Chapter 2: The Higher-Dimensional "OR" State: The Matrix of the Law of Excluded Middle

Author: Zhang Suhang (Luoyang, Henan, China)

Abstract

This paper is the second installment in the research series Geometric Origin of the Law of Excluded Middle. Based on the MOC Multi-Origin Geometry framework, this study constructs the fundamental substrate that gives rise to the Law of Excluded Middle — the higher-dimensional "OR" state. Classical logic presupposes a closed bivalent domain where the truth value of any proposition is unique and mutually exclusive, rendering the Law of Excluded Middle P\lor\neg P a universally valid axiom. This paper demonstrates that in the uncollapsed higher-dimensional fundamental layer, a proposition and its negation allow for topological possibilities including coexistent truth values, truth-value gaps and undetermined states, with no inherent binary partition or mandatory exclusive mechanism.

The higher-dimensional "OR" state is a pre-logical ontological structure that precedes logical judgment and binary differentiation, rather than a disjunctive operation within classical logic. Through formal definition of the truth-value space, deduction of three types of non-classical truth-value configurations and comparative analysis with other logical schools, this paper rigorously proves that the Law of Excluded Middle does not necessarily hold in the higher-dimensional fundamental space.

This paper ultimately establishes a hierarchical relationship: the higher-dimensional "OR" state acts as the geometric matrix of the Law of Excluded Middle, while the classical Law of Excluded Middle emerges as a low-dimensional special case formed via dimensional projection, truth-value fixation and coarse-graining constraints imposed on the higher-dimensional space. The findings lay a core foundation for subsequent research on projection mapping mechanisms, the generation of binarization and the modeling of higher-dimensional truth-value systems.

Keywords: Higher-dimensional "OR" state; Law of Excluded Middle; MOC Multi-Origin Geometry; Coexistent truth values; Logical emergence; Pre-logical ontology

1. Introduction

Classical logic is built upon a closed domain where binary differentiation has been accomplished. Under this premise, propositions are strictly assigned only two mutually exclusive and exhaustive truth values: true and false. Accordingly, the Law of Excluded Middle acquires absolute formal validity.

Nevertheless, classical logic evades a fundamental question: what form does a proposition take prior to truth-value judgment, semantic division and binary assignment? By treating the binary structure as an innate given, classical logic fails to interpret the origin, scope of application and emergent conditions of the Law of Excluded Middle.

Within the MOC Multi-Origin Recursive Geometry framework, this paper argues that logical judgment is not the most fundamental primitive structure. Beneath low-dimensional bivalent logic lies a more fundamental higher-dimensional geometric space. This space is not bound by binary exclusive rules and permits the coexistence of P and \neg P. We define this primitive topological state as the higher-dimensional "OR" state.

The structure of this paper is arranged as follows. Chapter 2 elaborates the geometric origin and formal definition of the higher-dimensional "OR" state, and clarifies its ontological differences from classical disjunction. Chapter 3 formally proves the non-necessity of the Law of Excluded Middle in the higher-dimensional space. Chapter 4 establishes the hierarchical relation of "matrix-projection-emergence". Chapter 5 summarizes the core arguments and research value of this paper within the whole series.

2. Definition and Geometric Essence of the Higher-Dimensional "OR" State

2.1 Geometric Origin of the Higher-Dimensional "OR" State

In the hierarchical structure of MOC, the higher-dimensional fundamental layer refers to the deepest geometric state where spatial primitives remain uncoarsened, coordinate origins are non-unique and the topological structure is uncollapsed. Free from the exclusive constraints of low-dimensional logic, this layer possesses three ontological properties:

1. State superposition: Opposing truth-value primitives can exist topologically in parallel, with no mutual exclusion constraints as imposed in classical logic.

2. Undifferentiated truth values: The space is not divided into two isolated subsets of true and false by definite boundaries, and the truth-value system remains continuous and unfixed.

3. No mandatory selection: No endogenous or exogenous operator compels the system to perform binary screening and unique truth-value assignment.

