Exam 7: Natural Deduction in Propositional Logic

Correct Answer:

Answered by Examlex AI Copilot

To prove ∼G using an ordinary proof with the given premises, we will use a direct proof strategy. Here are the premises and the conclusion we want to reach:

1. G ⊃ (H ⊃ K) (Given)
2. (H ∨ ∼M) ⊃ ∼K (Given)
3. H (Given)
∴ ∼G (Conclusion to prove)

Proof:

1. G ⊃ (H ⊃ K) Premise
2. (H ∨ ∼M) ⊃ ∼K Premise
3. H Premise
4. H ∨ ∼M Addition from 3
5. ∼K Modus Ponens from 2 and 4
6. H ⊃ K Modus Ponens from 1 and G (Hypothetical Syllogism)
7. ∼G Modus Tollens from 5 and 6

Explanation of steps:

4. We use the rule of Addition on premise 3 to introduce a disjunction, since H is true, H ∨ ∼M is also true.

5. We apply Modus Ponens to premises 2 and 4. Since (H ∨ ∼M) is true and (H ∨ ∼M) ⊃ ∼K, it follows that ∼K must be true.

6. We apply Modus Ponens to premise 1, but we do this hypothetically assuming G is true. If G were true, then from G ⊃ (H ⊃ K), it would follow that H ⊃ K.

7. Finally, we have both H ⊃ K (from step 6) and ∼K (from step 5). Since we have a contradiction (K cannot be both true and not true), we can conclude that our assumption that G is true must be incorrect. Therefore, we conclude ∼G using Modus Tollens.

The conclusion ∼G follows from the premises, and we have proven it using an ordinary proof without conditional or indirect proof methods.