Tag: derive the conclusions

Baby Logic Practice 2: Derive the Conclusions Using Reductio

Difficulty: What the hell

Today’s baby logic practice is on reductio ad absurdum, also known as indirect proof. The basic idea is simple: if assuming the opposite of what you want to prove leads to a contradiction, then the thing you wanted to prove must be true.

So, for each of the following exercises, derive the indicated conclusion using reductio. No quantifiers. No modal logic. No funny business. Just contradiction, negation, and a little patience.

Exercise 1

1. P → Q

2. ¬Q

∴ ¬P

Exercise 2

1. P ∨ Q

2. ¬P

∴ Q

Exercise 3

1. (P ∧ Q) → R

2. P

3. ¬R

∴ ¬Q

Exercise 4

1. P → Q

2. Q → R

3. ¬R

∴ ¬P

Exercise 5

1. P → (Q ∨ R)

2. ¬Q

3. ¬R

∴ ¬P

Exercise 6

1. P ↔ Q

2. ¬Q

∴ ¬P

Exercise 7

1. P ∨ Q

2. P → R

3. Q → R

4. ¬R

∴ ¬P ∧ ¬Q

Exercise 8

1. (P ∨ Q) → R

2. ¬R

∴ ¬P

In some of these, reductio will feel a little artificial because there are quicker ways to get the conclusion. Too bad. The point today is to practice reductio, not to be efficient.

I’ll probably post the answers tomorrow. Have fun.

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.

Baby Logic Practice 1: Derive the Conclusions

Difficulty: What the hell

Today’s post is simple: a few symbolic-logic proof exercises. No quantifiers. No modal logic. Just derive the indicated conclusions from the premises supplied. These are not one-step gimmies, so slow down and actually work through them.

I’ll likely post solutions tomorrow.

Exercise 1

1. P → Q

2. Q → R

3. P ∨ S

4. ¬R

∴ S

Exercise 2

1. P → (Q ∧ R)

2. ¬R ∨ S

3. P

∴ S

Exercise 3

1. P → Q

2. R → S

3. ¬Q

4. P ∨ R

∴ S

Exercise 4

1. (P ∨ Q) → R

2. ¬R

∴ ¬P ∧ ¬Q

Exercise 5

1. P → Q

2. Q → (R ∨ S)

3. ¬R

4. P

∴ S

Exercise 6

1. P ↔ Q

2. Q → R

3. ¬R

∴ ¬P

Exercise 7

1. (P ∧ Q) → R

2. P

3. ¬R

∴ ¬Q

Exercise 8

1. P → Q

2. Q → R

3. R → S

4. ¬S

∴ ¬P

If these still feel too easy, good. Basic logical competence should eventually feel boringly obvious. That’s the point.