let alone a country's constitution it's the only rational way to proceed.

TRANSCENDENCY AND IMMANENCY

We saw above that "reality" defined as all what's experiential, is entirely mental and immanent, encompassed by sensations and by percepts of Imagery.

However, Sensorium maps Imagery as a map of some hypothetical territory - "Transcendency". And, Imagery appearing as an ordered reality, temptation arises to regress it to its transcendental territory and to posit thus called forth transcendency as the "real world out there". Yet, no matter how tenacious and persistent this temptation may be, the transcendental "reality" of regressed mental constructs is nothing, but a "transcendental illusion".

We saw above that one attributes to the secondary constructs, viz. "physical bodies" a more unshakable "reality" than to the percepts which gave rise to them and that finally, one constructs the "physical world" by populating it with them.

We are now in position to precise it further:

One may give in to the temptation and fall prey to the transcendental illusion by regressing mental secondary constructs - the "physical bodies" to "real objects" of transcendency. To use the fashionable philosophical term, the transcendental illusion is tantamount to reification.

Once reified, the transcendental "real object" maintains the illusory speculative reality independently of its originating percepts and sensations, as a "container" which may, but does not have to contain them i. e. functions as autonomous "thing in itself" aka "noumenon".

Contrariwise, the bodies of imagery are indissociable from their originating percepts and sensations, i. e. function as phenomena.

We may conclude: IMMANENCY IF PHENOMENAL AND TRANSCENDENCY - NOUMENAL.

The transcendental illusion applies also to logic. The transcendental "world out there" being populated with noumenal "real objects", their interrelations and associations with "contained" sensations fall into the province of the reified, noumenal, absolute logic.

While the ER structures of the rational, immanent logic represent the relative causal order of events, the transcendental, irrational pseudo-logic posits absolute structures of transcendental phantasmagoric "real objects" related by the illusory, "deterministic causality".

FUZZINESS

All imagery constructs are affected by fuzziness stemming from two principal sources:

1. Time approximation by discrete tick intervals of involved clocks ("TIME, AWARENESS AND EVENTS").

2. Arbitrariness of bodies constructed by collecting similar patterns of sensations. It affects, of course, the bodies themselves, via bodies - the space, and, together with time - the space-time.

The general, ontological fuzziness of the physical reality should not be confused with particular uncertainty cases stemming from detectors affecting the detected, from statistic procedures, or from singularity areas of continua.

EXISTENCE

"Existence" is one of the most muddled, multivalued and meaningless terms. It usually refers to phantasmagoric "objects" of the transcendental World Out There. We shall apply it exclusively to empirically verified events, i. e. to such events, whose symbolic representations get high plausibility within explicit or implicit ERN logical procedures. Due to the general fuzziness of the physical reality we refute absolute existence and restrict it to more or less plausible fuzzy events.

POSTSCRIPT 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.

Diagnostics, 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 diagnostics, 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 "diagnostics" theory "diagnostics" recursively implies "sickness", "syndrome" and "symptom", which may be noted:

"diagnostics" imp("sickness" imp("syndrome" imp("symptom")))

or in the indented format:

1 diagnostics
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 diagnostics", i. e. the top-down recursive enumeration of its parts. ("Flat Explosion" displays just one level of parts.)

Fig. 1. Deep Explosion of "diagnostics".

1 diagnostics 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 diagnostics" 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 diagnostics = 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 diagnostics 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 "diagnostics".

1 diagnostics 77 axiom
2 sickness_1_3 oof 80 thrm
2 sickness_2_4 oof 1 thrm
2 sickness_3_4 oof 6 thrm

"diagnostics" 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 "diagnostics" as 77.

Plausibility of the Axiom "diagnostics" (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 diagnostics oof 77 axiom
2 syndrome_4_aeg and 12 thrm
3 sickness_2_4 and 1 thrm
4 diagnostics oof 77 axiom
3 sickness_3_4 and 6 thrm
4 diagnostics 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 Diagnostics 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 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 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.