Chapter 6 ：Reconstruction of the Three Fundamental Laws of Logic: The Law of Identity, the Law of Contradiction and the Law of Excluded Middle as Products of Low-Dimensional Projection

Author: Zhang Suhang, Luoyang, Henan

Abstract

This paper is the sixth installment in the series Geometric Origin of the Law of Excluded Middle. Within the MOC geometric framework, it systematically reconstructs the three fundamental laws of classical logic: the Law of Identity, the Law of Contradiction and the Law of Excluded Middle. Previous chapters have demonstrated that the Law of Excluded Middle emerges as a low-dimensional structure from the collapse of high-dimensional "OR" states via projective mapping. This paper further proves that neither the Law of Identity nor the Law of Contradiction possesses a priori universal validity. All three laws arise collectively from the binarization constraints imposed by projective mapping on high-dimensional truth-value structures.

These logical laws hold strictly only within the bivalent image space \{0,1\} generated by projection, while they can be relaxed, suspended or even invalidated in the primordial high-dimensional truth space. This paper elaborates the unified geometric mechanism behind the three laws, defines the generating conditions and effective scope of the axiomatic system of classical logic, and realizes the paradigm extension from classical logic to high-dimensional MOC geometric logic.

Keywords: Law of Identity; Law of Contradiction; Law of Excluded Middle; Projective Mapping; Low-Dimensional Emergence; MOC Geometric Framework

1. Introduction

Classical formal logic has long regarded the Law of Identity, the Law of Contradiction and the Law of Excluded Middle as three absolute axioms of rational thinking, which underpin all mathematical logic and daily reasoning. Their standard formulations are presented as follows:

- Law of Identity: P \to P, stating that any proposition is strictly equivalent to itself.

- Law of Contradiction: \neg(P \land \neg P), holding that a proposition and its negation cannot both be true.

- Law of Excluded Middle: P \lor \neg P, asserting that either a proposition or its negation must be true.

Prior works in this series have verified that the Law of Excluded Middle is not an absolute truth, but a low-dimensional constraint generated by the projective collapse of the high-dimensional truth space. On this basis, this chapter puts forward a generalized unified proposition: the three fundamental laws of classical logic are not innate rules of thought. Instead, they are low-dimensional normative conditions emerging from different aspects after the projective mapping \Pi enforces binarization on high-dimensional "OR" states.

Without projective constraints in the primordial high-dimensional domain, the bivalent opposing structure dissolves, and the rigid restrictions of the three laws are lifted naturally. This chapter reconstructs the geometric generation mechanism of each law and establishes a unified theoretical framework for their common origin and interdependence.

2. Reconstruction of the Law of Identity: Label Solidification Induced by Projective Mapping

2.1 Implicit Presuppositions of the Classical Law of Identity

The classical Law of Identity assumes that propositions, objects and truth values maintain absolute constancy across different contexts, hierarchical levels and time sequences. Its validity relies on an implicit premise: the system under discussion is static, single-layered and a closed bivalent system. This prerequisite is not explicitly stated in classical logic, yet it is essential for the Law of Identity to hold.

2.2 Non-identical Structures in the High-Dimensional Truth Space

In the high-dimensional truth space \mathcal{T} defined by the MOC framework, the system resides in an uncollapsed high-dimensional "OR" state, where a proposition P and its negation \neg P may coexist, remain undetermined or present truth-value gaps, representing various non-classical states.

Furthermore, the high-dimensional space features a multi-layered recursive structure. An object perceived as identical at the superficial layer may decompose into multiple components and superposed states at deeper fundamental levels. Through coarse-grained hierarchical projection, structural differences in deep layers are smoothed out, and the object is recognized as a single homogeneous entity at the superficial layer. It follows that identity is not an inherent property of objects, but a representational effect dependent on hierarchical projection. No absolute and unconditional self-identity exists in the primordial high-dimensional system.

