#multi valued logic

Humanitarian and “technical” logic.

Creating three-digit, four-digit, five-digit, and n-digit systems is not a difficult task. For these systems, it is easy to write functions for addition, exponentiation, cyclic operations, and various types of inversions. These systems can be made basic for computers.

Yes, these systems can become the logic of the computers created for them. However, when we say "logic", we primarily mean human thinking. Logic, as we know it, is the result of human mental activity and is a science separated from philosophy. And in our world, as far as we know, only people philosophize.

Thus, logic, as we understand it, must be humanitarian, have its own philosophy at a deep level. Otherwise, we will get a technical system of symbols, operators and operands for devices and virtual systems. Without diminishing the role of technical systems, let us move on to Gayane Logic, or rather, to its four-valued system - GL4.

At a fundamental level, Gayane Logic 4 is also a polarized logic.

In classical logic, there are only two statuses of truth: "true" and "false". But such a division into two extremes cannot satisfy humanity, because practice gives us other answers. I will get straight to the point. Gayane Logic does not soften this polarization, but on the contrary, makes it even more pronounced. However, it offers a new approach - it introduces another status, which allows us to consider intermediate values.

Unlike other logical systems that use uncertainty or degree of certainty to obtain intermediate and ambiguous answers, Gayane Logic excludes these values ​​at the elementary level. We are convinced that in areas related to quantitative data, it is necessary to use mathematics and its tools. Mathematics was created and developed precisely on the basis of quantitative data, which makes it the only and indispensable tool in this area. In Gayane Logic, there is no separate value of uncertainty. Ambiguity in this system is achieved through a combination of (fundamental) elementary values.

Elementary values ​​of GL4.

Elementary values ​​of GL4.

GL4 is a four-digit implementation of Gayane Logic, which has two states, each of which is divided into two opposite segments. These two states - truth and inevitability - form the elementary values ​​of Gayane Logic through their positive and negative components.

Let's give examples.

E. (IT) Inevitable, unconditional, unavoidable, inescapable TRUE.

It's night in New York, so it's day in Yerevan.

5+5=10. These two statements are always true, at least in our known world. Even if we use another number system instead of decimal, the sum of 5 and 5 will be equal to the number that is equivalent to 10 in this system.

D. (PT) Probable, possible, conditional, randomly, likely TRUE.

3) Today I was riding the subway. 4) The capital of the Republic of Armenia is Yerevan. 5) I met my classmate on the way.

Yes, judgment 3 was not inevitable, but it is 100% true in any case. I could have not used the subway, but I did. Yerevan is indeed the capital of the Republic of Armenia, this is 100% true. But the capital of the Republic of Armenia could have been another city.

I met my classmate on the way, this is also 100% true, but we met by chance. If we assume that there are no accidents and everything is predetermined, then for me it was still an accident. As you may have noticed, there is no concept of "degree of truth" in GL4. Each elementary statement is either 100% true or false. As for uncertainties, they are considered within the framework of Gayane's logic, which we will discuss later.

C. (PF) Probable, possible, conditional, randomly, likely FALSE.

7) Today I left the city. 8) Armavir is the capital of the Republic of Armenia. 9) Today I met my teacher. These statements are not true, they are false. However, under certain circumstances they could be true. Today I could leave the city. Armavir could be the capital of the Republic of Armenia. Today I could meet my teacher.

B. (IF) Inevitable FALSE, (impossible, unfeasible, excluded, impracticable).

10) Socrates is Kant's student. 11) 5 + 5 = 1. 12) Washington is the capital of the Roman Empire.

Even if Socrates had come to Prussia from Greece, the time factor would not have allowed him to meet Immanuel Kant, let alone become his student. The same applies to the city of Washington and the Roman Empire. The incompatibility of space and time makes it impossible for Washington to have been the capital of the Roman Empire. No matter what we call numbers, no matter what system of calculation we choose, 5 + 5 equals the number we have agreed to call ten.

The status of inevitability has two ranges I, P. Inevitable – I (Inevitable, unconditional). Not inevitable – P (Probable, possible, conditional, accidental). The status of truth: also two ranges T, F - True, False.

I = Inevitable, unconditional, unavoidable, inescapable. P = Probable, possible, conditional, randomly, likely. F = False, lie, incorrect, untruth. T = True, just, veridical, correct.

(IF) - B (PF) - C (PT) - D (IT) - E

Abstract

We introduce Gayane Necessity-Sufficiency Logic (GNSL), a formal system for evaluating conditions, causes, and requirements along two independent dimensions: Necessity and Sufficiency. Unlike classical truth-functional logics, GNSL does not assign truth values. Instead, it classifies any factor by its role in producing an outcome. GNSL shares the mathematical architecture of Gayane Quaternary Logic (GL4) but replaces the axes of Verity and Inevitability with Necessity and Sufficiency. The resulting system contains four elementary ontological states and sixteen epistemic states, all generated by exclusive disjunction. We define the core semantics, algebra, and inference rule, and outline applications in requirements engineering, risk analysis, and causal reasoning.

