Checking the validity of both logical consequences:.
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∃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
L(c2 → G(c2) :T
L(c2) :F G(c2) :T
×
A counter-model extracted from the tableau: a structure with domain{ a1, a2}
wherec1 is interpreted asa1,c2 is interpreted asa2, and bothLandGare interpreted as{ a1}.
Therefore,A ≡ B (c) Using predicatesT(x) for “xis talking” andL(x) for “xis 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: