Baby Logic Practice 1: Derive the Conclusions (Answers)

Difficulty: What the hell

Here are the answers to yesterday’s baby logic practice.

Exercise 1

1. P → Q

2. Q → R

3. P ∨ S

4. ¬R

∴ S

---

5. ¬Q modus tollens 2, 4

6. ¬P modus tollens 1, 5

7. S disjunctive syllogism 3, 6

Exercise 2

1. P → (Q ∧ R)

2. ¬R ∨ S

3. P

∴ S

---

4. Q ∧ R modus ponens 1, 3

5. R simplification 4

6. ¬¬R double negation 5

7. S disjunctive syllogism 2, 6

Exercise 3

1. P → Q

2. R → S

3. ¬Q

4. P ∨ R

∴ S

---

5. Q ∨ S constructive dilemma 1, 2, 4

6. S disjunctive syllogism 3, 5

Exercise 4

1. (P ∨ Q) → R

2. ¬R

∴ ¬P ∧ ¬Q

---

3. ¬(P ∨ Q) modus tollens 1, 2

4. ¬P ∧ ¬Q De Morgan’s Law 3

Exercise 5

1. P → Q

2. Q → (R ∨ S)

3. ¬R

4. P

∴ S

---

5. P → (R ∨ S) hypothetical syllogism 1, 2

6. R ∨ S modus ponens 4, 5

7. S disjunctive syllogism 3, 6

Exercise 6

1. P ↔ Q

2. Q → R

3. ¬R

∴ ¬P

---

4. P → Q biconditional elimination/material equivalence 1

5. P → R hypothetical syllogism 2, 4

6. ¬P modus tollens 3, 5

Exercise 7

1. (P ∧ Q) → R

2. P

3. ¬R

∴ ¬Q

---

4. ¬(P ∧ Q) modus tollens 1, 3

5. ¬P ∨ ¬Q De Morgan’s Law 4

6. ¬¬P double negation 2

7. ¬Q disjunctive syllogism 5, 6

Exercise 8

1. P → Q

2. Q → R

3. R → S

4. ¬S

∴ ¬P

---

5. P → R hypothetical syllogism 1, 2

6. P → S hypothetical syllogism 3, 5

7. ¬P modus tollens 4, 6

So there you have it–the answers to our first baby logic quiz.