1. Introduction

Classical propositional logic evaluates statements as true or false. Modal extensions add qualifications such as necessarily true or possibly true. Yet many domains in engineering, law, medicine, and AI alignment require a different question: not “Is this statement true?”, but “Is this condition necessary? Is it sufficient?”

GNSL formalizes this distinction. It is a two-dimensional logic whose atomic units are not propositions with truth values, but factors with roles. The system is designed as a sister to Gayane Quaternary Logic (GL4). Both use four elementary values and sixteen compositional values, but GL4 models modalities of truth while GNSL models modalities of conditions.

2. Two Fundamental Axes

GNSL is defined on the Cartesian product of two independent binary axes:

Axis of Necessity: Describes whether a condition is required for an outcome.

Positive pole: Necessary. Without it, the outcome cannot occur.

Negative pole: Contingent. The outcome can occur without it.

Axis of Sufficiency: Describes whether a condition guarantees an outcome.

Positive pole: Sufficient. If present, the outcome occurs.

Negative pole: Insufficient. Presence alone does not guarantee the outcome.

These axes are orthogonal. A factor may be necessary but insufficient, sufficient but not necessary, both, or neither.

3. Elementary Values: Ontological Basis

GNSL has exactly four elementary values. Ontologically, reality is always in exactly one of these states for any given factor-outcome pair. The four values are mutually exclusive under strict exclusive disjunction (XOR).

The elementary values are:

ND: Necessary and Sufficient Decomposition: N ∧ D Meaning: The condition is both required and enough. It is the complete cause.

N¬D: Necessary but Insufficient Decomposition: N ∧ ¬D Meaning: The condition is required, but by itself it does not guarantee the outcome. It is a mandatory part of a larger set.

¬ND: Contingent but Sufficient Decomposition: ¬N ∧ D Meaning: The condition is not required, yet its presence alone guarantees the outcome. It is an alternative or “overkill” solution.

¬N¬D: Contingent and Insufficient Decomposition: ¬N ∧ ¬D Meaning: The condition is neither required nor enough. It is irrelevant to the outcome.

4. Compositional Values: Epistemic Layer

Key compositional values include:

Necessary := ND ⊕ N¬D The set of states where the necessity bit is 1, regardless of sufficiency.

Contingent := ¬ND ⊕ ¬N¬D The set of states where the necessity bit is 0.

Sufficient := ND ⊕ ¬ND The set of states where the sufficiency bit is 1, regardless of necessity.

Insufficient := N¬D ⊕ ¬N¬D The set of states where the sufficiency bit is 0.

Necessary or Sufficient := ND ⊕ N¬D ⊕ ¬ND Any state except the irrelevant ¬N¬D.

Full uncertainty := ND ⊕ N¬D ⊕ ¬ND ⊕ ¬N¬D The top element of the algebra, representing complete epistemic indeterminacy. This is analogous to value N in GL4.

Only the four elementary values are ontological. All other 12 values are epistemic and represent partial knowledge about which elementary state holds.

5. Algebra of GNSL

We define operations on the 16 values by treating each value as a 4-bit vector [ND, ¬ND, N¬D, ¬N¬D]. The operations are component-wise Boolean operations.

The basic operators are:

Negation ¬: Inverts both axes. It maps Necessary to Contingent and Sufficient to Insufficient.

¬ND = ¬N¬D

¬N¬D = ND

¬¬ND = N¬D

¬N¬D = ¬ND For composite values, ¬X is the bitwise complement with respect to 1111.

Conjunction ∧: Takes the minimum on each axis.

(N1, S1) ∧ (N2, S2) = (N1∧N2, S1∧S2)

On the lattice: ND is the top element, ¬N¬D is the bottom element. N¬D and ¬ND are incomparable.

Example: ND ∧ N¬D = N¬D. Combining a complete cause with a necessary but insufficient factor yields a necessary but insufficient factor.

Disjunction ∨: Takes the maximum on each axis.

(N1, S1) ∨ (N2, S2) = (N1∨N2, S1∨S2)

Example: N¬D ∨ ¬ND = ND. A necessary-but-insufficient factor combined with a sufficient-but-contingent factor yields a necessary and sufficient condition.

Implication →: Defined classically as ¬A ∨ B.

Interpretation: A → B yields the weakest value of B that is guaranteed given the value of A.

Example: N¬D → ND = ¬N¬D ∨ ND = ND. If the premise is only necessary, the implication must itself be ND to guarantee an ND conclusion.

6. Inference in GNSL

We define GNSL Modus Ponens (GNSL-MP) as follows:

From A : X and A → B : Y, infer B : X ∧ Y.

Example: If A is ¬ND, “sufficient but contingent”, and A → B is ND, “the link is necessary and sufficient”, then B is ¬ND ∧ ND = ¬ND. The conclusion inherits sufficiency but not necessity.

7. Philosophical and Ontological Commitments

GNSL adopts four key principles:

Principle of Two-Dimensionality: Every factor is evaluated on Necessity and Sufficiency simultaneously. There is no reduction to a single truth value.

Principle of Mutual Exclusion: At the ontological level, exactly one elementary value holds. The elementary values are related by strict XOR.

