Natural Deduction Systems: A Step-by-Step Guide to Proofs

Natural Deduction Systems: A Step-by-Step Guide to Proofs

Natural deduction systems are a powerful and popular approach to proving theorems in formal logic. Unlike traditional deductive systems, natural deduction eschews the use of formal axioms and focuses instead on building proofs in a more intuitive manner, following the flow of logical reasoning.

In this article, we'll explore the key components of natural deduction systems—proof rules, assumptions, propositional logic, and inference rules—to better understand how natural deduction works.

Proof Rules

Natural deduction systems are founded on a set of proof rules that determine how we may move from one statement to another in a valid proof. These rules are derived from the logical connectives, such as:

• Conjunction: (A ∧ B \to A) and (A ∧ B \to B).
• Disjunction: (A \lor B, ¬A \to B), and (B, ¬B \to A).
• Implication: (A, A \to B \to B).
• Negation: (A \to ¬(¬A)).

Natural deduction employs these rules to build proofs step by step, adhering to a set of inference rules, which we'll discuss below.

Assumptions

Natural deduction systems rely on the use of assumptions, which are temporary statements that can be used to derive new conclusions. These assumptions are denoted by a horizontal line ((\ \vdash\ )) followed by a statement (or multiple statements). For example:

[ A \vdash B \to A ]

This indicates that (A) is an assumption, and we can use this assumption to prove (B \to A). Once an assumption is no longer needed, it is canceled, and its truth value is unchanged.

Propositional Logic

Propositional logic is the foundation of natural deduction, providing the basic rules and structures for building proofs. Propositional logic deals with statements that are either true or false, and it employs the logical connectives mentioned above to construct more complex statements from simpler ones.

Inference Rules

Inference rules are the procedures used to move from one statement to another in a natural deduction proof. These rules can be roughly divided into two categories: elimination rules and introduction rules.

Elimination rules remove a particular logical connective, while introduction rules introduce a new logical connective. For instance, the elimination rule for the (\wedge) (and) connective is:

[ \dfrac{A \vdash B}{A \wedge B \vdash C} ]

This rule allows us to eliminate the (\wedge) connective by proving a statement based on two assumptions. On the other hand, the introduction rule for the (\wedge) connective is:

[ \dfrac{A \vdash C}{A \wedge D \vdash C} \qquad \dfrac{D \vdash C}{A \wedge D \vdash C} ]

This rule allows us to introduce the (\wedge) connective by proving a statement based on two independent assumptions.

Examples and Practice

Let's consider a proof in natural deduction using the following assumptions and rules:

[ \begin{align*} A \vdash A \to B \ B \vdash B \to C \ A \vdash C \end{align*} ]

Our goal is to prove (A \vdash C). Here's a proof using the rules and inference rules mentioned above:

[ \begin{align*} & A \vdash A \to B \ & \downarrow \ & A, B \vdash C \ & \downarrow \ & A \vdash B \to C \ & \downarrow \ & A, A \to B \vdash B \to C \ & \downarrow \ & A \vdash (A \to B) \to (B \to C) \ & \ & A \vdash A \to C \ & \downarrow \ & A, A \to C \vdash C \ & \downarrow \ & A \vdash (A \to C) \to C \ & \ & A \vdash (A \to B) \to (B \to C), (A \to C) \to C \ & \downarrow \ & A \vdash (A \to B) \to ((B \to C) \to C) \ & \downarrow \ & A \vdash (A \to B) \to (A \to (B \to C) \to C) \ & \downarrow \ & A, A \to B \vdash A \to (B \to C) \to C \ & \downarrow \ & A \vdash A \to (A \to C) \ & \downarrow \ & A \vdash A \to C \end{align*} ]

This proof demonstrates how to use natural deduction to prove a statement step by step, starting with our assumptions and using the elimination and introduction rules of propositional logic.

Conclusion

Natural deduction systems provide a powerful and intuitive approach to proving theorems in formal logic. By understanding proof rules, assumptions, propositional logic, and inference rules, we can build formal proofs following a logical flow, effectively communicating our reasoning and demonstrating the validity of our conclusions.