1. P ⊃ (Q ⊃ R)
2. ∼(R ⊃ S) ⊃ ∼T
3. (U $ \lor $ ∼T) $ \lor $ ∼(Q ⊃ S) / ∼U ⊃ ∼(P • T)
-Which of the following propositions is a likely last line of the indented sequence for an indirect proof nested within an indented sequence for conditional proof for the given argument?
A) P • ∼P
B) Q • ∼Q
C) T • ∼T
D) All of the above
E) Options B and C, but not option A

Question

Quizplus · Accepted Answer

This argument requires both an indirect proof nested within a conditional proof.

Assume ¬U:

From (2), if ~(R ⊃ S) then ~T, and by contraposition, if T, then (R ⊃ S).
From (1), if P, then (Q ⊃ R), and we have (R ⊃ S) by (4).
This gives us, if P, then (R ⊃ S), which contraposes to ~(R ⊃ S) ⊃ ~P.

Assume U or ~T:
We must show ~P ･ T, from which ¬U → ~P ･ T, which by contraposition is ~P → U v ~T
There are different ways to do this, but one possibility is:

Case 1: U 
Then we have (U v ~T) second disjunct inherited, and so ~P becomes the needed contradiction.

Case 2: ~T 
Then we have ~T disjunct inherited, which works the same as in Case 1 to contradict ~P that we get from the premises.
 
Thus, the most appropriate last line(s) to close the subproof(s) is ~P ⊃ (U v ~T).

Therefore, the answer is option E: Options B and C, but not option A.

1 P ⊃ (Q ⊃ R) 2 ∨ \lor ∨ ∼T) ∨ \lor ∨

1 P ⊃ (Q ⊃ R)
2 $\lor$ ∼T) $\lor$