Answers and Solutions to Selected Exercises 265 1.1.9 (d) If ((p → q ) ∨ (p → ¬q )) → ¬p is not a tautology, then: ((p → q ) ∨ (p → ¬q )): T and ¬p: F [p → q : T or p → ¬q : T] and p: T [p → q : T and p : T] or [p → ¬q : T and p: T] [q : T and p: T] or [q : F and p: T] The formula is not a tautology Either of the assignments p : T, q : T or p : T, q : F will render the formula false (j) In order to falsify the formula ((p ∨ q ) → r) → ((p → r) ∧ (q → r)), we must have (p ∨ q ) → r : T and (p → r) ∧ (q → r) : F For (p ∨ q ) → r to be true, there are two possible cases: Case 1: p ∨ q : F Then p : F and q : F But then p → r : T and q → r : T so that (p → r) ∧ (q → r ) : T, a contradiction with (p → r) ∧ (q → r) : F Case 2: r : T Then p → r : T and q → r : T so that (p → r) ∧ (q → r) : T, a contradiction with (p → r) ∧ (q → r ) : F All attempts to falsify the formula end in contradictions, so the formula is a tautology (l) If ((p → r) ∨ (q → r )) → ((p ∨ q ) → r) is not a tautology then: ((p → r) ∨ (q → r)) : T and (p ∨ q ) → r: F [p → r: T or q → r: T] and [p ∨ q : T and r: F] [p → r: T and p ∨ q : T and r : F] or [q → r: T and p ∨ q : T and r : F] [p: F and r : F and p ∨ q : T] or [q : F and r: F and p ∨ q : T] [p: F and q : T and r: F] or [p: T and q : F and r: F] 1.1.10 (a) (b) (c) (d) The formula is not a tautology For example, the assignment p : F, q : T, r : F will render it false P cannot be a liar, otherwise he would be telling the truth So, P is a truth-teller and therefore Q must be a liar If the answer was “yes,” the inhabitant he asked is a truth-teller and the other inhabitant a liar If the answer was “no,” then either both are truth-tellers or the one who answered was a liar, so in this case the stranger would not be able to determine who is a liar and who is not The answer must therefore have been “yes” If A is a liar, then his claim must have been false, therefore B must be a liar while A is not So, A cannot be a liar Therefore, B cannot be a liar either Therefore both A and B are truth-tellers Note that no local can say that he is a liar, as this would result in the Liar’s paradox; Y’s statement must therefore be false, making him a liar But this means that Z’s statement is true, so he is a truth-teller Since X’s answer invariably must be “No, I’m a truth-teller,” it is not possible to determine whether X is a truth-teller or liar