Chapter 4: Projection Mapping \boldsymbol{\Pi} and State Collapse

Mechanism: \boldsymbol{\Pi: \mathcal{T}\to\{0,1\}}, Forced Binarization and Emergence of the Law of Excluded Middle

Author: Zhang Suhang, Luoyang, Henan

Abstract

This paper is the fourth installment of the series Geometric Origin of the Law of Excluded Middle. It aims to construct the projection mapping \Pi from the high-dimensional truth space \mathcal{T} to the classical two-valued set \{0,1\}, and elaborate the geometric mechanism behind the emergence of the Law of Excluded Middle. As demonstrated in the second and third chapters of this series, the Law of Excluded Middle does not hold universally under high-dimensional "OR" states, and non-classical forms including truth-value gap, truth-value glut and indeterminate state have been defined. This paper further states that when a system projects from the fundamental high-dimensional layer to the superficial low-dimensional layer, the projection mapping \Pi implements forced binarization — filling truth-value gaps, resolving truth-value gluts and stabilizing indeterminate states — which triggers state collapse and makes the Law of Excluded Middle P \lor \neg P universally valid in low-dimensional spaces. This paper presents the rigorous mathematical definition of the projection mapping, proves its inevitability and uniqueness, and clarifies that the Law of Excluded Middle is not an a priori logical law, but a derivative result of projection and state collapse.

Keywords: Projection Mapping; State Collapse; Forced Binarization; Emergence of the Law of Excluded Middle; MOC Geometry

1. Introduction

As illustrated previously, in the "OR" states of the fundamental high-dimensional layer, a proposition P and its negation \neg P can coexist. Truth values may present as gaps, gluts or indeterminate states, so the Law of Excluded Middle is not necessarily valid. Nevertheless, the logical systems adopted by human daily cognition and classical science are all two-valued, where the Law of Excluded Middle holds universally. This contradiction can be explained by the projection mechanism.

According to the MOC geometric framework, the low-dimensional superficial logical space where we reside does not exist independently. It is the image space generated by projecting the fundamental high-dimensional space via the projection mapping. The projection process triggers state collapse, which constrains the coexistent high-dimensional states into two-valued forms. Consequently, the Law of Excluded Middle emerges and becomes a seemingly universal rule from a low-dimensional perspective.

This chapter strictly defines the projection mapping \Pi, analyzes its effects on the three types of non-classical truth states, and proves that the Law of Excluded Middle is inevitably valid in the image space of projection.

2. Definition of Projection Mapping \boldsymbol{\Pi}

2.1 From High-Dimensional Truth Space to Two-Valued Set

Let \mathcal{T} denote the high-dimensional truth space defined in the previous chapter. Its elements are truth assignment pairs (v(P), v(\neg P)), where v(P), v(\neg P) \in \{0,1,\bot\}. The space contains non-classical states such as the truth-value gap (0,0) and the truth-value glut (1,1). The classical two-valued set is \{0,1\}, in which 0 stands for false and 1 stands for true.

Definition 1 (Projection Mapping)

The projection mapping \Pi: \mathcal{T} \to \{0,1\} is a function that maps high-dimensional truth states to classical two-valued truth values, satisfying:

\Pi\big( (v(P), v(\neg P)) \big) = b,\quad b \in \{0,1\}

The domain of this mapping covers all high-dimensional truth states, and its codomain is the classical two-valued set. The basic definition alone allows arbitrary mapping results. Therefore, additional rationality axioms are introduced to endow the mapping with geometric inevitability.

2.2 Axioms of Rationality

The construction of the projection mapping follows the three axioms below. These axioms are formulated based on the requirement of MOC geometry that low-dimensional projection shall minimize information loss and be compatible with the original topological structure.

- Axiom 1 (Potential Preservation): If a proposition tends to be true in the high-dimensional space (e.g. truth-value glut state), the projection result is preferentially true; if it tends to be false (e.g. truth-value gap state), the projection result is preferentially false. Let the truth potential function be \tau: \mathcal{T} \to [0,1], then the value of \Pi is monotonically correlated with \tau.