We define the primitive state satisfying the above geometric features and existing prior to logical judgment as the higher-dimensional "OR" state, denoted as \mathcal{O}.

The higher-dimensional "OR" state is not a logical operation in the classical sense, nor is it equivalent to the disjunctive normal form P\lor\neg P. Classical disjunction serves as an inference rule within the bivalent system, whereas the higher-dimensional "OR" state represents the geometric ontology that predates the birth of bivalent logic.

2.2 Ontological Differences between the Higher-Dimensional "OR" State and Classical Disjunction

To fully distinguish between surface-level logical rules and underlying geometric ontology, the comparative relationships are presented as follows:

Comparison Dimension Classical Disjunction Higher-Dimensional "OR" State 

Premise of truth values Presupposes a closed bivalent domain with uniquely determined truth values Open and unfixed truth-value space, allowing simultaneous truth, simultaneous falsity and undetermined states 

Constraint mechanism Restricted by the Law of Excluded Middle, forcing an exclusive choice Free from exclusive constraints; multi-state coexistence is the intrinsic norm 

Logical hierarchy Inference tool in the low-dimensional projection layer Pre-logical and pre-judgment geometric matrix 

Sequence of existence Defined only after the establishment of bivalent logic Exists prior to binarization and formal logical rules 

Core Proposition: The classical "OR" is the formal expression of the Law of Excluded Middle, while the higher-dimensional "OR" state is the generative matrix of the Law of Excluded Middle.

3. Proof of the Invalidation of the Law of Excluded Middle under the Higher-Dimensional "OR" State

3.1 Sufficient Premises for the Validity of the Law of Excluded Middle

The standard formal expression of the Law of Excluded Middle is:

\vdash P \lor \neg P

Its universal truth relies on two indispensable underlying premises:

1. The truth-value domain is strictly dichotomized as \mathcal{T}=\{T,F\}, which is mutually exclusive and exhaustive.
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2. Every proposition is assigned a unique truth value, leaving no room for truth-value gaps, truth-value overflow or fluctuating undetermined states.

The above conditions are artificial boundary constraints generated after low-dimensional projection, rather than inherent attributes of the higher-dimensional fundamental space.

3.2 Higher-Dimensional Truth-Value Space and Three Types of Non-Classical Configurations

We define the higher-dimensional truth-value mapping:

v_{\mathcal{O}}: \mathcal{P}\to \mathcal{T}_{\mathcal{O}}

where \mathcal{P} denotes the universal set of propositions, and \mathcal{T}_{\mathcal{O}} stands for the open topological space of higher-dimensional truth values.

Breaking through the binary limitation, \mathcal{T}_{\mathcal{O}} contains three fundamental truth-value structures:

1. Truth-value gap (simultaneous falsity)

v_{\mathcal{O}}(P)=0,\quad v_{\mathcal{O}}(\neg P)=0

A proposition and its negation lack valid truth support simultaneously. The disjunctive formula fails to hold, and the Law of Excluded Middle becomes invalid.
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2. Truth-value overflow (simultaneous truth)

v_{\mathcal{O}}(P)=1,\quad v_{\mathcal{O}}(\neg P)=1

Two opposing truth values coexist in superposition, breaking the complementary and exclusive nature of the negation operator. Although the formula is formally true, its semantics deviate completely from the "mutually exclusive binary choice" defined by the classical Law of Excluded Middle, resulting in trivial and non-substantive truth.
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3. Evolving undetermined state

v_{\mathcal{O}}(P)=\bot

The truth value evolves within the topological space without a fixed assignment. The truth or falsity of the disjunctive expression cannot be judged, and the Law of Excluded Middle loses its applicability entirely.

In conclusion, the Law of Excluded Middle is not universally true in the higher-dimensional "OR" state space, and only holds as a special case.

