Construct proofs for the following valid arguments. The first fifteen proofs can be complete in three or less additional lines. The next five proofs will be a bit longer. It is important to note that there is always more than one way to construct a proof. If your proof differs from the answer key, that doesn’t mean it is wrong.

Question

#1
1. A ? B
2. (A v C) ? D /? A ? D
#2
1. A
2. B /? (A v C) ? B
#3
1. D ? E
2. D ? F /? E
#4
1. J ? K
2. J /? K v L
#5
1. A v B
2. ~A ? ~C /? B
#6
1. A ? B
2. ~B ? ~C /? ~A
#7
1. D ? E
2. (E ? F) ? (F? D) /?D ? F
#8
1. (T ? U) ? (T ? V)
2. T /? U v V
#9
1. (E ? F) v (G ? H)
2. I ? G
3. ~(E ? F) /? I ? H
#10
1. M ? N
2. O ? P
3. N ? P
4. (N ? P) ? (M v O) /?N v P
#11
1. A v (B ? A)
2. ~A ? C /? ~B
#12
1. (D v E) ? (F ? G)
2. D /? F
#13
1. T ? U
2. V v ~U
3. ~V ? ~W /? ~T
#14
1. (A v B) ? ~C
2. C v D
3. A /? D
#15
1. L v (M ? N)
2. ~L ? (N ? O)
3. ~L /? M ? O
#16
1. A ? B
2. A v (C ? D)
3. ~B ? ~E /? C
#17
1. (F ? G) ? (H ? I)
2. J ? K
3. (F v J) ? (H v L) /? G v K
#18
1. (E v F) ? (G ? H)
2. (G v H) ? I
3. E /? I
#19
1. (N v O) ? P
2. (P v Q) ? R
3. Q v N
4. ~Q /? R
#20
1. J ? K
2. K v L
3. (L ? ~J) ? (M ? ~J)
4. ~K /? M

Answer 1

1.
1. A ? B
2. (A v C) ? D /? A ? D
3. A Simplification 1
4. A v C Addition 3
5. D Modus ponens 2, 4
6. A ? D Conjunction 3, 5
2.
1. A
2. B /? (A v C) ? B
3. A v C Addition 1
4. (A v C) ? B Conjunction 2, 3
3.
1. D ? E
2. D ? F /? E
3. D Simplification 2
4. E Modus ponens 1, 3
4.
1. J ? K
2. J /? K v L
3. K Modus ponens 1, 2
4. K v L Addition 3
5.
1. A v B
2. ~A ? ~C /? B
3. ~A Simplification 2
4. B Disjunctive syllogism 1, 3
6.
1. A ? B
2. ~B ? ~C /? ~A
3. ~B Simplification 2
4. ~A Modus tollens, 1, 3
7.
1. D ? E
2. (E ? F) ? (F? D) /? D ? F
3. E ? F Simplification 2
4. D ? F Hypothetical syllogism 1, 3
8.
1. (T ? U) ? (T ? V)
2. T /? U v V
3. T v U Addition 2
4. U v V Constructive dilemma 1, 2, 3
9.
1. (E ? F) v (G ? H)
2. I ? G
3. ~(E ? F) /? I ? H
4. G ? H Disjunctive syllogism 1, 3
5. I ? H Hypothetical syllogism 2, 4
10.
1. M ? N
2. O ? P
3. N ? P
4. (N ? P) ? (M v O) /?N v P
5. M v O Modus ponens 3, 4
6. N v P Constructive dilemma 1, 2, 5
11.
1. A v (B ? A)
2. ~A ? C /? ~B
3. ~A Simplification 2
4. B ? A Disjunctive syllogism 1, 3
5. ~B Modus tollens 3, 4
12.
1. (D v E) ? (F ? G)
2. D /? F
3. D v E Addition 2
4. F ? G Modus ponens 1, 3
5. F Simplification 4
13.
1. T ? U
2. V v ~U
3. ~V ? ~W /? ~T
4. ~V Simplification 3
5. ~U Disjunctive syllogism 2, 4
6. ~T Modus tollens 1, 5
14.
1. (A v B) ? ~C
2. C v D
3. A /? D
4. A v B Addition 3
5. ~C Modus ponens 1, 4
6. D Disjunctive syllogism 2, 5
15.
1. L v (M ? N)
2. ~L ? (N ? O)
3. ~L /? M ? O
4. N ? O Modus ponens 2, 3
5. M ? N Disjunctive syllogism 1, 3
6. M ? O Hypothetical syllogism 4, 5
16.
1. A ? B
2. A v (C ? D)
3. ~B ? ~E /? C
4. ~B Simplification 3
5. ~A Modus tollens 1, 4
6. C ? D Disjunctive syllogism 2, 5
7. C Simplification 6
17.
1. (F ? G) ? (H ? I)
2. J ? K
3. (F v J) ? (H v L) /? G v K
4. F ? G Simplification 1
5. F v J Simplification 3
6. G v K Constructive dilemma 2, 4, 5
18.
1. (E v F) ? (G ? H)
2. (G v H) ? I
3. E /? I
4. E v F Addition 3
5. G ? H Modus ponens 1, 4
6. G Simplification 5
7. G v H Addition 6
8. I Modus ponens 2, 7
19.
1. (N v O) ? P
2. (P v Q) ? R
3. Q v N
4. ~Q /? R
5. N Disjunctive syllogism 3, 4
6. N v O Addition 5
7. P Modus ponens 1, 6
8. P v Q Addition 7
9. R Modus ponens 2, 8
20.
1. J ? K
2. K v L
3. (L ? ~J) ? (M ? ~J)
4. ~K /? M
5. L Disjunctive syllogism 2, 4
6. ~J Modus tollens 1, 4
7. L ? ~J Conjunction 5, 6
8. M ? ~J Modus ponens 3, 7
9. M Simplification 8