- Axiom 2 (Uniqueness and Determinacy): The projection result must be a definite two-valued state, excluding probabilistic or fuzzy values. Projection is a deterministic function rather than a random process.

- Axiom 3 (Symmetry Compatibility): The projection mapping shall conform to the low-dimensional symmetry characteristics of MOC geometry. For states with dual symmetry in the high-dimensional space, their projection results satisfy corresponding symmetric constraints.

Combining geometric intuition and simplification principles, the standard projection rule is established as follows:

\Pi\big( (v(P), v(\neg P)) \big) =

\begin{cases}

1, & v(P)=1,\ v(\neg P)=0 \quad (\text{classically true state})\\

0, & v(P)=0,\ v(\neg P)=1 \quad (\text{classically false state})\\

1, & v(P)=1,\ v(\neg P)=1 \quad (\text{truth-value glut, forced assignment as true})\\

0, & v(P)=0,\ v(\neg P)=0 \quad (\text{truth-value gap, forced assignment as false})\\

1, & v(P)=\bot,\ v(\neg P)=\bot \quad (\text{indeterminate state, default assignment as true})

\end{cases}

Note: For indeterminate states, the default value can be selected following the principle of minimum rule conflict. In this paper, indeterminate states are uniformly projected to true. Extended models can introduce parameters for projection directions. Regardless of the selection, the logical self-consistency of the whole system remains intact, and the observation result in the low-dimensional space is always unique.

2.3 Geometric Interpretation of Projection

In the MOC geometric system, the projection from high dimension to low dimension is essentially a coarse-graining process. Multiple discrete primitives are integrated into continuous units, and diverse superposed truth states undergo state collapse and transform into single definite truth values. This process is inevitably accompanied by information loss. Unique features of the high-dimensional space, including coexistent truth values, vacant truth values and undefined truth values, are eliminated, leaving only the binary judgment of true or false. Hence, the forced binarization implemented by \Pi is an inevitable consequence of geometric coarse-graining and state collapse.

3. Forced Binarization: Manipulation of Three Non-Classical States

3.1 Filling the Truth-Value Gap

3.1 Filling the Truth-Value Gap

The truth-value gap corresponds to the state (0,0), where both P and \neg P are false. In this case, the disjunction P \lor \neg P evaluates to false, which violates the classical Law of Excluded Middle. Under the standard projection, \Pi\big((0,0)\big)=0, meaning the proposition P is judged to be false. In a two-valued system, the truth value of a negation is complementary to that of the original proposition, so \neg P is automatically judged to be true.

The original state where both propositions are false is converted into the classical form of " P is false and \neg P is true", and the disjunction evaluates to true.

Conclusion: The projection mapping fills the truth-value gap and validates the Law of Excluded Middle.

3.2 Resolving the Truth-Value Glut

The truth-value glut corresponds to the state (1,1), where both P and \neg P are true. This is a unique coexistent form in high-dimensional spaces. Under the standard projection, \Pi\big((1,1)\big)=1, P remains true while \neg P is constrained to be false. The coexistence feature of "being both true and false" in high dimensions is removed, and the system returns to the classical logical rule of "either one or the other".

Conclusion: The projection mapping eliminates redundant truth values and resolves the truth-value glut, restoring the classical semantics and validity of the Law of Excluded Middle.

3.3 Stabilizing the Indeterminate State

The indeterminate state is denoted as (\bot,\bot), where the truth value of the proposition is undefined. The projection process triggers state collapse and assigns a definite two-valued state. The standard projection uniformly sets the result to true. Whether the indeterminate state is stabilized as true or false, it will eventually form a classical pair of one true and one false, so the disjunction is necessarily true.

Conclusion: The projection mapping assigns definite values to indeterminate states and completes state collapse, which facilitates the emergence of the Law of Excluded Middle.

3.4 Invariance of Classical Truth States

For the states (1,0) and (0,1) that already comply with classical two-valued rules, the projection mapping keeps their values unchanged, and the Law of Excluded Middle holds naturally.

