‍Abstract

This paper explores the philosophical and computer-theoretical evolution from formal logic to dialectical logic, and uses Bitcoin as an example to reveal how it transcends consistency systems and tends toward completeness systems.

The article first compares formal logic with dialectical logic, then introduces the correspondence between the Turing machine system and the Turing ordinal logic system, and finally analyzes how Bitcoin, through transaction verification, block selection, and the longest chain principle, achieves the unification of consistency and completeness, thereby constructing a “super-formal system.”

1. Logic Layer: Formal Logic and Dialectical Logic

• Formal Logic: Aims at consistency, follows the law of identity, the law of excluded middle, and the law of non-contradiction. It rejects the existence of contradictions and emphasizes the reliability of determinate reasoning.
• Dialectical Logic: Aims at completeness, allows the existence of contradictions, and emphasizes the unity of opposites. Contradictions are not regarded as errors but as the driving force of system evolution.
• Relation: Formal logic is only a proper subset of dialectical logic. Dialectical logic not only encompasses contradiction-free consistent logic, but also includes inconsistent logic of contradictory opposition and dynamic unity.
  

Illustrative diagram of inclusion relations:

2. Computer Theory Layer: Turing Machine System and Ordinal Logic System

• Turing Machine System: As a formalized computational model, it solves consistency-computable problems. Although it can describe deterministic computation, it is always constrained by Gödel’s incompleteness, unable to resolve undecidable problems within the system.
• Ordinal Logic System: Proposed by Turing in his doctoral dissertation, it extends the reasoning capacity of the Turing machine through “transfinite iteration” and the “oracle machine.” This system allows continued evolution and adjudication within the boundaries of contradiction and undecidability, embodying the spirit of dialectical logic.
• Comparison:
  • Turing machine system corresponds to the “consistency” of formal logic.
  • Oracle machine system corresponds to the intuitive contradiction of non-formal systems.
  • Ordinal logic system corresponds to the “completeness” of dialectical logic.
    

Illustrative diagram of inclusion relations:

3. Blockchain: Bitcoin as a Super-Formal System

Consistency Layer:

• Transaction verification (TX) = operation of the Turing machine system, ensuring that each transaction locally satisfies consistency.
• The block-producing process follows the principles of formal logic, avoiding contradictions and errors.
  

Completeness Layer:

• The block selection mechanism (fork resolution) is equivalent to the function of the oracle machine, resolving fork problems at the same height.
• The longest chain principle and PoW (Proof of Work) together form the process of “transfinite iteration,” enabling the system to achieve unity and evolution amidst contradictions (forks).
• The timestamp chain not only maintains consistency but also continuously generates new “Gödel information,” embodying the self-expansion of completeness.
  

Conclusion: Bitcoin is not merely a consistency verification system; by introducing mechanisms that transcend formal logic, it constructs a “super-formal system” with dialectical logic characteristics.

Summary

Through a three-layer comparison (logic, computational theory, and blockchain practice), this paper reveals the relationship between formal logic and dialectical logic, and their concretization in computer science and the Bitcoin system.

Formal logic guarantees consistency but is insufficient to explain evolution and the unity of contradictions; dialectical logic pursues completeness by accommodating contradictions. Bitcoin, through transaction verification, oracle-like fork resolution, and longest-chain iteration, unifies these two logics, becoming the first “super-formal system” to operate in reality.