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On the Essential Identity of Binary and the Law of Excluded Middle and Its Core Significance

Abstract

This paper aims to clarify the intrinsic relationship between binary (the binary numeral system) and the law of excluded middle (LEM). Breaking away from the traditional view that merely treats their relationship as a matter of instrumental convenience, this paper demonstrates, from the three aspects of logical essence, structural core, and underlying origin, the essential identity of binary and LEM, and clarifies the core academic significance behind this identity. The study abandons superficial application-level interpretations, returns to logical ontology and geometric origins, and reveals that binary is not an artificially invented independent encoding rule but rather a symbolic and materialized expression of LEM. The two share the same underlying logical kernel. This thesis serves as a key entry point for reconstructing the hierarchical relationship between mathematical logic and encoding systems.

Keywords

Binary; law of excluded middle; two-valued logic; essential identity; MOC framework

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I. Introduction

In mathematical logic and digital encoding systems, binary, as a universal fundamental encoding form, has always shown a high degree of compatibility with the law of excluded middle (LEM) in classical two-valued logic. However, academia has long defined their relationship only at the instrumental level, i.e., "using the symbols 0 and 1 to represent logical truth and falsehood," without ever deeply exploring the essential reason for their compatibility, let alone uncovering the core academic value behind this relationship.

In fact, binary and LEM are by no means coincidentally compatible or merely borrowing from each other; they are a logical symbiosis sharing the same origin and identical kernel. Based on logical ontology and the MOC multi-origin geometric logic framework, this paper systematically explains the essential mechanism of their identity, and argues that revealing this essential identity is the core significance for clarifying the origin of two-valued logic, defining the jurisprudential positioning of binary, and reconstructing the relationship between the underlying logic and encoding systems.

II. The Core Essence of the Law of Excluded Middle

The law of excluded middle is the core axiom of classical two-valued logic. Its essential connotation can be distilled as follows: within the same logical system, a proposition is either true or false; there is no intermediate state that is neither true nor false; there is no third logical value that is fuzzy, indeterminate, or superposed.

LEM delimits the boundary of classical logic and establishes an absolute two-valued rule of "either this or that." It is the foundational cornerstone of all classical mathematical logic and deterministic reasoning. It rejects all intermediate, fuzzy, or undecidable states, and is the ultimate expression of two-valued logical rules.

III. The Inner Logic of Binary and Its Essential Identity with LEM

(I) The Formal Kernel of Binary

Binary is a counting and encoding system containing only two basic symbols, 0 and 1. Its core rule is: any coding digit can take only one of the two values, 0 or 1; there is no third symbol or third state between 0 and 1; no intermediate value, no fuzzy interval, no superposition. It is an absolute two-valued symbolic system.

(II) Core Manifestations of Their Essential Identity

1. Identical Kernel Rules
LEM's "either true or false, no intermediate state" and binary's "either 0 or 1, no intermediate value" are two expressions of the same logical rule: LEM is an abstract logical axiom, binary is a concrete symbolic encoding. They share the underlying logical kernel of "absolute two-valuedness, rejection of intermediate states." There is no essential difference.
2. No Temporal or Ontological Priority, Only Morphological Differentiation
It is not that LEM came first and binary was subsequently adapted to it, nor that binary came first and logical rules were subsequently applied to it. They are symbiotically co-originating: LEM is the abstract-level two-valued logical law, while binary is the symbolic implementation and material carrier of this law. They are different concretizations of the same logical essence in different domains.
3. Complete Synchrony in Failure
When LEM fails (giving rise to undecidability, superposition states, intermediate truth values), binary simultaneously fails to represent the new logical state with 0 and 1. Conversely, when binary can only describe a system by 0 and 1, LEM must strictly hold. Their validity and failure are perfectly synchronized, further confirming their essential identity and co-existence and co-extinction relationship.

IV. The Underlying Origin of the Essential Identity of Binary and LEM

Relying on the MOC multi-origin geometric logic framework, the ultimate origin of their identity can be precisely revealed:

Both binary and LEM are born in a single-origin, low-curvature, flat geometric space. This space possesses absolute symmetry and regularity, with no structural breakdown or local bending. It naturally gives rise to the absolute two-valued logical rule of "either this or that," which is abstractly expressed as LEM and symbolically expressed as binary.

In other words, the single-origin, low-curvature geometric structure is the common original matrix of the two. LEM is its abstract logical expression, and binary is its symbolic encoding expression. This is the ultimate reason for their essential identity.

V. The Greatest Significance of Revealing Their Essential Identity

Revealing the essential identity of binary and LEM is by no means a trivial clarification of logical associations. Its greatest significance lies in the following:

1. Breaking Traditional Misconceptions
It completely overturns the fragmented view that "binary is a human invention, while LEM is a transcendental axiom," establishing that they are "co-originating and share the same kernel." It puts an end to the superficial, instrumental interpretation of their relationship in academia.
2. Clarifying Their Jurisprudential Positioning
It makes clear that binary is not an independent, fundamental encoding truth, but merely a symbolic derivative of LEM; and that LEM is not a universal logical axiom, but only a local logical product of a specific geometric space. This provides the core theoretical basis for de-sacralizing and de-absolutizing both.
3. Establishing a Unified Path for Tracing the Origins of Logic and Encoding
For the first time, it establishes a common origin-tracing channel for mathematical logic and digital encoding systems, bringing the previously separate disciplines of logic and encoding under the same underlying geometric framework. This lays the essential theoretical foundation for interdisciplinary system reconstruction and for the construction of higher-dimensional non-two-valued logics and novel encoding systems.
4. Highlighting the Paradigm-Shifting Value of Higher-Dimensional Frameworks
Through their identity, it conversely demonstrates that a higher-dimensional, multi-origin, high-curvature logical system that transcends the two-valued limitation is the more universal underlying rule, while binary and LEM are merely low-dimensional local special cases. This provides crucial support for paradigm innovation.

VI. Conclusion

Binary and the law of excluded middle are not in a merely instrumental relationship as long assumed by academia. They possess an absolute identity, sharing the same origin, the same kernel, and co-existing and co-extincting. Both are products of two-valued logic derived from a single-origin, low-curvature, flat geometric space, differing only in form as abstract logical axiom versus concrete symbolic encoding, with no essential logical distinction.

Through ontological logical argumentation and geometric tracing within the MOC framework, this paper not only clarifies for the first time the essential mechanism of their identity, but also confirms their core positioning as low-dimensional, local, special cases of logic. It both affirms the applied value of binary in the encoding field and of LEM in classical mathematical logic, and clarifies their jurisprudential boundaries and scope of application.

From the perspective of academic development, the explorations of binary and its logical connection by pioneers such as Leibniz remained at the level of superficial association and naive conjecture. This paper achieves a leap from "intuitive association" to "essential demonstration," providing a unified underlying origin for two-valued logic and digital encoding systems. It also supplies key empirical support for the MOC multi-origin geometric logic framework to subsume and integrate traditional logic and encoding systems, and thus has fundamental theoretical significance for reconstructing underlying mathematical logic and breaking through the limitations of the two-valued paradigm.

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