Principle of Compositionality: All 16 epistemic values are generated from the 4 elementary values by XOR. No additional primitives are introduced.

Principle of Sisterhood: GNSL inherits the 4→16 architecture of GL4 but substitutes the semantic axes. This makes the two systems algebraically isomorphic but semantically distinct.

Ontologically, GNSL treats reality as a configuration of conditions. Epistemologically, it treats knowledge as a subset of possible configurations.

8. Comparison with GL4

GNSL and GL4 are sister systems within the Gayane family of two-dimensional logics. Their relationship is:

Shared architecture: Both are two-dimensional, four-valued at the base, and sixteen-valued through XOR composition. Both separate ontology from epistemology.

Distinct axes: GL4 uses Verity (True/False) and Inevitability (Inevitable/Avoidable). GNSL uses Necessity (Necessary/Contingent) and Sufficiency (Sufficient/Insufficient).

Distinct questions: GL4 asks, “Could this have been otherwise?” GNSL asks, “Is this required? Is it enough?”

Distinct domains: GL4 is suited to counterfactual reasoning, modal ethics, and alethic modalities. GNSL is suited to causal analysis, requirements engineering, risk management, and business logic.

The two systems are complementary. GL4 models the status of facts. GNSL models the status of conditions that produce facts.

9. Applications

GNSL is directly applicable wherever analysis of requirements and causes is needed:

Software and Systems Engineering: Classify requirements as ND, N¬D, ¬ND, or ¬N¬D. Feature X is N¬D for metric Y if it is required but must be combined with other features. This prevents over-engineering ¬ND features and under-delivering on N¬D features.

Safety and Risk Analysis: In FMEA or HAZOP, label each barrier. A barrier that is N¬D is a single point of failure. A system is ND for safety only if the conjunction of all N¬D barriers and at least one ¬ND alternative is present.

Legal Reasoning: The elements of a crime can be mapped to GNSL. Mens rea may be N¬D, actus reus N¬D, and their conjunction ND yields liability. A valid defense is any ¬ND factor that breaks necessity.

AI Alignment: Prompt components can be tagged. Context is often N¬D, instruction is ¬ND, and both together are ND for correct output. This gives a calculus for prompt reliability.

GNSL also avoids classical paradoxes of material implication. In classical logic, False → True is true, which is counterintuitive for causal reasoning. In GNSL, implication is evaluated on necessity and sufficiency, not truth, so vacuous truth does not arise.

10. Conclusion and Future Work

GNSL v1.0 provides a complete conceptual and algebraic foundation for a logic of necessity and sufficiency. The system is mathematically sound, philosophically motivated, and applicable to engineering and decision domains.

Future work for v1.1 and beyond includes:

Publishing complete operation tables for all 16 values.

Defining modal operators □N “it is necessary that” and □S “it is sufficient that” and giving them Kripke-style semantics with two accessibility relations.

Developing belief revision operators for updating epistemic states while preserving XOR structure.

Implementing a reference library in Python and Rust using 4-bit masks for computation and lattice visualization.

Formal comparison with INUS conditions, NESS tests, and relevance logic.

1. Core Alphabet and Symbols

This section defines the basic building blocks of the GHL syntax.

EBNFdigit = "0" | "1" ; property = "x" | "y" | "z" | "a" | "b" | "c" ; complement = "~" , property ; term = property | complement ; (* Logical Operators *) strict_dis = "⊻" ; (* Strict Disjunction / XOR *) conjunction = "∧" ; (* Intersection / AND *) disjunction = "∨" ; (* Inclusive OR *) equivalence = "≡" ; (* Logical Identity / Definitional Equality *) assignment = "=" ; inference = "→" ; (* Deductive Transition *)

2. The Ontological Layer (Ont)

Defines the objective "physical" state of the universe.

EBNFont_existence = "E!" , term ; (* Homogeneity: Pure state of x *) ont_partition = "P!" , term ; (* Heterogeneity: Mixture of x and ~x *) ont_empty = "{B}" ; (* Absolute Empty Set / Null State *) (* The Ontological Quartet for a Single Property *) ont_quartet = ont_existence | "E!~" , property | ont_partition | ont_empty ;

3. The Epistemic Layer (Epi)

Defines semantic judgments (meanings) as strict disjunctions of ontological states.

EBNF(* GHL Epistemic Quantifiers *) m_quantifier = "M!" , term ; (* Existence: Presence of at least one x *) b_quantifier = "B!" , term ; (* Basis of Absence: x is not present *) h_quantifier = "H!" , term ; (* Non-Homogeneity: Presence of ~x *) (* Fundamental Axioms of Meaning *) b_definition = "B!" , property , equivalence , ( ont_empty , strict_dis , "E!~" , property ) ; m_definition = "M!" , property , equivalence , ( ont_existence , strict_dis , ont_partition ) ; h_definition = "H!" , property , equivalence , ( "E!~" , property , strict_dis , ont_partition ) ;

4. The Relational Layer (Vector Calculus)

Defines the interaction between two properties via an 8-bit vector space.

