Natural Deduction Systems: Assumptions, Implications, and Proofs

Natural Deduction Systems: Unraveling Assumptions, Implications, and Proofs

Natural deduction (ND) is a powerful approach to proving theorems in logic, particularly in the realm of propositional logic and beyond. In this article, we'll explore the fundamentals of natural deduction systems, focusing on assumptions and implications, inference rules, and proof strategies.

Assumptions and Implications

In ND, we make use of assumptions, also known as hypothesis or premises, which are considered true for the duration of a given proof. Implications play a central role in ND, representing the "if... then" relationship. The key inference rule for implications is modus ponens, which states that if you have an implication (P → Q) and know that the antecedent (P) is true, then the consequent (Q) must also be true.

Inference Rules

Natural deduction systems do not rely on truth tables, but rather on a set of rules that dictate when we can draw conclusions based on assumptions and implications. Some common inference rules for propositional logic include:

1. Modus Ponens: P → Q, P ⊸ Q (if P implies Q, and P is true, then Q is true)
2. Conjunction Introduction (∧-I): P, Q ⊸ P ∧ Q (if P and Q are both true, then P ∧ Q is true)
3. Conjunction Elimination (∧-E): P ∧ Q ⊸ P, P ∧ Q ⊸ Q (if P ∧ Q is true, then both P and Q must be true)
4. Disjunction Introduction (∨-I): P ⊸ P ∨ Q, Q ⊸ P ∨ Q (if P is true, or Q is true, then P ∨ Q is true)
5. Disjunction Elimination (∨-E): P ∨ Q, ¬P ⊸ Q (if P ∨ Q is true, and not-P is true, then Q is true)

Proof Strategies

When using natural deduction, we aim to construct a proof that is valid and complete, meaning that it correctly demonstrates the truth of the desired conclusion given the assumptions. There are several strategies that we can use to achieve this.

1. Use of introduction and elimination rules: We introduce new propositions using introduction rules and eliminate them using elimination rules.
2. Use of discharging assumptions: We assume premises and use them to derive conclusions, discharging them once we no longer need them.
3. Use of subproofs: We can create subproofs to demonstrate parts of a proof and then use the conclusions of those subproofs in the main proof.
4. Use of proof by contradiction (reductio ad absurdum): We assume the negation of the conclusion and try to derive a contradiction, which, by the principle of explosion, implies the desired conclusion.

Examples

Let's consider a simple example to illustrate these concepts.

Proof: To prove (P → Q) → (¬Q → ¬P)

1. Assume: P → Q 2. Assume: ¬Q

3. Assume: P (from 1, using ∨-E with P ∨ ¬P) 4. Derive: Q (from 1 and 3, using modus ponens) 5. Conclude: ¬Q (from 2, using ¬-I with Q ∸ ¬Q) 6. Contradiction: P & ¬P (from 3 and 5, using ∧-I) 7. Conclude: ¬(P & ¬P) (from 6, using ¬-I with P ∸ ¬(P & ¬P))

8. Conclude: (¬(P & ¬P) → (P → Q)) (from 3–7, using modus ponens)

9. Conclude: (¬Q → ¬(P & ¬P)) (from 2 and 8, using modus ponens)

10. Conclude: (¬Q → (P → ¬P)) (from 9, using ¬-E with ¬(P & ¬P ∸ P → ¬P))

11. Conclude: (P → Q) → (¬Q → ¬P) (from 1 and 10, using modus ponens)

With this simple example, we've demonstrated how natural deduction can be used to construct proofs in propositional logic. By building on this foundation, we can then explore more complex proofs in first-order logic and beyond.