72 ¬B → ¬A Logic as a Tool ND A → B: [¬B ]1 , ¬B → ¬A , [A]2 [¬A] ⊥ B (→I ) A→B Again, this derivation requires the use of (RA) A ∨ B ND ¬A → B : [¬A]1 , [A]3 ⊥ B [¬A]2 , [B ]3 A∨B ¬A → B ¬A → B ¬A → B Note the application of the rule (∨E ) (reasoning per cases) in the last step of this derivation References for further reading For more details on the theory and examples of derivations in Natural Deduction see van Dalen (1983), Jeffrey (1994) (who referred to “synthetic trees”), Smullyan (1995), Fitting (1996), Huth and Ryan (2004), Nederpelt and Kamareddine (2004), Prawitz (2006, the original development of the modern version of Natural Deduction), Chiswell and Hodges (2007), and van Benthem et al (2014), as well as Kalish and Montague (1980) and Bornat (2005), who present Natural Deduction derivations in a boxed form rather than in tree-like shape Exercises 2.4.1 Prove that all inference rules of ND are logically sound 2.4.2 Construct logically sound rules for the introduction and elimination of the biconditional ↔ 2.4.3 Show that the following tautologies are theorems of ND (Hint: for some of these you will have to use the rule (RA).) (a) (p ∧ (p → q )) → q (b) p → ¬¬p (c) ¬¬p → p (d) p ∨ ¬p (e) ((¬p → q ) ∧ (¬p → ¬q )) → p (f) ((p → q ) ∧ (p → ¬q )) → ¬p (g) ((p → q ) ∧ (q → r)) → (p → r) (h) ((p → q ) ∧ (p → r )) → (p → (q ∧ r)) (i) ((p → r) ∧ (q → r)) → ((p ∨ q ) → r) (j) p → ((q → r) → ((p → q ) → (p → r))) (k) (p → (q ∨ ¬r )) → (((q → ¬p) ∧ r ) → ¬p)