Translate each of the following arguments into quantified form and prove that each is valid using natural deduction. The letters that follow each argument give the predicate letters to use in symbolizing the argument.

5. If all store supervisors are wise, then some employees benefit. If there are some store supervisors who are not wise, then some employees benefit. As you can see, either way, some employees benefit. (S, W, E, B)

6. If someone studies philosophy, then all students benefit. If someone studies literature, then there are some students. So if someone studies philosophy and literature, then someone benefits. (P, B, L, S)

7. Everyone is a Democrat or a Republican, but not both. If someone is a Democrat, then she is a liberal or a conservative. All conservatives are Republican. So all Democrats are liberal. (D, R, L, C)

8. If there are any mavericks, then all politicians are committed to change. If there are any politicians, then anyone who is committed to change is pandering. So, if there are any mavericks, politicians are pandering. (M, P, C, A (for “pandering”))

9. If everyone is a liberal, then no one is a conservative. There is a governor of Alaska and she is a conservative. So at least someone is not a liberal. (L, C, G)