Chapter 3: Three Non-Classical States of the High-Dimensional Truth Space \mathcal{T}

Further elaboration: Definitions of truth-value gap, truth-value glut and indeterminate state

Author: Zhang Suhang (Luoyang, Henan)

Abstract

This paper is the third installment in the series Geometric Origin of the Law of Excluded Middle. It further elaborates the logical structure of high-dimensional "OR" states, and formally defines the high-dimensional truth space \mathcal{T} along with its three non-classical states: truth-value gap, truth-value glut and indeterminate state. In classical two-valued logic, truth values are restricted to the set \{0,1\}. By contrast, the high-dimensional truth space \mathcal{T} allows a proposition and its negation to coexist, with unfixed or even undefined truth values. This paper rigorously demonstrates that the three non-classical states are concrete manifestations of the failure of the Law of Excluded Middle in high dimensions, and constitute the fundamental reason why the projection mapping \Pi is required for "forced fixation". This chapter lays a solid semantic foundation for the subsequent research on projection mechanisms and binarization processes.

Keywords: high-dimensional truth space; truth-value gap; truth-value glut; indeterminate state; Law of Excluded Middle; MOC geometry

1. Introduction

In the second paper of this series, we introduced the basic concept of high-dimensional "OR" states: a proposition P and its negation \neg P can coexist with unfixed truth values, so the Law of Excluded Middle does not necessarily hold. Nevertheless, such "coexistence" is not a single uniform state. The truth structure at the fundamental high-dimensional level is far richer than that of classical two-valued logic, and more complex than a simple "both being true and false".

To accurately describe this structure, we introduce the high-dimensional truth space \mathcal{T} and identify three typical non-classical states within it:

- Truth-value Gap: Both P and \neg P are false, meaning neither holds true.

- Truth-value Glut: Both P and \neg P are true, meaning they hold true simultaneously.

- Indeterminate State: The truth values of P and \neg P remain undetermined, in a state of evolution or being undefined.

These states cannot be expressed within the framework of classical two-valued logic, yet they prevail in high-dimensional "OR" states. An understanding of them is essential to explain why the projection mapping \Pi has to implement forced binarization.

2. Definition of the High-Dimensional Truth Space \mathcal{T}

2.1 From Classical Truth Values to High-Dimensional Truth Values

The truth value set of classical two-valued logic is \{0,1\}, where 0 stands for false and 1 for true. The high-dimensional truth space \mathcal{T} is an extension of \{0,1\}. Its elements are not individual truth values, but paired truth assignments or truth state vectors.

Definition

\mathcal{T} = \big\{ \big(v(P), v(\neg P)\big) \,\big|\, P \in \mathcal{P} \big\}

where \mathcal{P} denotes the set of all propositions. The values of v(P) and v(\neg P) are not confined to \{0,1\}, and may include a third value or an undefined symbol.

For the sake of discussion, three special symbols are adopted herein:

- 0: False

- 1: True

- \bot: Undefined / Undetermined

Accordingly, the high-dimensional truth space \mathcal{T} contains at least the following states (non-exhaustive):

\mathcal{T} \supseteq \big\{ (0,0),\,(1,1),\,(\bot,\bot),\,(0,\bot),\,(1,\bot),\,(\bot,0),\,(\bot,1),\,(0,1),\,(1,0) \big\}

Among these states, (0,1) and (1,0) correspond to assignments in classical two-valued logic, while all others are excluded from classical logic.

2.2 Topological Interpretation of the High-Dimensional Truth Space

From the perspective of MOC geometry, the high-dimensional truth space serves as the logical counterpart of the fundamental high-dimensional layer. At this layer, the spatial structure remains unpartitioned and unprojected, so the truth of a proposition has not been divided into two mutually exclusive regions. The richness of \mathcal{T} is precisely the logical reflection of high-dimensional geometric structures.

