Given the argument: S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K This argument is: A) Invalid; fails in 3^rd line. B) Invalid; fails in 2^nd line. C) Invalid; fails in 1^st line. D) Invalid; fails in 4^th line. E) Valid.

Question

Quizplus · Accepted Answer

The argument is valid. We can prove it through a series of conditional proofs. 
1. Assume K. 
2. From premise S ⊃ (K ∨ ∼S), we can infer K ∨ ∼S. 
3. We now have two subcases: 
   a) Assume K. From 1, we have K, which is the same as the conclusion S. 
   b) Assume ∼S. From 2, we have ∼S, which contradicts our assumption. 
4. Since in either subcase our assumption leads to the conclusion S, we have proven K ⊃ S. 
5. Assume S. 
6. From premise S ⊃ (K ∨ ∼S), we can infer K ∨ ∼S. 
7. We now have two subcases:
   a) Assume K. We already have the conclusion, which is K. 
   b) Assume ∼S. From our assumption S, we have a contradiction. Therefore, ∼S leads to anything, including the conclusion K. 
8. Since in either subcase our assumption leads to the conclusion K, we have proven S ⊃ K. 
9. From 4 and 8, we have proven S ≡ K.

Given the Argument: S ⊃ (K ∨ ∼ S)