Owen will be happier if, and only if, he either practices the cello or quits music lessons. It is not that case that Owen quits music lessons only if he'll be happier. So, it is not the case that he both never sleeps and doesn't practice.
-Which of the following is the best translation into PL of this argument?
A) H ≡ (P $ \lor $ Q) ∼(H ⊃ Q) / ∼(N • ∼P)
B) H ≡ (P $ \lor $ Q) ∼(Q ⊃ H) / ∼N • ∼P
C) H ≡ (P $ \lor $ Q) H ⊃ ∼Q / ∼(N • ∼P)
D) H ≡ (P $ \lor $ Q) ∼(Q ⊃ H) / ∼(N • ∼P)
E) H ≡ (P $ \lor $ Q) ∼(H ⊃ Q) / ∼N • ∼P

Question

Quizplus · Accepted Answer

First, we assign the following symbolization key:
H - Owen will be happier
P - Owen practices the cello
Q - Owen quits music lessons
N - Owen never sleeps

We then translate the premises of the argument using this key:
1. H ≡ (P $ \lor $ Q) - Owen will be happier if, and only if, he either practices the cello or quits music lessons.
2. ∼(H ⊃ Q) - It is not the case that Owen quits music lessons only if he'll be happier.

From these premises, we can conclude that:
3. ∼(N • ∼P) - So, it is not the case that he both never sleeps and doesn't practice.

This matches the form of answer choice D, so it is the correct choice.

Owen Will Be Happier If, and Only If, He Either ∨ \lor ∨

Owen Will Be Happier If, and Only If, He Either $\lor$