Introduction to Natural Deduction in Propositional Logic

Natural Deduction System in Propositional Logic

The Natural Deduction (ND) system is a powerful and systematic approach to proving theorems in logic. It originated in the 1930s, offering an attractive alternative to other deductive systems like Hilbert-style proof theory and Gentzen's sequent calculus. In this article, we'll dive into the core elements and strategies of the Natural Deduction system, primarily in propositional logic, to help you better understand and engage with this method of logical reasoning.

Propositional Logic

Propositional logic (PL) forms the foundation of Natural Deduction. It involves the manipulation of propositions, which are expressions that can be either true or false, using basic operators such as conjunction (∧), disjunction (∨), negation (¬), and implication (→). The ultimate goal in propositional logic is to find valid arguments, which are sequences of propositions in which each line of the argument is true, given that the premises are true.

Proof Strategies

Natural Deduction proofs are constructed by starting with an initial set of assumptions, or premises, and applying inference rules to deduce new propositions. There are three primary strategies employed in Natural Deduction: direct, indirect, and mixed.

1. Direct strategy: In a direct proof, we aim to show that the conclusion is true under the given assumptions. We start by assuming premises, apply rules to derive intermediate steps, and eventually reach the conclusion without any assumptions.

2. Indirect strategy: Indirect proofs are used to prove that a statement is false by showing that it leads to a contradiction. To prove a statement P, we assume that it is false (¬P) and derive a contradiction from this assumption. We then conclude that the original statement P must be true.

3. Mixed strategy: A mixed strategy combines direct and indirect approaches to prove that a statement P is true. We first assume ¬P and derive a contradiction from this assumption. Since we've shown that ¬P leads to a contradiction, we conclude that P must be true.

Inference Rules

Natural Deduction relies on a collection of inference rules to formally derive true conclusions. These rules are designed to mirror the intuitive logical connectives, providing a rigorous and systematic framework for logical reasoning.

1. Assumption: We can introduce a proposition in a line of a derivation assuming that it is true.

2. Implication elimination (→E): From A → B and A, we can deduce B.

3. Implication introduction (→I): Given A and B, we can derive A → B.

4. Disjunction elimination (∨E): From A ∨ B and ¬A, we can deduce B; from A ∨ B and ¬B, we can deduce A.

5. Disjunction introduction (∨I): Given A and B, we can derive A ∨ B.

6. Conjunction elimination (∧E): From A ∧ B, we can deduce A and B.

7. Conjunction introduction (∧I): Given A and B, we can derive A ∧ B.

8. Negation elimination (¬E): From ¬A, we can deduce A → B (for any B).

9. Negation introduction (¬I): Given A → B, we can derive ¬(A ∧ ¬B).

Assumptions and Premises

Assumptions and premises serve as the foundations of our Natural Deduction proofs. An assumption is a proposition that we temporarily assume to be true in a particular line of a derivation, while a premise is a proposition that we have already established as true in the context of our proof. We use the same symbols for both assumptions and premises, but we distinguish between them based on when they are introduced in our derivation.

For example, in a proof using the indirect strategy, we might assume that a statement P is false (¬P) and derive a contradiction. We then conclude that P must be true. In this case, ¬P is an assumption, while P is a premise (since we've shown it to be true).

In summary, Natural Deduction is a powerful tool for proving theorems in propositional logic. By understanding the basic strategies, inference rules, and the role of assumptions and premises, you'll be well-equipped to construct sound and rigorous Natural Deduction proofs. And remember, when working with Natural Deduction, less is more. Avoid unnecessary steps and embrace clarity and simplicity in your proofs.