1. (∀x)Ix $ \lor $ ∼(∃x)Hx
2. (∃x)Jx ⊃ ∼(∀x)Ix
3. Hc / (∀x)∼Jx
-Which of the following propositions is not a likely last line of the indented sequence for an indirect proof of the given argument?
A) Hc • ∼Hc
B) (∃x)Hx • ∼(∃x)Hx
C) Id • ∼Id
D) (∃x)Jx • ∼(∃x)Jx
E) (∀x)Ix • ∼(∀x)Ix

Question

Quizplus · Accepted Answer

Assuming ∼(∀x)Ix and Hc, we can use Universal Instantiation to get ∼Ic. Using Modus Tollens with the first premise, we can conclude (∃x)Hx. Then, assuming (∃x)Hx, we can use Existential Instantiation to get Hd. Using Modus Tollens with the second premise, we can conclude ∼(∀x)Ix. Combining this with our previous assumption, we get a contradiction of ∼Ic and Ic. Therefore, we can conclude (∀x)∼Jx.

Option B, (∃x)Hx • ∼(∃x)Hx, is not a likely last line of the indented sequence for an indirect proof because it does not follow directly from the steps taken in the proof, and it is not a contradiction that would result in a proof by contradiction.

1 (∀X)Ix ∨ \lor ∨ ∼(∃X)Hx 2

1 (∀X)Ix $\lor$ ∼(∃X)Hx
2