Baby Logic Practice 3: Conditional Proofs (Answers)

Difficulty: What the hell

Here are the answers to Baby Logic Practice 3. As promised, each proof uses conditional proof, even in cases where there are quicker or more boring ways to get the conclusion.

Exercise 1

1. P → Q

∴ P → Q

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2. Assume P

3. Q modus ponens 1, 2

4. P → Q conditional proof 2–3

Exercise 2

1. P → Q

2. Q → R

∴ P → R

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3. Assume P

4. Q modus ponens 1, 3

5. R modus ponens 2, 4

6. P → R conditional proof 3–5

Exercise 3

1. P ∧ Q

∴ P → Q

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2. Assume P

3. Q simplification 1

4. P → Q conditional proof 2–3

Exercise 4

1. P → (Q → R)

2. P

∴ Q → R

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3. Assume Q

4. Q → R modus ponens 1, 2

5. R modus ponens 3, 4

6. Q → R conditional proof 3–5

Exercise 5

1. P → Q

2. R → S

3. P ∧ R

∴ P → S

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4. Assume P

5. R simplification 3

6. S modus ponens 2, 5

7. P → S conditional proof 4–6

Exercise 6

1. P → Q

2. Q → R

3. R → S

∴ P → S

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4. Assume P

5. Q modus ponens 1, 4

6. R modus ponens 2, 5

7. S modus ponens 3, 6

8. P → S conditional proof 4–7

Exercise 7

1. P → Q

2. P → R

∴ P → (Q ∧ R)

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3. Assume P

4. Q modus ponens 1, 3

5. R modus ponens 2, 3

6. Q ∧ R conjunction 4, 5

7. P → (Q ∧ R) conditional proof 3–6

Exercise 8

1. P → Q

2. Q → (R ∨ S)

3. ¬R

∴ P → S

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4. Assume P

5. Q modus ponens 1, 4

6. R ∨ S modus ponens 2, 5

7. S disjunctive syllogism 3, 6

8. P → S conditional proof 4–7

So there you have it—the answers to our third baby logic quiz. Conditional proof is simple in principle, but it gets nicer once assuming the antecedent starts to feel automatic.