2.3 Mechanism for the Generation of the Law of Identity via Projection

The projective mapping \Pi: \mathcal{T} \to \{0,1\} collapses high-dimensional superposed states and uniquely assigns each high-dimensional truth value to a definite value b\in\{0,1\} within the bivalent set.

Subject to the constraints of the image space:

1. Every proposition is assigned a unique and fixed truth-value label;

2. Truth values satisfy the elementary reflexivity b=b;

3. All structural differences in high dimensions are eliminated by projection, locking the state of objects from the observational perspective.

This mechanism directly gives rise to the classical Law of Identity: any projected proposition is necessarily self-consistent and identical to itself.

Conclusion of this section: The Law of Identity is an emergent property in low dimensions resulting from the solidification of truth-value labels and the elimination of high-dimensional discrepancies, rather than a universal a priori axiom.

3. Reconstruction of the Law of Contradiction: Mutual Exclusivity Constraints Induced by Projective Mapping

3.1 Systematic Functions of the Classical Law of Contradiction

The Law of Contradiction forbids a proposition and its negation from being true simultaneously. As a core constraint preventing logical explosion and maintaining systemic self-consistency, it lays the foundation for mutual exclusivity in bivalent logic.

3.2 Coexistence of Truth Values in High-Dimensional Space

In the unprojected high-dimensional "OR" states, this series has defined the truth-value overflow state, namely v(P)=1 and v(\neg P)=1.

This state characterizes the high-dimensional coexistence of affirmative and negative truth values. Since the high-dimensional truth space is not a Boolean algebra, it does not follow the rule that a contradiction implies all propositions. Accordingly, the coexistence of P and \neg P is a valid and self-consistent topological state in the high-dimensional system, where the Law of Contradiction naturally fails.

3.3 Mechanism for the Generation of the Law of Contradiction via Projection

Projective mapping features strong selective mutual exclusivity. It performs forced bivalent selection on high-dimensional overflow and superposed states, collapsing coexistent affirmative and negative truth values into a single definite value.

The post-projection image space strictly conforms to:
\neg 0=1,\quad \neg 1=0
A truth value and its negation are strictly complementary and mutually exclusive with no overlapping intervals. Hence P\land\neg P is always false, and the Law of Contradiction inevitably holds.

Conclusion of this section: The Law of Contradiction arises when projection eliminates high-dimensional coexistent states and establishes a mutually exclusive structure of truth values in low dimensions. Coexistence is permitted in high dimensions, while mutual exclusivity is enforced in low dimensions.

4. Reconstruction of the Law of Excluded Middle: Completeness of Truth Values Induced by Projective Mapping

4.1 Review of Non-complete States in High Dimensions

As concluded in Chapters 2 to 5 of this series, the high-dimensional truth space generally contains truth-value gaps (0,0), truth-value overflow (1,1) and undetermined superposed states. In the primordial high-dimensional structure, affirmative and negative truth values cannot cover the entire state domain, so the Law of Excluded Middle has no universal validity.

4.2 Completeness Mechanism of Projection

The projective mapping \Pi regularizes incomplete high-dimensional structures in a forced manner: it fills truth-value gaps, eliminates truth-value overflow and fixes all undetermined superposed states, reducing all high-dimensional states uniformly to the full bivalent set \{0,1\}.

The image space achieves complete coverage of truth values: every proposition is definitely true or false with no vacancies or overflow, which leads to the universal validity of P\lor\neg P, namely the Law of Excluded Middle.

4.3 Unified Geometric Framework of the Three Laws

The three classical axioms are not independent rules, but three structural dimensions derived from the same binarization process via projection.

