I have a problem here that I'm trying to solve. The question asks me to check whether the argument is valid or not without using a truth table. I used equivalence laws to reduce the statement down to
B ∨ Q. My conclusion now is that this argument is not valid because
B ∨ Q isn't always true itself. My questions were, Is this a proper way to prove the validity? and, are my reductions correct from the equivalence laws? Thank you very much.
((B → ¬Q) ∧ ¬B) → Q ⇔ ((¬B ∨ ¬Q) ∧ ¬B) → Q (conditional law) ⇔ ¬((¬B ∨ ¬Q) ∧ ¬B) ∨ Q (conditional law) ⇔ (¬ (¬B ∨ ¬Q) ∨ B) ∨ Q (DeMorgan’s law) ⇔ ((B ∧ Q) ∨ B) ∨ Q (DeMorgan’s law) ⇔ (B ∨ (B ∧ Q)) ∨ Q (commutative law) ⇔ B ∨ Q (absorption law)