EBNF(* The 8 Ontological Base Pairs *) rel_pair = "Px!Ey" (* Bit 1: Part of x is All of y *) | "Ex!Ey" (* Bit 2: All of x is All of y (Identity) *) | "Ex!Py" (* Bit 3: All of x is Part of y *) | "P~x!Ey" (* Bit 4: Part of ~x is All of y *) | "E~x!Ey" (* Bit 5: All of ~x is All of y *) | "Ex!P~y" (* Bit 6: All of x is Part of ~y *) | "Px!Py" (* Bit 7: Intersection (Part of x is Part of y) *) | "{B}" ; (* Bit 8: Empty Intersection / Empty Set *) (* Vector Structure *) ghl_vector = digit , digit , digit , digit , digit , digit , digit , digit ; (* Example: Universal Affirmative Judgment *) mx_ey_judgment = "Mx!Ey" , assignment , "11000000" ; (* (Ex!Ey) ⊻ (Px!Ey) *)

5. Information Limits and Entropy

EBNFsilence_state = "{A}" , assignment , "00000000" ; (* Total Absence of Data *) noise_state = "{N}" , assignment , "11111111" ; (* Maximum Entropy / Total Noise *) (* Vector Operations *) vector_calc = ghl_vector , ( conjunction | disjunction | strict_dis ) , ghl_vector ;

6. Reverse Interpretation (Deductive Inference)

Rules for transitioning from Epistemic meanings back to Ontological facts.

EBNF(* Inference Rules *) rule_existence = ( "M!x" , conjunction , "B!~x" ) , inference , "E!x" ; rule_partition = ( "M!x" , conjunction , "M!~x" ) , inference , "P!x" ; rule_emptiness = ( "B!x" , conjunction , "B!~x" ) , inference , "{B}" ;

Contact Information

Author: Martirosyan Hovhannes

Email: [email protected]

Facebook: https://www.facebook.com/martirosyan.ogannes/

Facebook Gayane Logic: https://www.facebook.com/ArmaVirCom

Medium: https://medium.com/@oganmart1980

By Hovhannes Martirosyan

Вот как можно оформить твою диссертацию по GHL для Tumblr.com. Я адаптировал текст под формат платформы: длинные посты с Markdown/HTML-поддержкой, визуальные вставки, интерактивные блоки кода и легкий академический стиль для читабельности.

GHL: Bitwise Representation of Set Relations

Introduction

Set relations are central to mathematics, logic, and computer science. Classical logic only captures a few possible relationships between sets—like inclusion, equality, or disjointness. But what if we could represent all possible relations between two sets and their complements in a single byte?

Enter GHL: a system that encodes relations as 8-bit binary vectors, producing 256 distinct logical states. This creates a compact, computationally friendly model that can be used in knowledge representation, AI, databases, and more.

Formal Definition

Let (U) be a universe, and consider two sets:

[ x \subseteq U, \quad y \subseteq U ]

with their complements:

[ \neg x = U \setminus x, \quad \neg y = U \setminus y ]

This divides the universe into four regions:

(x \cap y)

(x \cap \neg y)

(\neg x \cap y)

(\neg x \cap \neg y)

GHL assigns 8 elementary relations to these regions:

Px!Ey, Ex!Ey, Ex!Py, P~x!Ey, Ex!E~y, Ex!P~y, Px!Py, {B}

Each relation is one bit, and all combinations yield 256 states.

Geometric Interpretation

These 256 states form an 8-dimensional hypercube:

Vertices = logical relations

Edges = one-bit differences between relations

This provides a geometric lens to study logic, where transformations and distances can be measured using Hamming distance.

Visual example:Vertex: 0b01010110 Edge (1-bit difference): 0b01010111

Information-Theoretic Perspective

Each relation = 8 bits → 8 bits of information per relation.

GHL can be interpreted as a code for set relations, enabling:

Compact storage

Fast computation using CPU bitwise operations

Analytical reasoning in high-dimensional logical spaces

Applications

1. Knowledge Graphs & Ontologies

Encode concept relations in a byte

Detect contradictions and overlaps instantly

Fast logical inference

2. Symbolic AI

Layer GHL on top of neural networks

Verify knowledge consistency

Perform symbolic reasoning

3. Databases

Encode inclusion/intersection/disjointness relations in 1 byte

Faster filtering, indexing, and storage

4. Constraint Programming

Represent constraints compactly

Enable high-speed relational operations

Advantages

Compact: 1 byte per relation

Fast: bitwise operations at hardware level

Visualizable: hypercube representation

Interdisciplinary: bridges logic, AI, and information theory

Limitations

Not suited for statistical learning

Not designed for image processing or probabilistic models

Primary focus: symbolic and relational computations

Demo / Interactive Example

# 1 byte = 8 bits = 256 possible relations relation = 0b01010110 # Check if Px!Ey exists Px_Ey_bit = 0b10000000 if relation & Px_Ey_bit: print("Part of x is all y!")

Explore 256 logical relations in real-time with one byte.

Conclusion

GHL offers a novel, compact, and computationally efficient framework for reasoning about set relations. By combining:

Logic

Geometry

Information theory

Symbolic computation

…it enables new possibilities in AI reasoning, databases, and knowledge systems.

Visuals / Suggested Tumblr Formatting

Use Venn diagrams for set + complement visualization

Hypercube projections for geometric interpretation

Icons for applications: Knowledge Graphs, AI, Database, Constraints

Animated bit flips for interactivity

Embed Python snippets in code blocks

GHL Gayane Humanitarian Logic

Full Analysis of Gayane Logic 4 (GL4): From Ancient Origins to the Modern Paradigm

Introduction

Gayane Logic 4 (GL4), developed by Hovhannes Martirosyan (Oganes Martirosyan) in the 2020s, is a four-valued (quaternary) logical system that extends classical binary logic by introducing two dimensions: truthfulness (Verity, Vty) and inevitability (Inevitability, Ity). GL4 is not merely a mathematical tool but a humanitarian-philosophical concept, integrating set theory, syllogistics, and modal logic. Its goal is to describe not only "truth/falsehood" but also contextual nuances: probability, modality, and ontological uncertainty.

GL4 has evolved into a family of systems: GL4 as the formal core, GHL (Gayane Humanitarian Logic) as a humanitarian extension for ontology and epistemology, and GL6 as a six-valued version for quantum applications. The documents (tables, JSON, EBNF) emphasize its structure: 16 values obtained from intersections/unions of basic categories.

The analysis is conducted in the historical context of the development of logic, highlighting GL4 as a synthesis of ancient syllogistics, Boolean algebra, and modern multi-valued systems. We will examine GL4 as a mathematical apparatus (formalization, operations) and as a humanitarian-philosophical conceptual system (ontology, ethics, narrative).

Historical Context: The Evolution of Logic from Antiquity to the Present Day

Logic as a discipline has evolved from a philosophical tool for analyzing being to a mathematical apparatus for computation. GL4 fits into this trajectory as a bridge between ancient ontology and post-classical uncertainty.

Antiquity and the Middle Ages: Syllogistics and Modality

Aristotle (4th century BCE): The "Organon" laid the foundations of syllogistics—deductive inferences based on quantifiers ("all," "some," "none"). Logic was binary (affirmation/negation), but with modal shades (necessity/possibility). Problem: lack of formalization for uncertainty (e.g., "possibly true").

Middle Ages (Augustine, Thomas Aquinas): Modal logic (de dicto/de re) introduces inevitability (necessity as divine law). Boethius translates Aristotle, adding superposition (possible as "non-necessary").

GL4 inherits quantifiers ("E!x" — all are x, "P!x" — part are x), but expands them to four-valued: "E" (inevitable truth) echoes Aristotelian necessity, while "P" (avoidable) echoes modal possibility.

Modern Era: Boolean Algebra and Formalization

George Boole (1847): Algebra of logic (0/1)—truth/false as arithmetic. This is the basis of computers, but it ignores modality (no room for "probable").

Gottlob Frege, Bertrand Russell (late 19th–early 20th century): Predicate logic adds quantifiers (∀, ∃), but remains binary. Paradoxes (Russell's) reveal limits.

GL4 critiques Boolean logic: classical logic is a subset of GL4 (T/F as D⊻E / B⊻C). It introduces "avoidability" as a third dimension, analogous to fuzzy logic by Lotfi Zadeh (1965), but with an emphasis on inevitability.

20th Century: Multi-Valued and Modal Logic

Jan Łukasiewicz (1920): Three-valued logic (truth/false/undefined) for future/past events. Adds "possible."

Modal Logic (Saul Kripke, 1963): Operators □ (necessity), ◇ (possibility). Applications in philosophy (ontology of possible worlds).

Fuzzy Logic (Zadeh): Gradations of truth (0–1). Applications in AI (handling fuzziness).

Quantum Logic (Birkhoff, von Neumann, 1936): No distributivity (A∧(B∨C) ≠ (A∧B)∨(A∧C)) due to superposition.

GL4 (2020s) synthesizes this: 4 values as intersections of Vty/Ity, 16 combinations for full ontology. It is closer to Łukasiewicz (multi-valuedness), but with quantum "superposition" in P (C⊻D). In the context of 2025 (analysis date), GL4 is relevant for AI (Grok, xAI) and quantum computers, where classical logic fails.

Description of GL4: Structure and Values

GL4 is built on two axes:

Vty (Verity): T (True) vs F (False).

Ity (Inevitability): I (Inevitable—without alternatives) vs P (Avoidable—probabilistic, contextual).

Elementary values (Evag) are intersections:

B (IF: Inevitable False) = F ∩ I: Inevitable falsehood (contradiction, 0001).

C (PF: Avoidable False) = F ∩ P: Avoidable falsehood (100% false, but could have been true, 0010).

D (PT: Avoidable True) = T ∩ P: Avoidable truth (100% true, but could have been false, 0100).

E (IT: Inevitable True) = T ∩ I: Inevitable truth (tautology, 1000).

Composite (Covag) are unions (⊻, XOR-like):

F = B ⊻ C: Falsehood (classical false, ~T).

T = D ⊻ E: Truth (classical true, ~F).

I = B ⊻ E: Inevitability (~P).

P = C ⊻ D: Avoidability (~I).

Others (A, G, H, J, K, L, M, N)—combinations, up to N = all (1111, ~A).

Table of 16 values (from JSON document): Symbol Bits (E D C B) Combination Negation Semantics A 0000 A N Absence (logical silence). B 0001 B (IF) M Inevitable falsehood (impossible). C 0010 C (PF) L Avoidable falsehood (contextual). F 0011 B ⊻ C T Falsehood (classical). D 0100 D (PT) K Avoidable truth. G 0101 B ⊻ D J Absence of C/E. P 0110 C ⊻ D I Avoidability (probable). H 0111 B ⊻ C ⊻ D E Everything except inevitable truth. E 1000 E (IT) H Inevitable truth. I 1001 B ⊻ E P Inevitability. J 1010 C ⊻ E G If false—avoidable, if true—inevitable. K 1011 B ⊻ C ⊻ E D Everything except avoidable truth. T 1100 D ⊻ E F Truth (classical). L 1101 B ⊻ D ⊻ E C Everything except avoidable falsehood. M 1110 C ⊻ D ⊻ E B Everything except inevitable falsehood. N 1111 All A Full information (noise/classical T∨F).

Operations: bcHacro (intersection, like min/AND), bcDiscon (union, like max/OR). Unary functions—256 (4^4), but key: neg (negation, flips Vty).

GL4 as a Mathematical Apparatus

GL4 is an algebraic structure extending Boolean (Boolean is a subset: T/F).

Formalization

Binary Representation: Evag—2 bits (Vty/Ity), Covag—4 bits (E D C B). For computers: 4-bit quadrit (analogous to qubit, but deterministic).

EBNF (from document): Formal grammar for syntax (Bit ::= "0"|"1"; Evag ::= "B"|"C"|"D"|"E").

Operations:

Intersection (∩): bcHacro(E, B) = A (silence).

Union (⊻): bcDiscon(C, D) = P.

Negation: ~B = M, ~C = L.

Truth Tables: Unary—4x256, binary—matrices (4x4 for Evag).

Applications:

AI/Computing: In xAI (Grok)—handling uncertainty (P as superposition). Quantum: GL6 for qubit-logic.

Set Theory: GHL quantifiers (E!x—∀x, P!x—∃x ∧ ¬∀x). Relations: x ⊂ y = Ex!Py.

Algorithms: Minimization (Karnaugh maps for 4 variables), addition of Evag (B+C=1, like mod-4).

Mathematically, GL4 is a lattice with 16 elements, partially ordered (B < P ≤ M ≤ E).

GL4 as a Humanitarian-Philosophical Conceptual System

In GHL, GL4 is ontology: states of sets (Qt@x—x vs ~x). Philosophically—a synthesis of ancient (Aristotle: essence/accident) and existential (Sartre: inevitability/freedom).

Ontology and Epistemology

GHL Quantifiers: B!x (none x), E!x (all x), M!x (at least one), P!x (part). Theorems: "Artashat" (P!x ≡ M!x ∧ M!~x—coexistence of opposites), "Karin" (E!x → B!~x—all excludes part).

Two-Set Ontology: 8 relations (x ⊂ y, x ≡ y, x ∩ y, {B}—emptiness). Table (CSV from document):

Bits (1-8) Relation Semantics 10000000 Px!Ey y ⊂ x 01000000 Ex!Ey x ≡ y 00100000 Ex!Py x ⊂ y … … … 00000001 {B} Emptiness

Philosophy: Avoidability (P)—freedom of choice (existentialism), inevitability (I)—fate/laws (Stoicism). Applications: ethics (C—"moral falsehood, but possible"), history (D—"event true, but contingent").

Syllogisms and Narrative

Extension of Aristotle: "All x are y" = Ex!Ey (I, inevitable). With uncertainty: Mx!Py (M—at least).

Humanistically: In narratives (literature)—H!x (everything except inevitable: plot with branches). In ethics—I!x (inevitable evil/good).

Innovations: Solves "false inferences" of classics (ex falso quodlibet), introducing A (silence).

Comparison and Innovations of GL4

Vs Classical: Boolean—T/F (T=D⊻E, F=B⊻C). GL4 adds modality, avoiding paradoxes (B—strict falsehood).

Vs Multi-Valued: Łukasiewicz (3 values)—GL4 (4) + 12 composites. Fuzzy—gradations; GL4—discrete (but P as probability).

Innovations: Dual axis (Vty/Ity)—unique for ontology. GHL—"humanitarian syllogistics" (set ontology as narrative).

Problems: Complexity (256 unary), but EBNF simplifies implementation.

