Chapter 5 The Emergence of Binary: Symbolization of the Law of Excluded Middle

— Binary is the minimal symbol set of the law of excluded middle, and numeral systems are counting modes after projection

Author: Zhang Suhang (Luoyang, Henan)

Abstract

This is the fifth paper in the series Geometric Origin of the Law of Excluded Middle. It mainly demonstrates that binary is the unique minimal symbol set naturally emerging from the law of excluded middle via projective mapping. Based on the high-dimensional "OR" state, high-dimensional truth space \mathcal{T}, projective mapping \Pi (state collapse) and the emergence of the law of excluded middle in the image space established in the previous four papers, this paper strictly proves that the image space after projection constitutes a two-valued judgment system, which must and can only be symbolized with two symbols, namely binary. Binary is not a man-made invention, but an inevitable result of the symbolization of the law of excluded middle. No other numeral systems are discussed herein.

Keywords: Binary; Law of Excluded Middle; Symbolization; Projective Mapping; Minimal Symbol Set

1. Introduction

In classical logic, symbols 0 and 1 are used to denote "false" and "true", which is generally regarded as a conventional convenience. Nevertheless, the MOC geometric framework reveals that this convention is essentially an inevitable outcome of geometric projection.

Conclusions from previous papers are summarized as follows:

- The law of excluded middle does not hold in the high-dimensional original space.

- The projective mapping \Pi collapses the high-dimensional truth space \mathcal{T} into the two-valued image space \{0, 1\}.

- The law of excluded middle P \lor \neg P is universally valid in the image space.

This paper focuses on the symbolization of the two-valued image space. It is proven that any symbol set capable of fully representing a two-valued judgment system must contain exactly two elements. Such minimal symbol set is binary. Binary serves as the symbolic embodiment of the law of excluded middle, rather than an optional auxiliary tool.

2. Structure of the Two-Valued Image Space

2.1 Algebraic Properties of the Image Space

The projective mapping \Pi: \mathcal{T} \to \{0, 1\} has the range \{0, 1\}. The classical negation operation is defined on this set as:

\neg 0 = 1,\quad \neg 1 = 0

It forms a two-order Boolean algebra, which is the simplest non-trivial Boolean algebra. Its truth table is completely determined by the two elements and the negation operation.

2.2 Necessity of Symbolization

The image space is the intrinsic representation of logical truth values. For recording, communication and computation, it is necessary to map intrinsic truth values onto physical symbols such as carvings, voltage levels and optical signals. This mapping process is defined as symbolization:

\sigma: \{0,1\} \to S

where S denotes a symbol set, and \sigma is a bijection that preserves the original truth value structure.

3. Uniqueness of the Minimal Symbol Set

3.1 Lower Bound of Symbol Quantity

If the cardinality of the symbol set satisfies |S| = 1, the single symbol cannot distinguish the two states 0 and 1. Hence we have |S| \geq 2.

3.2 Upper Bound under the Minimal Completeness Constraint

The image space contains only two logical states. If |S| > 2, redundant symbols will arise: multiple symbols map to the same truth value, or some symbols remain unused. This setting does not violate logical rules, yet it fails to satisfy the definition of a minimal complete set. Under the constraint of zero redundancy, the cardinality of the minimal symbol set is |S| = 2.

3.3 Isomorphism of All Binary Symbol Sets

Let S = \{a,b\}, and define the bijection \sigma(0)=a,\ \sigma(1)=b. For any other binary symbol set S' = \{c,d\}, an equivalent mapping can be realized by relabeling elements. From the perspective of logical representation, all binary symbol sets are isomorphic.

3.4 Selection of Specific Symbols

Symbol forms including 0/1, T/F and +/- are products of historical contingency. The mathematical and computational community adopts 0 and 1 primarily because they are compatible with arithmetic operations and correspond to physical states of circuits. The core research focus lies in the binary nature rather than specific symbol shapes. The two-valued judgment derived from the projected law of excluded middle inherently requires a minimal binary symbol set.

4. Homology and Symbiosis between Binary and the Law of Excluded Middle

4.1 Homology

- Law of Excluded Middle: P \lor \neg P holds universally in the image space.

- Binary: the minimal symbol set \{0,1\} of the two-valued image space together with the negation operation.

Both originate from the collapse of high-dimensional "OR" states induced by the projective mapping \Pi. Without projection, there would be no two-valued image space, and accordingly no law of excluded middle or binary system.

4.2 Symbiosis

The validity of the law of excluded middle relies on the two-valued structure of the image space, and the application of binary symbol sets is premised on the existence of such image space. The two are interdependent and inseparable. Retaining the law of excluded middle while adopting a ternary symbol set such as \{0,1,2\} will introduce redundancy and break the concise algebraic form of the negation operation. Applying binary to encode non-two-valued logic will disconnect symbols from the negation operation, and eliminate the natural link between binary and classical logic.

In essence, binary and the law of excluded middle are two manifestations of the same geometric projection in the symbol domain and the logical domain respectively.

5. Conclusions

This paper completes the core arguments of the fifth chapter in the series Geometric Origin of the Law of Excluded Middle, with main conclusions as follows:

1. The projective mapping generates the two-valued image space \{0, 1\}, where the law of excluded middle holds universally.

2. A symbol set that fully characterizes the two-valued image space must have exactly two elements, which ensures completeness.

3. Such minimal binary symbol set is binary. Specific symbol forms are arbitrary choices, while the binary attribute is logically inevitable.

4. Binary and the law of excluded middle share the same origin, both evolving from the projection and collapse of high-dimensional "OR" states.

Binary is not an optional tool invented by humans, but an inevitable symbolic form derived from the law of excluded middle via projective mapping. Every application of 0 and 1 records a collapse process from coexistent high-dimensional states to definite low-dimensional states.

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.

End of Chapter 5