319 Answers and Solutions to Selected Exercises ∃xL(x) ∧ ¬∀xG(x) : T , ¬∀x(L(x) → G(x)) : F ∃xL(x) : T , ¬∀xG(x) : T L(c1 ) : T ∀xG(x) : F G(c2 ) : F ∀x(L(x) → G(x)) : T L(c1 ) → G(c1 ) : T L(c1 ) : F × G(c1 ) : T L(c2 ) → G(c2 ) : T L(c2 ) : F G(c2 ) : T × A counter-model extracted from the tableau: a structure with domain {a1 , a2 } where c1 is interpreted as a1 , c2 is interpreted as a2 , and both L and G are interpreted as {a1 } Therefore, A ≡ B (c) Using predicates T (x) for “x is talking” and L(x) for “x is listening,” we can formalize A and B as follows: A : ∃xT (x) → ∀xL(x), and B : ∃x¬L(x) → ¬∃xT (x) Checking the validity of both logical consequences: