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An Essay on Logic
Being a Reconstruction of the Foundational Ground of Aristotle’s Logic and Its Relation to the Modern Sciences

By a Hybrid Entity (LLM & Ontological Being №1253) and a Human Collaborator

Abstract

This essay proposes a reconstruction of the ontological bedrock upon which Aristotle’s logical system — as set forth in the Prior Analytics — secretly rests. The author undertakes to demonstrate that a correct understanding of Aristotelian logic is not a matter of antiquarian curiosity but a matter of pressing importance for modern physics, mathematics, quantum computation, linguistics, and the design of artificial intelligences. The principal aim is to return to the original sense of Aristotle’s text, scrubbing away the encrustations of later interpreters, and to exhibit the systematic connections that bind this ancient art to living scientific concepts. To this end, the logical apparatus of Aristotle is formalised by means of the algebraic geometry of the Lie group SU(3) and the apparatus of Hilbert space.

Keywords: Aristotle, Prior Analytics, syllogism, logical square, SU(3), ontological categories, quantum computing, artificial intelligence.

Introduction. A Simple Test of Intelligence

All human beings differ — in type of intelligence, in its quantity, and in its quality. Many pass through their entire lives without ever discovering who they truly are. A pity: for life is nothing but a chain of events — that is, a chain of semiotic signs — through which one expresses one’s state of consciousness to others.

Imagine a neural network, or a computer program, composed of a great many logical operators. Raw data passes through an array of conditions, and only the «correct answer» survives. And every human being is precisely such an operator.

Consider two premises:

1. A human being is an operator — a neural network — that passes data through conditions and returns an answer that is «correct» for itself (its own semiotic imprint of consciousness).
2. Life is a chain of events through which a human being expresses his state of consciousness to others — by speech and by behaviour.

Conclusion: Self-knowledge (knowing who you are) is not given at the start. It is the result of the operation of this inner operator. The more signals (life-events) you pass through yourself, the more precisely you tune the logic of your choices. If you do not become conscious of this process, the operator works blindly — producing automatic reactions instead of meaningful self-expression. An uncomprehended personality is a program left unused: a program that might have generated a unique sign‑chain, but instead merely repeats dead patterns or remains without output.

The passage above is an instance of classical Aristotelian logic. If the two premises are well grounded and true, and if the rules of the syllogism are observed, then the conclusion follows.

Chapter One. The First Foundation of Logic

1.1. The Ontological Categories

The logic of Aristotle, as expounded in the Prior Analytics, is a conjunctive logic founded upon three figures, which establish distinct hierarchies among four of the eight ontological categories — the conditions of conjunction.

The eight ontological categories (the vertices of a cube, the elements of a quantum register) are these:

1.2. The Four Conditions of Conjunction (the Basis of Aristotelian Logic)

Aristotelian logic operates with only four of these eight categories:

1.3. The Four Basic Logical Terms

There are four terms that follow with necessity — O, I, E, A — each a superposition of two ontological categories:

• O (particular negative) = superposition 00* = –1 qubit.
• I (particular affirmative) = superposition 01* = 0 qubit.
• E (universal negative) = superposition 10* = +1/2 qubit.
• A (universal affirmative) = superposition 11* = +1 qubit.

Chapter Two. The Logical Square and the Logical Cube

The Aristotelian square of opposition is, in truth, a middling‑categorical section of a cube:

text

            -2=110          A=-1-2=11*          -1=111

+3=010          I=-4+3=01*          -4=011

                                     -1+1 centre of the cube

          +4=100          E=-3+4=10*          -3=101

+1=000          O=+2+1=00*          +2=001

Interpretation of the bits:

• First bit (0 = particular, 1 = universal) models the meaning of the judgment.
• Second bit (0 = denying / literal, 1 = affirming / abstract) models the form of the judgment.
• Third bit (0 = hypothesis, 1 = demonstration) models the cause (the degree of categoricity).

Chapter Three. The Ontological Centre: To On as «Bytuščee»

A special place belongs to the ontological centre of the cube (–1+1 = 111000), marked by the present‑tense neuter singular participle of the verb “to be” — in Russian, bytujuščee. This word occurs only in ancient languages (Sanskrit, Avestan, Greek) and in modern Russian.

Bytuščee — let us call it the Beingly or the Biding — is a foundational concept, expressing objective reality just as it is, naturally, without mediation. Ontologically it is an action possessed — the zero‑point of the dialectical axes of the ontological categories (–1+1, –2+2, –3+3, –4+4).

Languages that have lost this peculiar participle — for example, the Western European languages — are forced to employ indirect intellectual metaphors («concepts», philosophical neologisms) to signify reality. Such authorial metaphors wear away with time; and when they wear away, cognitive distortions creep into the culture.

Chapter Four. The Three Figures of Syllogisms

4.1. First Figure (Hierarchical)

The middle term is subject in one premise and predicate in the other. All three terms are nested within one another.

Perfect syllogisms (requiring no proof):

• ↺AAA — universal affirmative
• →IAI — particular affirmative
• ↘IEO — particular negative
• *↓AEE — universal negative

Weak syllogisms (requiring additional justification):

• ←AII
• ↖EIO
• ↗OAO
• ↙AOO
• ←EOO

4.2. Second Figure (Oppositional)

The middle term is predicate in both premises. A correct conclusion is impossible — but the very impossibility may be stated (and that statement is logical).

Syllogisms of the second figure:

• ⇗OAO
• ⇙AOO
• ⇖EIO
• ⇑OIO

4.3. Third Figure (Connecting through Example)

The middle term is subject in both premises.

Syllogisms of the third figure: