Question

Use the Direct Proof method and the eighteen rules of inference to derive the conclusions of the following arguments. You may not use IP or CP.

(11) Find: ~A

1. ~C ⊃ (A ⊃ C)

2. ~C

(12) Find: ~K

1. J ⊃ (K ⊃ L)

2. L v J

3. ~L

(13) Find: ~X

1. Y ⊃ W

2. ~W v ~Y

3. X ⊃ (Y • Z)

(14) Find: ~F

1. (F ⊃ G) v H

2. ~G

3. ~H

Answer 1

11) ~a can be derived using direct proof as follows:

1. ~c ⊃ (a ⊃ c) (Premise)

2. ~c (Premise)

3. Assume a (for indirect proof)

4. From 2 and 3, we have ~c ∧ a (conjunction introduction)

5. From 1 and 4, we have ~c (Modus Ponens)

6. From 2 and 5, we have a contradiction (¬ Intro)

7. Therefore, ¬a (¬ Intro)

13) ~x can be derived using direct proof as follows:

1. y ⊃ w (Premise)

2. ~w v ~y (Premise)

3. Assume x (for indirect proof)

4. y • z (From 3 and 1, Modus Ponens)

5. y (Simplification from 4)

6. ~w (Disjunctive Syllogism from 2 and 5)

7. ~y (Disjunctive Syllogism from 2 and 6)

8. From 7, we have ~y ∧ z (Conjunction introduction)

9. Therefore, ~x (¬ Intro)

12) ~k can be derived using direct proof as follows:

1. j ⊃ (k ⊃ l) (Premise)

2. l v j (Premise)

3. ~l (Premise)

4. Assume k (for indirect proof)

5. From 1 and 4, we have j ⊃ l (Modus Ponens)

6. From 2 and 3, we have j (Disjunctive Syllogism)

7. From 5 and 6, we have l (Modus Ponens)

8. From 3 and 7, we have a contradiction

9. Therefore, ~k (¬ Intro)

14) ~f can be derived using direct proof as follows:

1. (f ⊃ g) v h (Premise)

2. ~g (Premise)

3. ~h (Premise)

4. Assume f (for indirect proof)

5. f ⊃ g (Assumption 1 from 1)

6. g (Modus Ponens from 4 and 5)

7. From 2 and 6, we have a contradiction

8. Therefore, ~f (¬ Intro)

Step-by-step explanation: