Question: Use conditional proof or indirect proof and the eighteen rules of inference to establish the truth of the following tautologies. 1. P [(P

Use conditional proof or indirect proof and the eighteen rules of inference to establish the truth of the following tautologies.

1. P ⊃­ [(P ⊃ Q) ⊃­ Q]

2. (∼P ⊃­ Q) ∨ (P ⊃­ R)

3. P ≡ [P ∨ (Q ⊃­ P)]

4. (P ⊃­ Q) ⊃­ [(P ∙­ R) ⊃­ (Q ∙­ R)]

5. (P ∨ ∼ Q) ⊃­ [(∼P ∨ R) ⊃­ (Q ⊃­ R)]

6. P ≡ [P ∙­ (Q ∨ ∼ Q)]

7. (P ⊃­ Q) ∨ (∼Q ⊃­ P)

8. (P ⊃­ Q) ≡ [P ⊃ (P ∙­ Q)]

9. [(P ⊃­ Q) ∙­ (P ⊃­ R)] ⊃­ [P ⊃­ (Q ∙­ R)]

10. [∼(P ∙ ∼ Q) ∙ ∼ Q] ⊃ ∼ P

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