a) Prove that (p ⊃ q) ⊃ [(p ⊃ ~q) ⊃ (p ⊃ r)] is a theorem by giving a proof in our natural deduction
system
Sure! To prove that (p ⊃ q) ⊃ [(p ⊃ ~q) ⊃ (p ⊃ r)] is a theorem using natural deduction, we can start by assuming the antecedent of the implication and then proceed step by step to establish the consequent. Here's a step-by-step proof:
1. Assume (p ⊃ q) (Assumption)
2. Assume (p ⊃ ~q) (Assumption)
3. Assume p (Assumption)
4. From 1 and 3, infer q (→ Elimination)
5. Assume ~q (Assumption)
6. From 3 and 5, infer r (→ Elimination)
7. From 4 and 6, infer r (→ Elimination)
8. From 5 and 7, infer r (→ Elimination)
9. From 3-8, infer (p ⊃ r) (→ Introduction)
10. From 2 and 9, infer (p ⊃ r) (→ Introduction)
11. From 1 and 10, infer [(p ⊃ ~q) ⊃ (p ⊃ r)] (→ Introduction)
12. From 11, infer (p ⊃ q) ⊃ [(p ⊃ ~q) ⊃ (p ⊃ r)] (→ Introduction)
Therefore, we have proven that (p ⊃ q) ⊃ [(p ⊃ ~q) ⊃ (p ⊃ r)] is a theorem in our natural deduction system.