3. Formal Definitions of the Three Non-Classical States

3.1 Truth-Value Gap

Definition: A proposition P and its negation \neg P are both false, i.e.

v(P) = 0,\quad v(\neg P) = 0

Denoted as \text{Gap}(P).

Interpretation: P is neither true nor false. This differs fundamentally from the notion of "unknown" in classical logic — an unknown proposition can still be binarized, whereas a truth-value gap refers to the absence of truth value, placing the proposition in a "vacant region" within the truth space.

Example in the context of MOC: A proposition at the fundamental high-dimensional layer may possess no definite attributes before projection. For instance, the proposition "This discrete primitive element lies to the left of the origin" is neither true nor false when no coordinate system has been established.

Implications for the Law of Excluded Middle

The classical Law of Excluded Middle P \lor \neg P requires at least one constituent proposition to be true.

In the case of a truth-value gap, v(P \lor \neg P) = \max(0,0) = 0, so the disjunction is false. The Law of Excluded Middle therefore fails.

3.2 Truth-Value Glut

Definition: A proposition P and its negation \neg P are both true, i.e.

v(P) = 1,\quad v(\neg P) = 1

Denoted as \text{Glut}(P).

Interpretation: P is both true and false. This does not constitute an explosive contradiction prohibited by the Law of Non-Contradiction. Instead, it represents another form of coexistence in high-dimensional states, where two truth values are present concurrently.

Example in the context of MOC: In a high-dimensional "OR" state, P and \neg P can coexist without conflict. For example, the statement "This point lies within region A and outside region A" is impossible in classical geometry. In an unprojected high-dimensional space, however, the position of a point is not uniquely defined, so both descriptions can hold true simultaneously.

Implications for the Law of Excluded Middle

Here v(P \lor \neg P) = \max(1,1) = 1, so the disjunction evaluates to true.

Nevertheless, this "truth" does not follow the classical rule of "either one or the other", but rather presents a state of "both being the case". Although the Law of Excluded Middle holds in terms of truth value, its semantic connotation is altered. Its validity in this scenario is merely trivial and deviates from classical meaning.

3.3 Indeterminate State

Definition: The truth values of P and \neg P are undetermined and undefined:

v(P) = \bot,\quad v(\neg P) = \bot

Denoted as \text{Indet}(P).

Interpretation: Distinct from a truth-value gap, an indeterminate state does not mean falsehood, but indicates that no truth value has been assigned yet. It corresponds to an intermediate stage of recursive evolution prior to the completion of projection.

 Example in the context of MOC: In a high-dimensional "OR" state, P and \neg P can coexist without conflict. For example, the statement "This point lies within region A and outside region A" is impossible in classical geometry. In an unprojected high-dimensional space, however, the position of a point is not uniquely defined, so both descriptions can hold true simultaneously.

Implications for the Law of Excluded Middle

Here v(P \lor \neg P) = \max(1,1) = 1, so the disjunction evaluates to true.

Nevertheless, this "truth" does not follow the classical rule of "either one or the other", but rather presents a state of "both being the case". Although the Law of Excluded Middle holds in terms of truth value, its semantic connotation is altered. Its validity in this scenario is merely trivial and deviates from classical meaning.

3.3 Indeterminate State

Definition: The truth values of P and \neg P are undetermined and undefined:

v(P) = \bot,\quad v(\neg P) = \bot

Denoted as \text{Indet}(P).

Interpretation: Distinct from a truth-value gap, an indeterminate state does not mean falsehood, but indicates that no truth value has been assigned yet. It corresponds to an intermediate stage of recursive evolution prior to the completion of projection.

Example in the context of MOC: During hierarchical recursion, a proposition may be in transition from a high-dimensional space to a low-dimensional one, with no definitive outcome. For example, when multiple discrete primitive elements are merging in the process of coarse-graining and have not yet formed a distinct continuous point, any proposition concerning this point falls into an indeterminate state.

