If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable.
-Which of the following propositions is an immediate (one-step) consequence in M of the given premises?
A) (Jx • Ix) • ∼Ax
B) (∀x)∼(∼Ax • Wx)
C) ∼Aa • Wa
D) (Ja • ∼Ra) ⊃ (∀x)[(Jx • Ix) ⊃ Wx]
E) (∃x)(Jx • ∼Rx) ⊃ ∼(∀x)∼[(Jx • Ix) ⊃ Wx]

Question

Quizplus · Accepted Answer

The given premises can be symbolized as:

1. Some journalists are not respectable: ∃x(Jx ∧ ¬Rx)
2. If some journalists are not respectable, then all journalists on the web have work to do: ∀x[(Jx ∧ ¬Rx) → Wx]
3. Some journalists on the web lack assignments: ∃x(Jx ∧ Wx ∧ ¬Ax)
4. It is not the case that something without an assignment has work to do: ∀x[¬(Wx ∧ ¬Ax) → ¬Wx]
Conclusion: All journalists are respectable: ∀x(Jx → Rx)

By combining premises 1 and 2, we can derive ¬(∀x(Jx → Rx)). This can be simplified to (∃xJx) ∧ (∃x¬Rx), which also follows from premise 3. From here, we can apply De Morgan's law to the second part of the derived sentence to get ∃x(Jx ∧ ¬Rx). But this is just what our first premise stated. Therefore, it is not possible to derive any new conclusion from the given premises using one step of deduction. However, by examining the form of the answer choices, we can see that only option B is a universally quantified statement that involves the negation of an antecedent and the denial of a consequent, which matches a common form of the contrapositive. We can show that it is true by contraposition:

¬(∀x)[¬(¬Ax ∧ Wx)]
∃x ¬[¬Ax ∧ Wx]
∃x (Ax ∨ ¬Wx)
∀x ¬(Ax ∨ ¬Wx)
∀x (¬Ax ∧ Wx)

The last statement means that anyone without an assignment is a journalist on the web, which is true given that our premises already imply that some journalists on the web lack assignments. Therefore, the best choice is B.

If Some Journalists Are Not Respectable, Then All Journalists on the Web