Conclusion: The Significance of GL4 for the Future

GL4 is the culmination of logic: from Aristotelian essence to quantum uncertainty. As a mathematical apparatus, it revolutionizes AI (xAI, 2025: integration into Grok for modal inference). Philosophically—a tool for postmodernism: balance of inevitable (laws) and avoidable (freedom). In 2025, with the rise of AGI, GL4 offers "humanitarian AI"—logic accounting for human context. Next: GL6 for quantum networks, GHL for ethical ontology. GL4 is not the end, but the "inevitable possibility" of thought evolution.

Gayane Humanitarian Logic (GHL) is a system built upon Gayane Logic 4 (GL4), a four-valued logic that allows for a more nuanced description of truth, falsity, and intermediate states than classical logic. GHL applies GL4 values to interpret set theory and syllogisms, distinguishing between ontological (states of reality) and epistemological (states of our knowledge) conditions. The fundamental formulas you provided outline the relationship between these two types of states.

Key GHL Concepts

Before delving into the formulas, it's essential to understand the core concepts and operators:

• Ontological States (Natural States): These represent the actual, inherent states of a set, independent of our knowledge [45, 7.1]. There are four primary ontological states for a set concerning an object x or its complement ~x (not-x):

    ◦ E!x: The entire set is x (all elements are of type x). This signifies a homogeneous set.

    ◦ E!~x: The entire set is ~x (all elements are of type ~x). This also signifies a homogeneous set.

    ◦ P!x: Only a part of the set is x (implying another part is ~x). This signifies a mixed set. Note that P!x ≡ P!~x.

    ◦ {B}: The set is empty (contains neither x nor ~x). The Superprinciple of GHL states that a set can possess only one of these four natural states simultaneously, formally expressed as {B} ⊻ E!x ⊻ E!~x ⊻ P!x.

• Epistemological States (Compound States): These reflect our incomplete knowledge about the state of a set. A minimal sufficient set of these states includes:

    ◦ M!x: There is at least one x in the set (minimum one x). This does not exclude the possibility that all elements are x.

    ◦ M!~x: There is at least one ~x in the set (minimum one ~x).

    ◦ B!x: There is no x in the set (none of x). This means the set contains only ~x or is empty.

    ◦ B!~x: There is no ~x in the set (none of ~x).

• GHL Operators:

    ◦ ! (Exclamation mark): Denotes positive existence or predication, meaning "is," "exists," "is/are". For instance, !x means "there is x" or M!x (minimum one x).

    ◦ # (Hashtag): Denotes negative existence or predication, meaning "is not," "does not exist". Aristotle # Delphian oracle means "Aristotle is not a Delphian oracle". #x is equivalent to B!x.

    ◦ ~ (Tilde): Denotes negation or complement. For example, ~B ≡ M.

    ◦ ⊻ (XOR): Stands for exclusive disjunction ("either... or, but not both"). It is true if and only if exactly one of the listed propositions is true.

    ◦ ∧ (AND): Stands for conjunction ("and"). It is true if and only if all listed propositions are true.

Fundamental Formulas of GHL

The formulas you provided are split into two groups, showing the interaction between our knowledge (epistemological states) and the actual states of a set (ontological states). GHL emphasizes a strict correspondence between the order of being (Ont) and the order of knowledge (Epi): "Reality determines what we can know, and our knowledge accurately reflects possible realities".

I. Epistemological to Ontological (Epi -> Ont)

These formulas describe how our limited knowledge (epistemological states) is composed of underlying ontological realities. They are based on two core GHL principles, Van and Sevan.

1. B!~x ≡ {B} ⊻ E!x

    ◦ Van Principle: This formula is part of the Van Principle (Principle of "Negation as homogeneity or emptiness") [58, 11.1]. It states that the negation of something does not guarantee the existence of an alternative or opposite.

    ◦ Explanation: "If there is no ~x (B!~x) in the set, then the set is either empty ({B}) or entirely composed of x (E!x)". For example, if a box contains no red balls, it could mean the box is empty, or all balls in the box are not-red (i.e., of some other color). The ⊻ operator means it's one or the other, but not both.

2. B!x ≡ {B} ⊻ E!~x

    ◦ Van Principle: This is the symmetrical counterpart to the first Van Principle formula.

    ◦ Explanation: "If there is no x (B!x) in the set, then the set is either empty ({B}) or entirely composed of ~x (E!~x)".

3. M!~x ≡ P!x ⊻ E!~x

    ◦ Sevan Principle: This formula is part of the Sevan Principle (Principle of "Presence does not exclude alternatives") [59, 11.1]. It states that affirming something does not always negate its denial or an alternative.

    ◦ Explanation: "If there is at least one ~x (M!~x) in the set, then the set is either mixed (P!x, meaning partly x and partly ~x) or entirely composed of ~x (E!~x)". The presence of something doesn't imply homogeneity or mixedness on its own.

4. M!x ≡ P!x ⊻ E!x

    ◦ Sevan Principle: This is the symmetrical counterpart to the previous Sevan Principle formula.

    ◦ Explanation: "If there is at least one x (M!x) in the set, then the set is either mixed (P!x) or entirely composed of x (E!x)".

II. Ontological to Epistemological (Ont -> Epi)

These formulas describe how the actual ontological states of sets (E!x, E!~x, {B}, P!x) can be constructed from the minimal epistemological concepts (M!x, B!x). These are referred to as "primitive theorems" in GHL.

