285 Answers and Solutions to Selected Exercises 2.4.6 (a) [¬r] ¬r ∨ p [¬p] , [¬p → r] r ⊥ p p → ¬q ¬q ¬q (¬p → r) → ¬q [p]2 p → ¬q ¬q 2.4.7 We formalize each of the propositional arguments by identifying the atomic propositions in them and replacing them with propositional variables For a selection of them, we then will prove the soundness of the resulting inference rule by deriving it in ND (b) Denote “Nina will go to a party” by p, and “Nina will go to office” by q Then the argument becomes: p ∨ ¬q, ¬p ∨ ¬q ¬q The rule is derivable in ND and therefore sound, so the argument is correct ¬q ∨ p [¬q ]2 ¬q ¬q ∨ ¬p [¬q ] ¬q ¬q [¬p]1 , [p]2 ⊥ ¬q ¬q (d) Denote “Socrates is happy” by p, “Socrates is stupid” by q , and “Socrates is a philosopher” by r Then the inference rule on which the argument is based is: p ∨ ¬q, p → ¬r r → ¬q The rule is derivable in ND and therefore sound, so the argument is correct p ∨ ¬q [p]1 , p → ¬r r]2 ¬r, ⊥ ¬q ¬q r → ¬q [¬q ]1 ¬q