275 Answers and Solutions to Selected Exercises (d) The logical consequence is not valid (p ∧ q) → r) : T , (p → r) ∧ (q → r) : F p→ r:F q→ r:F p : T, r : F q : T, r : F r:T × r:T × p∧q :F p:F × q:F p∧q :F p:F q:F × (g) The logical consequence is valid p → (q ∧ r) : T , (p → q) ∧ (p → r) : F p:F × p→ q:F p→ r:F p : T, q : F p : T, r : F q∧r:T p:F × q∧r:T q : T, r : T × q : T, r : T × 2.3.3 For each of these problems we construct the tableau with all premises assigned the required truth values and the conclusion assigned the opposite truth value If the tableau closes, then such truth assignment is impossible; the stated consequence therefore holds If the tableau does not close, then the stated consequence does not necessarily hold (a) The tableau closes, so the consequence holds: P → Q : T, Q → R : T, P ∧ S : T, R ∧ S : F P : T, S : T P :F × Q:T Q:F × R:T R:F × S :F ×