4. Proof for the Emergence of the Law of Excluded Middle

4.1 The Law of Excluded Middle in the Image Space

Define the projection image space \mathcal{B} = \Pi(\mathcal{T}). According to the projection rules, \mathcal{B} = \{0,1\}. Any proposition P in the image space has a unique truth value \Pi(P) \in \{0,1\}, and the negation operation is defined as:

\Pi(\neg P) = 1 - \Pi(P)

This is the standard semantics of classical two-valued logic.

Theorem (Emergence of the Law of Excluded Middle)

For any proposition P in the image space, \Pi(P) \lor \Pi(\neg P) = 1 holds universally. That is, the Law of Excluded Middle is valid throughout the image space.

Proof

Based on the definition of negation operation \Pi(\neg P) = 1 - \Pi(P), combined with the rule that disjunction takes the maximum value:

\Pi(P) \lor \Pi(\neg P) = \max\big(\Pi(P),\ 1-\Pi(P)\big) = 1

Thus the Law of Excluded Middle holds universally. Q.E.D.

4.2 Essence of Emergence

The core prerequisite of the above proof is that the negation operation is defined as truth value complementation in the image space. In the high-dimensional truth space \mathcal{T}, restricted by gaps and gluts, v(\neg P) and v(P) do not satisfy the complementary relation. The projection mapping forcibly revises the non-complementary negation relation in high dimensions into a complementary relation for two-valued states.

It follows that the Law of Excluded Middle is not an inherent logical rule of the high-dimensional space, but an emergent product derived from projection and state collapse. 

5. Physical Intuition and Interdisciplinary Analogy

5.1 Intuitive Description of State Collapse

In high-dimensional spaces, truth values of propositions exist as unrestricted free forms, including bilateral coexistence, vacant truth values and unformed states. As a spatial constraint mechanism, the projection mapping induces state collapse, converting diverse and uncertain high-dimensional truth forms into unique definite two-valued states.

After the completion of state collapse, the Law of Excluded Middle — stating that either a proposition or its negation must hold true — becomes a stable rule manifested in low-dimensional spaces. This rule, however, does not apply to the fundamental high-dimensional space before collapse.

5.2 Analogy with Quantum Measurement

The projection mapping defined in this paper is analogous to the measurement behavior in quantum mechanics: a microscopic superposition state collapses into a single definite eigenstate after measurement. The differences between the two theories are as follows: the projection mechanism of the MOC geometric framework covers logical states such as truth-value gaps and indeterminate states, which are not involved in quantum theory. Meanwhile, the standard projection is a deterministic process driven purely by geometry without randomness, which corresponds to the deterministic characteristics of the macroscopic low-dimensional world.

6. Conclusion

This chapter completes the core research of the fourth paper in the series Geometric Origin of the Law of Excluded Middle. The main achievements are summarized as follows:

1. The projection mapping \Pi is strictly defined, with its domain, codomain and standard rules clarified, together with the axioms for geometric compatibility.

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2. The mechanism of forced binarization is elaborated. Projection fills truth-value gaps, resolves truth-value gluts and stabilizes indeterminate states, eliminating non-classical high-dimensional features via state collapse.

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3. A formal proof for the emergence of the Law of Excluded Middle is presented. In the projection image space, the negation operation is transformed into truth value complementation, which makes the Law of Excluded Middle universally valid.

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4. The essential nature is clarified: the Law of Excluded Middle is not an innate a priori law. It is a low-dimensional logical rule emerging from dimensional reduction and state collapse of the high-dimensional space.

This chapter lays a foundation for subsequent research on binary symbolization and reconstruction of the three fundamental laws of logic. After projection, the low-dimensional two-valued logic is naturally compatible with the binary symbol system. The homologous relationship between two-valued logic and binary coding will be discussed in detail in Chapter 5.

References

[1] Zhang S H. Geometric Origin of the Law of Excluded Middle: Chapter 2 High-Dimensional "OR" States[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. Axiomatic System of the MOC Multi-Origin Geometric Logic Framework[R]. Preprint, 2026.

End of Chapter 4