表格
Logical Law Primordial High-Dimensional Features Projection Constraints Low-Dimensional Emergent Result
Law of Identity Multi-layer variation and indeterminate superposition; no absolute identity Solidify truth-value labels and eliminate hierarchical differences Self-identity structure:  
Law of Contradiction Coexistence of affirmative and negative truth values; allowance of overflow Enforce mutual exclusivity of truth values and eliminate superposition Mutual exclusivity structure:  
Law of Excluded Middle Truth-value gaps and undetermined states; incompleteness Enforce full coverage and closed bivalent range Completeness structure:  

Projection simultaneously imposes three structural constraints: unique labeling, mutually exclusive truth values and complete value range. Combined together, they form the complete axiomatic skeleton of classical bivalent logic.

5. Scope of Application: Classical Logic as a Special Case of Low-Dimensional Projection

5.1 Universal Validity within the Projected Image Space

In the fully collapsed single-layer bivalent image space \{0,1\}, the Law of Identity, the Law of Contradiction and the Law of Excluded Middle hold universally and rigorously. This logical system applies to human daily cognition, classical mathematics, Boolean algebra and binary logic in computer science, serving as an effective approximation for the world observed from a low-dimensional perspective.

5.2 Invalidation of the Three Laws outside the Projected Domain

The three classical laws can be broken in the primordial high-dimensional space, multi-layered recursive systems, partially projected systems and incompletely collapsed superposed systems:

1. In cross-hierarchical systems, the identity of objects varies with observational layers, weakening the Law of Identity.
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2. In high-dimensional superposed and overflow states, affirmative and negative truth values coexist, violating the Law of Contradiction.
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3. In incomplete and undetermined systems, persistent truth-value gaps render the Law of Excluded Middle invalid.

A core proposition is therefore established: classical formal logic is a special case of MOC high-dimensional geometric logic under the conditions of single layer, full projection and bivalent collapse.

6. Conclusions

This chapter completes the core arguments of the sixth chapter in the series Geometric Origin of the Law of Excluded Middle and formulates a unified theory on the geometric origin of the three fundamental laws of logic:

1. The Law of Identity is not a self-evident absolute axiom. It is a low-dimensional self-identity structure emerging when projective mapping solidifies truth-value labels and erases high-dimensional hierarchical differences.
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2. The Law of Contradiction is not an inevitable rule for self-consistency. It results from projective mapping eliminating the coexistence of high-dimensional truth values and constructing a mutually exclusive value range in low dimensions.
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3. The Law of Excluded Middle is not an innate form of thinking. It is a geometric product of projective mapping filling truth-value gaps, closing the full bivalent domain and realizing completeness in low dimensions.
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4. The three fundamental laws of logic share an identical origin. All stem from the structural constraints generated by the binarization of high-dimensional "OR" states through projection, acting as three logical manifestations of the same geometric process.
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5. Classical logic has a definite scope of application, limited to single-layered, fully collapsed and closed bivalent low-dimensional systems, and thus no longer possesses universal a priori validity.

The three cornerstone laws of classical logic do not represent the ultimate boundary of rationality. They are standardized logical paradigms shaped by geometric projection when observed from a low-dimensional perspective. Beyond the superficial constraints of bivalent projection, high-dimensional geometric logic can accommodate, interpret and transcend classical logic, providing a more fundamental geometric foundation for non-classical logic, superposition logic and reasoning for complex systems.

References

[1] Zhang S H. Geometric Origin of the Law of Excluded Middle: Chapter 2 High-dimensional "OR" State[R]. Preprint, 2026.
[2] Zhang S H. Geometric Origin of the Law of Excluded Middle: Chapter 3 Three Non-classical States of the High-dimensional Truth Space \mathcal{T}[R]. Preprint, 2026.
[3] Zhang S H. Geometric Origin of the Law of Excluded Middle: Chapter 4 Projective Mapping \Pi and State Collapse[R]. Preprint, 2026.
[4] Zhang S H. Geometric Origin of the Law of Excluded Middle: Chapter 5 The Emergence of Binary: Symbolization of the Law of Excluded Middle[R]. Preprint, 2026.

End of Chapter 6