If either Nationwide's running an ad or Progressive's paying a dividend implies that Allstate hires agents, then Safeco's opening new offices is a sufficient and necessary condition for Liberty's reducing premiums.
A) [A ⊃ (N ∨ P)] ⊃ (S ≡ L)
B) (S ≡ L) ⊃ [(N ∨ P) ⊃ A]
C) (N ∨ P) ⊃ [A ⊃ (S ≡ L)]
D) [(N ∨ P) ⊃ A] ⊃ (S ⊃ L)
E) [(N ∨ P) ⊃ A] ⊃ (S ≡ L)

Question

Quizplus · Accepted Answer

We can break down the given statements into propositional logic statements as follows:

- Nationwide's running an ad: N
- Progressive's paying a dividend: P
- Allstate hires agents: A
- Safeco's opening new offices: S
- Liberty's reducing premiums: L

From the given information, we can infer that:

- A ⊃ (N ∨ P): If Allstate hires agents, then either Nationwide is running an ad or Progressive is paying a dividend.
- S is both necessary and sufficient for L: Safeco's opening new offices is a necessary and sufficient condition for Liberty's reducing premiums.

To derive the conclusion that Safeco's opening new offices is a sufficient and necessary condition for Liberty's reducing premiums, we need to use the biconditional connective (≡). Thus, we can rewrite the second inference as:

- S ≡ L: Safeco's opening new offices is equivalent to Liberty's reducing premiums.

Now, we need to combine these two statements into a valid propositional logic argument. This can be done using the conditional connective (⊃):

- (N ∨ P) ⊃ A: If either Nationwide's running an ad or Progressive's paying a dividend, then Allstate hires agents.
- A ⊃ (S ≡ L): If Allstate hires agents, then Safeco's opening new offices is equivalent to Liberty's reducing premiums.

We can combine these two conditional statements using the transitive property of implication:

- (N ∨ P) ⊃ (S ≡ L): If either Nationwide's running an ad or Progressive's paying a dividend, then Safeco's opening new offices is equivalent to Liberty's reducing premiums.

This is equivalent to choice E, which is our answer.

If Either Nationwide's Running an Ad or Progressive's Paying a Dividend