# ChatGPT Answer Gentzen's natural deduction is a specific formal system of logic introduced by Gerhard Gentzen in the 1930s. It's often discussed in the context of mathematical logic and proof theory. Here's how it stands in relation to the general concept of natural deduction: ### General Concept of Natural Deduction Natural deduction is a logical framework where the emphasis is on deriving conclusions from premises in a way that closely resembles natural reasoning. The idea is to reflect the actual process of human logical thinking, emphasizing the structure of arguments more transparently than other formal systems like axiomatic systems. Natural deduction systems typically use a set of inference rules for introducing and eliminating logical connectives (like "and", "or", "if-then", etc.). ### Gentzen's Natural Deduction Gentzen's contribution to natural deduction was through his systems known as "Natural Deduction" (or "NJ" for intuitionistic logic and "NK" for classical logic). His approach is particularly noted for: - **Introduction and Elimination Rules**: Each logical operator has two main types of rules — introduction rules, which allow you to introduce the operator into logical proofs, and elimination rules, which allow you to remove or use the operator to derive further conclusions. - **Sequents**: Although sequents are more prominently used in his other system, "Sequent Calculus," they also help in understanding the flow of logic in natural deduction by showing how conclusions depend on sets of premises. - **Normalization Theorem**: One of Gentzen’s significant contributions through his natural deduction system was the normalization theorem, which states that any proof can be converted into a normal form where the introduction of a logical connective immediately precedes its elimination. ### Differences and Significance While the general concept of natural deduction aims at a broad method of logical reasoning that mimics natural thought processes, Gentzen’s natural deduction specifies a particular set of rules and structures designed to enhance the rigor and clarity of mathematical proofs. His methods help in understanding the underlying structure of proofs and contribute to meta-mathematical investigations such as consistency proofs for arithmetic systems. Gentzen's work on natural deduction has profoundly influenced the development of logic, particularly in areas like proof theory, where his ideas on the structuring of proofs and the handling of logical operations have laid foundational groundwork. His approach provides tools not just for theoretical investigations but also practical applications in computer science, especially in areas like type theory and programming languages semantics. # Gemini Answer There is a difference between Gentzen's Natural Deduction and the general use of the term "[[Natural Deduction]]." **Gentzen's Natural Deduction:** - **Origin:** Introduced by [[Gerhard Gentzen]] in 1934 in his landmark paper "Investigations into Logical Deduction." - **Formal System:** A specific proof calculus designed to closely resemble the way mathematicians naturally reason in proofs. - **Key Features:** - **Introduction and Elimination Rules:** Each logical connective (and, or, not, implies, etc.) has corresponding rules for introducing it into a proof and eliminating it from a proof. - **Tree-like Structure:** Proofs are often represented as tree-like structures, where the premises are at the top and the conclusion is at the bottom. - **Subproofs:** Allows for temporary assumptions within a proof, which are later discharged. - **Purpose:** - To provide a foundation for the study of logical deduction and proof theory. - To study the relationship between classical and intuitionistic logic. **General Use of "Natural Deduction":** - **Broader Meaning:** Refers to a family of proof systems that share the goal of modeling "natural" mathematical reasoning. - **Variations:** Different systems may have slightly different rules and conventions, but they generally adhere to the spirit of Gentzen's approach. - **Examples:** - Fitch-style natural deduction: A popular variation used in many introductory logic courses. - Sequent calculus: A related proof system that can be seen as a generalization of natural deduction. - **Purpose:** - To teach logic in a way that is intuitive and easy to understand. - To provide a framework for formalizing proofs in various areas of mathematics and computer science. **Key Differences:** - **Specificity:** Gentzen's Natural Deduction is a specific formal system, while the general term "Natural Deduction" encompasses a wider range of systems. - **Notation:** Gentzen's original notation used tree-like structures, but other systems may use different notations (e.g., Fitch-style uses vertical lines and indentation). - **Rules:** While all natural deduction systems share the concept of introduction and elimination rules, the specific rules may vary slightly between systems. **In Summary:** Gentzen's Natural Deduction was a groundbreaking development in logic and proof theory. It laid the foundation for the modern understanding of "natural" deduction and inspired the development of many related proof systems. While there are differences between Gentzen's original system and the broader use of the term, they all share the common goal of formalizing mathematical reasoning in a way that is intuitive and easy to understand. I hope this explanation is helpful! Please let me know if you have any other questions. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Natural Deduction") sort title, authors, modified, desc ```