3.3 Paradigm Distinction from Non-Classical Logical Systems

Existing non-classical logics follow the rule revision paradigm, while MOC geometric logic adopts the fundamental structure paradigm, representing an essential theoretical disparity:

1. Intuitionistic Logic: The invalidity of the Law of Excluded Middle stems from the lack of constructive proof, which is a limitation of epistemology. In contrast, the invalidity in the MOC framework is ontological and geometric, independent of human cognition.
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2. Quantum Logic: Its violation of the Law of Excluded Middle depends on physical measurement, observation collapse and the breakdown of distributive laws. The MOC "OR" state requires no physical carrier or measurement hypothesis and resides at a more primitive theoretical level.

The uniqueness of the MOC framework lies in that it does not revise logical axioms, but reveals the generative conditions and scope of application of logical axioms from a geometric perspective.

4. The Matrix Structure: Dimensional Projection and the Prelude to the Emergence of Bivalent Logic

4.1 Positioning of the Higher-Dimensional "OR" State

The higher-dimensional "OR" state itself is not a formal logical system, but a pre-existing geometric substrate upon which logical systems are constructed.

Before being acted upon by projection operators, the topological structure characterized by coexistence, superposition and indeterminacy always maintains its primitive form free from binarization, mutual exclusion and exhaustiveness.

4.2 Preparation for the Emergence Mechanism of Projection Mapping

When the dimensional projection operator \Pi acts on the fundamental space:

\Pi: \mathcal{T}_{\mathcal{O}} \to \{T,F\}

The diverse states in the higher-dimensional space are forcibly coarsened, uniquely assigned and enclosed by definite boundaries:

- Truth-value gaps are filled;
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- Truth-value overflow is split into mutually exclusive single truth values;
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- Evolving undetermined states are fixed as definite truth values.

The bivalent domain takes shape accordingly, and the Law of Excluded Middle emerges as a valid rule.

4.4 Hierarchical Conclusion on Matrix and Derivative

1. Geometric Matrix: The higher-dimensional "OR" state \mathcal{O}, where the Law of Excluded Middle does not apply and multi-state coexistence is permitted.
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2. Low-Dimensional Derivative: The closed space after projection, featuring mutually exclusive and exhaustive binary truth values, where the Law of Excluded Middle is valid.

The Law of Excluded Middle is not an a priori truth, but a local emergent law constrained by low-dimensional conditions derived from higher-dimensional geometry.

5. Conclusion

This paper completes the core demonstration of the second chapter in the series Geometric Origin of the Law of Excluded Middle, and establishes four definitive conclusions:

1. The higher-dimensional "OR" state is strictly defined as a pre-logical ontological structure in the higher-dimensional fundamental layer, where a proposition and its negation feature coexistent truth values, undifferentiated attributes and freedom from mandatory selection.

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2. Formal deduction verifies that the Law of Excluded Middle does not necessarily hold. The three fundamental configurations — truth-value gaps, truth-value overflow and evolving undetermined states — thoroughly break the preconditions of bivalent logic.

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3. The matrix-emergence relationship is confirmed: the higher-dimensional "OR" state serves as the geometric origin of the Law of Excluded Middle, while the Law of Excluded Middle is a derivative generated by low-dimensional projection.

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4. The boundary of classical logic is clarified: classical bivalent logic is only applicable to low-dimensional closed spaces with completed projection and value fixation, rather than the entire universal domain.

This research eliminates the a priori absoluteness of the Law of Excluded Middle, and reconstructs logic from an axiomatic a priori system into a system of emergence based on dimensional geometry. It provides an irreplaceable foundation for subsequent research including the modeling of projection operators, deduction of the binarization mechanism and the construction of hierarchical logic for the universal domain.

References

[1] Zhang Suhang. Geometric Origin of the Law of Excluded Middle: Chapter 1 Introduction[R]. Preprint, 2026.

[2] Zhang Suhang. Axiom System of MOC Multi-Origin Geometric Logic Framework[R]. Preprint, 2026.

[3] Zhang Suhang. Principle of Radix Generation via Dimensional Mapping[R]. Preprint, 2026.

End of Chapter 2