Baby Logic Practice 2: Derive the Conclusions Using Reductio (Answers)

Difficulty: What the hell

Here are the answers to Baby Logic Practice 2. As promised, each proof uses reductio ad absurdum, even in cases where there are quicker ways to get the conclusion.

Exercise 1

1. P → Q

2. ¬Q

∴ ¬P

---

3. Assume P

4. Q modus ponens 1, 3

5. ⊥ contradiction 2, 4

6. ¬P reductio 3–5

Exercise 2

1. P ∨ Q

2. ¬P

∴ Q

---

3. Assume ¬Q

4. ¬P ∧ ¬Q conjunction 2, 3

5. ¬(P ∨ Q) De Morgan’s Law, 4

6. ⊥ contradiction 1, 5

7. Q reductio 3–6

Exercise 3

1. (P ∧ Q) → R

2. P

3. ¬R

∴ ¬Q

---

4. Assume Q

5. P ∧ Q conjunction 2, 4

6. R modus ponens 1, 5

7. ⊥ contradiction 3, 6

8. ¬Q reductio 4–7

Exercise 4

1. P → Q

2. Q → R

3. ¬R

∴ ¬P

---

4. Assume P

5. Q modus ponens 1, 4

6. R modus ponens 2, 5

7. ⊥ contradiction 3, 6

8. ¬P reductio 4–7

Exercise 5

1. P → (Q ∨ R)

2. ¬Q

3. ¬R

∴ ¬P

---

4. Assume P

5. Q ∨ R modus ponens 1, 4

6. ¬Q ∧ ¬R conjunction 2, 3

7. ¬(Q ∨ R) De Morgan’s Law, 6

8. ⊥ contradiction 5, 7

9. ¬P reductio 4–8

Exercise 6

1. P ↔ Q

2. ¬Q

∴ ¬P

---

3. P → Q biconditional elimination, 1

4. Assume P

5. Q modus ponens 3, 4

6. ⊥ contradiction 2, 5

7. ¬P reductio 4–6

Exercise 7

1. P ∨ Q

2. P → R

3. Q → R

4. ¬R

∴ ¬P ∧ ¬Q

---

5. Assume P

6. R modus ponens 2, 5

7. ⊥ contradiction 4, 6

8. ¬P reductio 5–7

9. Assume Q

10. R modus ponens 3, 9

11. ⊥ contradiction 4, 10

12. ¬Q reductio 9–11

13. ¬P ∧ ¬Q conjunction 8, 12

Exercise 8

1. (P ∨ Q) → R

2. ¬R

∴ ¬P

---

3. Assume P

4. P ∨ Q addition 3

5. R modus ponens 1, 4

6. ⊥ contradiction 2, 5

7. ¬P reductio 3–6

So there you have it—the answers to our second baby logic quiz. Reductio can feel a little roundabout, but that’s part of what makes it useful: it forces you to see exactly where an assumption blows up.