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From Hilbert to Truth: How does natural deduction unlock the unsolved mysteries of logic?
In logic and proof theory, natural deduction is a proof calculation method that uses inference rules to express logical reasoning. These rules are closely related to human beings' "natural" way of thinking. This approach contrasts with Hilbertian systems, which rely as much as possible on axioms to express the laws of logical reasoning. The development process of natural deduction reflects the dissatisfaction of the mathematics and logic circles with the traditional logic system and promotes the emergence of new thinking.
The natural deduction method makes logical reasoning more intuitive and in line with the order of human thinking.
Historical background
The birth of natural deduction can be traced back to the 1930s. Dissatisfaction with the axiomatic methods of Hilbert, Frege, and Russell led scholars to explore more natural forms of proof. Jaskoski first proposed natural deduction in 1929, but proposals at that time mainly used graphical representations. It was not until 1933 that the German mathematician Gent Dehn independently proposed the modern expression of natural deduction in his paper and coined the term "natural deduction" (natürliches Schließen), which laid the foundation for subsequent research.
Günter Deyn's motivation was to verify the consistency of numerical theory, which led him to propose the natural deductive system.
The evolution of symbols and representations
Natural deduction methods of expression have evolved over time. Ghentdairn's tree-like proof form was later improved by Yaszowsky and transformed into various nested box representations, which laid the foundation for the later Fitch notation. Many mathematics textbooks include different notation systems, which makes it difficult for readers who are not familiar with these notations to understand proofs.
Various representations make learning logical proofs more complex, but also promote deeper understanding.
The structure of natural deduction
In natural deduction, a proposition is derived from a set of premises through the repeated application of inference rules. This process emphasizes the staged and systematic nature of logical reasoning and ensures rigor in every step of the reasoning process. Many modern logical systems still benefit from natural deduction, which demonstrates its importance in the study of logic.
Stability and consistency of logical reasoning
In logic, the stability and consistency of a theory are key indicators for evaluating its importance and applicability. A theory is inconsistent if it can be proven false from no hypothesis. In contrast, completeness means that every theorem or its negation can be proved by its logical inference rules. These concepts provide the basis for a deep understanding of how logical systems work.
Consistency and completeness are not only the verification standards of theory, but also the evaluation benchmark of logical systems.
Conclusion
The development of natural deduction not only changed our understanding of logical reasoning, but also opened up new research fields. Through a reasoning system that is closer to the way humans think, scholars can explore the deep structure of logic and its application scope. Logic is no longer just an abstract mathematical symbol, but an important tool for revealing the truth. With the in-depth study of natural deduction, we cannot help but ask, how will future logic further break through the current boundaries and create new ways of thinking?