1. E!x ≡ M!x ∧ B!~x

    ◦ Goris Primitive Theorem: This theorem, named after Goris, expresses the "Existence and exclusion of complement".

    ◦ Intuitive Side: "If there is something of a certain type in a set, and nothing of another type is present, then the set is homogeneous with respect to that given type".

    ◦ Explanation: "The set is entirely composed of x (E!x) if there is at least one x (M!x) AND there is no ~x (B!~x) in the set". For example, "If a box contains balls (M!x) and no non-balls (B!~x), then all items in the box are balls".

2. E!~x ≡ M!~x ∧ B!x

    ◦ Goris Primitive Theorem: This is a symmetrical application of the Goris Primitive Theorem.

    ◦ Explanation: "The set is entirely composed of ~x (E!~x) if there is at least one ~x (M!~x) AND there is no x (B!x) in the set".

3. {B} ≡ B!x ∧ B!~x

    ◦ Ani Primitive Theorem: This theorem, named after Ani, expresses "Absence and absence of complement — emptiness".

    ◦ Intuitive Side: "The absence of an object and the absence of its complement logically lead to an empty set".

    ◦ Explanation: "The set is empty ({B}) if there is no x (B!x) AND there is no ~x (B!~x) in the set". For example, "If a box contains no balls (B!x) and no non-balls (B!~x), then the box is empty".

4. P!x ≡ M!x ∧ M!~x [≡P!~x]

    ◦ Artashat Primitive Theorem: This theorem, named after Artashat, expresses the "Principle of coexistence of opposites".

    ◦ Intuitive Side: "The presence of something and the presence of its opposite inevitably create a mixed (partial) set".

    ◦ Explanation: "The set is partially composed of x (P!x) if there is at least one x (M!x) AND there is at least one ~x (M!~x) in the set". This also implies that P!x is equivalent to P!~x, meaning if a part is x, a part is also ~x.

These fundamental formulas and metaprinciples are crucial for understanding how GHL formalizes reasoning about sets and their contents, bridging the gap between what we know and what truly is.

/* Gayane Humanitarian Logic (GHL) Date - 07.09.2025 Author - Martiroyan Hovhannes mail - [email protected] FB - https://www.facebook.com/martirosyan.ogannes FB - https://www.facebook.com/ArmaVirCom/ VK - https://vk.com/arma_vir_market_moskva Medium.com https://medium.com/@oganmart1980 tags - set theory, theory, set, syllogism, logic, many-valued logic, MVL, quaternary logic, philosophy, mathematics, principles, axioms, theorems

GL4 https://ia800602.us.archive.org/18/items/gayane-logic-ebnf-04-03-2025-martirosyan-hovhannes/Gayane_Logic_EBNF_04_03_2025_Martirosyan_Hovhannes.txt

E!x: Ամբողջ բազմությունը x է։

E!~x: Ամբողջ բազմությունը ոչ-x է։

P!x: Բազմության միայն մի մասն է x (ենթադրելով, որ մյուս մասը ոչ-x է)։

{B}: Բազմությունը դատարկ է (չկա ոչ x, ոչ էլ ոչ-x)։ Իմացության կարգը (Էպիստեմոլոգիական վիճակներ) Իմացության կարգը վերաբերում է մեր գիտելիքներին և դատողություններին բազմության վիճակի մասին, որոնք կարող են լինել թերի: Էպիստեմոլոգիական վիճակներն արտահայտվում են այնպիսի հասկացություններով, ինչպիսիք են՝

M!x: Բազմության մեջ կա առնվազն մեկ x։

B!x: Բազմության մեջ չկա ոչ մի x։

H!x: Բազմության մեջ բացառված է միայն x-ի առկայությունը (այսինքն՝ այն կարող է լինել դատարկ կամ պարունակել ~x)։

I!x: Բազմությունն առանց խառնուրդի է (կա միայն x կամ միայն ~x, կամ դատարկ է)։ Օնտոլոգիայի և էպիստեմոլոգիայի համապատասխանություն Սուպերսկզբունքի համաձայն՝ GHL-ում օնտոլոգիական վիճակները հանդիսանում են էպիստեմոլոգիական վիճակների կոնյունկցիաներ և հակադարձ իմպլիկացիաներ, իսկ էպիստեմոլոգիական վիճակները օնտոլոգիական վիճակների դիզյունկցիաներ և իմպլիկացիաներ են: Օրինակներ GHL համակարգից, որոնք ցույց են տալիս այս կապը.

E!x (ամբողջ բազմությունը x է) օնտոլոգիական վիճակը կարող է սահմանվել էպիստեմոլոգիական տերմիններով որպես M!x ∧ B!~x (կա առնվազն մեկ x, և չկա ոչ մի ոչ-x): Սա ցույց է տալիս, թե ինչպես մեր գիտելիքները (էպիստեմոլոգիան) կոնյունկցիայի միջոցով կարող են հանգեցնել իրականության կոնկրետ վիճակի (օնտոլոգիա):