Implications for the Law of Excluded Middle

The Law of Excluded Middle presupposes that propositions carry definite truth values. When truth values are undefined, the law is neither verified nor falsified, and remains suspended. This indicates that the validity of the Law of Excluded Middle relies on the complete assignment of truth values.

4. Properties and Interrelations of the Three Non-Classical States

4.1 Summary Truth Table

 Table 1

State       Validity of the Law of Excluded Middle 

Classically True 1 0 1 Holds 

Classically False 0 1 1 Holds 

Truth-Value Gap 0 0 0 Fails 

Truth-Value Glut 1 1 1 Trivially holds (semantically altered) 

Indeterminate State       Suspended (meaningless) 

4.2 Transformational Relations Between States

At the fundamental high-dimensional layer, the three states can coexist and transform into one another, governed by recursion level, observation scale and degree of projection:

- When the system projects onto a lower-dimensional space, a truth-value gap may be filled and converted into a classical true or false state.

​

- A truth-value glut is resolved by forcing the selection of a single truth value during projection.

​

- An indeterminate state acquires definite truth values once projection is completed.

 These transformational rules provide direct logical objects for the subsequent study of the projection mapping \Pi.

5. Comparison with Classical Logic and Other Non-Classical Logics

Table 2

Logical System Allows Truth-Value Gap Allows Truth-Value Glut Allows Indeterminate State 

Classical Two-Valued Logic No No No 

Intuitionistic Logic Yes (different interpretation) No No 

Paraconsistent Logic No Yes (handles explosion) No 

Three-Valued / Multi-Valued Logic Yes (third value as indeterminate) No Yes (generally does not distinguish gap and indeterminacy) 

MOC High-Dimensional Truth Space Yes (absence of truth value) Yes (coexistent glut) Yes (evolutionary undefined state) 

Most existing non-classical logics are designed to address only one specific type of abnormal truth state. By contrast, the MOC framework unifies truth-value gaps, gluts and indeterminate states within a single geometric projection model, and explains their internal connections and mutual transformations. This constitutes the core distinction between MOC geometry logic and other logical theories.

6. Conclusion

This paper completes the core research of the third chapter in the series Geometric Origin of the Law of Excluded Middle, with the main outcomes summarized as follows:

1. The high-dimensional truth space \mathcal{T} is defined. As an extension of the classical two-valued set \{0,1\}, it takes truth value pairs \big(v(P),v(\neg P)\big) as its basic elements and adopts the symbol \bot for undefined states.

​

2. Three non-classical states are formally defined: the truth-value gap corresponds to (0,0) where the Law of Excluded Middle fails; the truth-value glut corresponds to (1,1) where the law holds only formally with altered semantics; the indeterminate state corresponds to (\bot,\bot) where the Law of Excluded Middle is suspended.

​

3. The transformational relations between states are revealed. The three non-classical states are ubiquitous at the fundamental high-dimensional layer, and will evolve into classical two-valued states under the action of the projection mapping \Pi.

​

4. Groundwork is laid for follow-up research. The "forced fixation" function of the projection mapping \Pi essentially eliminates truth-value gaps, resolves truth-value gluts and defines indeterminate states, so as to reconstruct the Law of Excluded Middle in low-dimensional spaces.

The three non-classical states in the high-dimensional truth space are natural manifestations when logic is extended to high-dimensional domains, rather than theoretical defects. The classical Law of Excluded Middle is not an inherently universal logical principle, but a low-dimensional rule formed after dimensional reduction and projection. Clarifying the characteristics and rules of the three non-classical states is a critical step toward exploring the geometric origin of the Law of Excluded Middle.

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. Axiomatic System of the MOC Multi-Origin Geometric Logic Framework[R]. Preprint, 2026.

[3] Kripke S. Outline of a theory of truth[J]. Journal of Philosophy, 1975.

[4] Priest G. What is so bad about contradictions?[J]. Journal of Philosophy, 1998.