Natural Deduction in Propositional Logic

Natural Deduction: A Powerful Tool for Logical Proofs

Natural deduction has emerged as a popular and intuitive system for proving theorems in formal logic. This article explores the key components of natural deduction, focusing on proof strategies, assumptions and implications, and the inference rules that form the framework of propositional logic.

Proof Strategies

Natural deduction operates from a set of assumptions, aiming to establish the truth of a statement through a series of logical steps. The strategies employed in natural deduction are designed to be natural and intuitive.

• Direct proof: Construct a proof based on a direct application of the given rules to derive the conclusion.
• Indirect proof: Demonstrate the negation of the conclusion through the use of contradiction.
• Proof by cases: Split the proof into multiple cases, each based on specific assumptions, and then combine the results.

Assumptions and Implications

In natural deduction, we make use of assumptions, which are temporarily considered true for the purpose of proving a statement or theorem. These assumptions are denoted using the symbol ⊸ (implies). To resolve an assumption, we must either prove that it is true based on the given rules or contradict it.

• Assumption: Introduces a statement as true for the duration of the proof.
• Contradiction: A statement is contradicted when its negation is proven to be true.
• Resolution of assumptions: A statement is resolved when it is either proven true or its negation is proven true.

Inference Rules

Natural deduction utilizes inference rules, which are statements that guide how we derive new statements from given premises.

• Modus ponens: Derive a statement A from statements B and (B → A).
• Modus tollens: Derive a statement A → B from the negation of statement B and A.
• Disjunction elimination: Derive statement A from statement (B ∨ A) and the negation of B.
• Disjunction introduction: Derive statement (B ∨ A) from a statement A or B.
• Conjunction elimination: Derive statement A from statements (A ∧ B) and the negation of B.
• Conjunction introduction: Derive statement (A ∧ B) from statements A and B.

Propositional Logic

Natural deduction is most commonly used in propositional logic, which is concerned with propositions as atomic units of truth. Propositional logic employs the use of truth tables, which represent the truth values of statements in a given set of propositions.

• Truth tables: A tool used to determine the truth values of compound statements based on the truth values of their constituent propositions.
• Tautologies: Statements that are true in all truth assignments for their propositions.
• Contradictions: Statements that are false in all truth assignments for their propositions.

Natural deduction is a powerful tool for proving theorems in formal logic because it is intuitive and well-suited to the construction of proofs. It allows us to express and reason about complex concepts in a clear and structured manner, making it a valuable resource for understanding and working with logic in various fields.