Question

prove that (p → (q → r)) and (q → (p → r)) are logically equivalent by deduction using a series of logical equivalences studied in class (truth tables will not be allowed)

Answer 1

To prove the logical equivalence of (p → (q → r)) and (q → (p → r)), we can use a series of logical equivalences. By assuming the truth of both statements and applying conditional elimination and introduction, we can show that they are equivalent.

Step-by-step explanation:

To prove that (p → (q → r)) and (q → (p → r)) are logically equivalent, we can use a series of logical equivalences. Let's start:

1. Assume p → (q → r) is true.
2. Using conditional elimination, we get q → r.
3. Assume q → (p → r) is true.
4. Using conditional elimination, we get p → r.
5. Now, let's prove (p → (q → r)) → (q → (p → r)).
6. Assume p → (q → r) is true.
7. Using conditional elimination, we get q → r.
8. Using conditional introduction, assuming q is true, we can prove p → r.
9. Using conditional introduction, we can prove (q → (p → r)).
10. Therefore, (p → (q → r)) and (q → (p → r)) are logically equivalent.