Also see [[Gentzen's Natural Deduction]]. # ChatGPT Answer Natural Deduction is a method or system in logic used to apply reasoning to reach conclusions from premises. It reflects the way humans naturally reason. This method involves a set of rules related to each logical operator and helps in determining the validity of arguments. Natural deduction systems are used in automated theorem proving, programming language semantics, and formal verification systems. The rules of natural deduction are usually classified into introduction rules and elimination rules. # Bard Answer In the realm of logic and proof theory, **[[natural deduction]]** is a particular **[[proof calculus]]** for exploring the validity of arguments. It stands in contrast to other systems like Hilbert-style axiomatic systems, which rely heavily on axioms to express logical laws. Natural deduction takes a more "natural" approach, focusing on **inference rules** that resemble how we reason in everyday life. These rules describe valid reasoning patterns, allowing us to **deduce** new propositions from given premises. Here's a breakdown of key aspects of natural deduction: **Principles:** - **Focus on inferences:** Instead of relying on axioms, the system uses a set of basic inference rules that represent valid reasoning patterns. - **Step-by-step progression:** Proofs in natural deduction are constructed by applying these rules repeatedly, building up a sequence of steps that justify the final conclusion. - **Explicit context:** Each step in a proof refers back to specific **premises** or earlier derived conclusions, maintaining clarity and transparency. **Benefits:** - **Intuitive structure:** The inference rules closely mirror natural reasoning patterns, making them easier to understand and apply compared to axiomatic systems. - **Flexibility and creativity:** Proof construction allows for different "paths" to reach the conclusion, promoting exploration and independent thinking. - **Focus on understanding:** Emphasis on the reasoning process behind each step fosters deeper understanding of the logical relationships between propositions. **Drawbacks:** - **Complexity for some cases:** Proving certain statements can be more intricate and laborious in natural deduction compared to axiomatic systems. - **Learning curve:** Understanding the inference rules and building proofs requires initial effort and practice. - **Formal language:** Though intuitive, the system still uses a specific format and terminology with which to express proofs. **Applications:** - **Formal logic education:** Natural deduction is a popular method for teaching logic concepts due to its intuitive nature and focus on reasoning processes. - **Formal proof verification:** Tools and systems can be built to automate proof checking in natural deduction, ensuring the validity of mathematical and logical arguments. - **Artificial intelligence research:** Studying natural deduction can help inform the development of reasoning and explanation capabilities in AI systems. If you'd like to explore specific aspects of natural deduction in more detail, such as the different inference rules or examples of proof construction, feel free to ask! I'm happy to dive deeper into this fascinating logical system. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Natural Deduction" ) ```