Natural Deduction in Propositional Logic: A Step-by-Step Guide

Natural Deduction in Propositional Logic: A Step-by-Step Guide

Natural Deduction systems are a popular approach to proving theorems in formal logic, particularly in Propositional Logic. We'll explore the basics of Natural Deduction, its proof rules, and the essential concepts of assumptions and inference rules.

Propositional Logic

Propositional logic deals with statements that can be either true or false, referred to as "propositions." It is the foundation of Natural Deduction, where we use symbols (\land) (and), (\lor) (or), (\lnot) (not), (\rightarrow) (implies), and (\leftrightarrow) (equivalent) to build logical expressions and derive conclusions.

Proof Rules

Natural Deduction employs a set of inference rules to construct formal proofs. Some fundamental rules include:

1. Assumption: A proposition can be assumed for the current sequence of inferences. This proposition can be used as a premise for other inferences, but it must be discharged (or eliminated) before the end of the proof.
2. Modus Ponens: If we have assumptions (A) and (A\rightarrow B), then we can infer (B).
3. Implication Introduction: If we prove (A) and (A\rightarrow B), then we can infer (B).
4. Disjunction Elimination: If we have a disjunction (A\lor B), we can either assume (A) (using the assumption rule) and prove (B) or assume (B) and prove (B).
5. Conjunction Elimination: If we have a conjunction (A\land B), then we can prove (A) and prove (B).
6. Negation Introduction: If we can prove that (A) leads to a contradiction, then we can infer (\lnot A).
7. Negation Elimination: If we have (\lnot A), then we can prove that (A) leads to a contradiction.

These rules, combined with the axioms of propositional logic, allow us to construct proofs.

Assumptions

Assumptions are fundamental to Natural Deduction. They are used to derive conclusions by making temporary assertions that can be discharged before the end of the proof. Assumptions are written as a horizontal line ((\hline)) followed by the proposition being assumed.

Inference Rules

Inference rules are the building blocks of Natural Deduction proofs. They help us manipulate our assumptions and our logical expressions to derive desired conclusions.

Natural Deduction proofs are typically constructed by using inference rules to manipulate assumptions, until we have proven a desired conclusion. The proof is presented in a step-by-step format, where each step shows how an inference rule was applied to the existing assumptions and conclusions.

Example Proof in Natural Deduction

Here is a simple example of a Natural Deduction proof:

1. (A \land B ) (Assumption)
2. (A ) (from 1, using Conjunction Elimination)
3. (B ) (from 1, using Conjunction Elimination)
4. (A \rightarrow C ) (Assumption)
5. (C ) (from 2, 4, and Modus Ponens)
6. (A \lor B ) (Assumption)
7. (A ) (from 6, using Disjunction Elimination)
8. (C ) (from 5 and 7, using Modus Ponens)
9. (B \rightarrow C ) (Assumption)
10. (C ) (from 3, 9, and Modus Ponens)
11. (C ) (from 6, using Disjunction Elimination)
12. (\Box) (Proof Complete, using 2, 5, 8, and 10 to derive (C))

In this example, we've used the Assumption rule to introduce (A \land B), (A \rightarrow C), (A \lor B), and (B \rightarrow C). We then used the Modus Ponens, Implication Introduction, and Disjunction Elimination rules to derive (C). The proof is complete when we have shown (C) without relying on any assumptions.

Natural Deduction provides a versatile, intuitive, and easily understood framework for proving theorems in Propositional Logic. With practice, you'll find that Natural Deduction can help you reason through complex logical problems with confidence and clarity.