71 Deductive Reasoning in Propositional Logic from the completeness of H by showing that every logical consequence that is derivable in H is also derivable in ND This will be sketched in Section 2.7 2.4.2 Examples of derivations in Natural Deduction Commutativity of the conjunction A ∧ B (∧E ) (∧I ) ND B ∧ A: A∧B A∧B (∧E ) B A B∧A Note that an assumption A ∧ B was used twice ND A → ¬¬A : [A]2 , [¬A]1 (¬E ) ⊥ (¬I ) ¬¬A (→ I ) A → ¬¬A In this derivation two additional assumptions have been made: A and ¬A The label indicates the application of the rule which allows the cancellation of the assumption ¬A, while label indicates the cancellation of A ND ¬¬A → A : [¬¬A]2 , [¬A]1 (¬E ) ⊥ (RA) A (→ I ) ¬¬A → A Note the application of (RA) in this derivation It can be proved that the derivation cannot be made without this or an equivalent rule ND (A → (B → C )) → ((A ∧ B ) → C ) : [A ∧ B ]1 , [A → (B → C )]2 A [A ∧ B ] (∧E ) (→ E ) B B→C (→ E ) C (→I ) (A ∧ B ) → C (→ I ) (A → (B → C )) → ((A ∧ B ) → C ) (∧E ) From this point onward I will usually omit the rule labels of the steps in the derivations, but the reader should be able to figure out which rule is being applied at each step A → B ND ¬B → ¬A : [¬B ]1 A → B , [¬B ]2 B ⊥ ¬A ¬B → ¬A