# Solutions to Conditional Proof exercises

Conditional Proof Exercises with solutions

Here are some exercises you can try:

1.

1. F ⊃ E

2. (F ∙ E) ⊃ R / F ⊃ R

| 3. F      acp

| 4. E      mp 1,3

| 5. F ∙ E    cn 3, 4

| 6. R      mp 2,5

5. F > R    cp 3-4

2.

1. G > T

2. (T v S) > K / G > K

| 3. G acp

| 4. T mp 1,3

| 5. T v S ad 4

| 6. K mp 2,5

7. G > K cp 3-6

3.

1. (G v H) > (S . T)

2. (T v U) > (C . D) / G > C

| 3. G acp

| 4. G v H ad 3

| 5. S . T mp 1,4

| 6. T cm, sm 5

| 7. T v U ad 6

| 8. C . D mp2,7

| 9. C sm 8

10. G > C cp 3-9

4.

1. A > ~(A v E) / A > F

| 2. A acp

| 3. ~(A v E) mp 1,2

| 4. ~A . ~E dm 3

| 5. A v F ad 2

| 6. ~A sm 4

| 7. F DS 5,6

8. A > F cp 2-7

5.

1. H > (I > N)

2. (H > ~I) > (M v N)

3. ~N / M

| 4. H acp

| 5. I > N mp 1,4

| 6. ~I mt3,5

7. H > ~ I cp 4-6

8. M v N mp 2,7

9. M cm, ds 8,3

6.

1. M ⊃ (K ⊃ L)

2. (L v N) > J / M > (K > J)

| 3. M acp

| | 4. K acp

| | 5. K > L mp1,3

| | 6. L mp 4,5

| | 7. L v N 6 ad

| | 8. J mp 2,7

| 9. K > J CP 4-8

10. M > (K > J) CP 3-9

And here are some word problems:

1. If high tech products are exported to Russia, domestic industries will benefit. If the Russians can effectively use high tech products, their standard of living will improve. So if high tech products are exported there, and they can effectively use them, their standard of living will improve and domestic industries will benefit.

1. E > D

2. U > S / (E ∙ U) > (S ∙ D)

| 3. E ∙ U acp

| 4. E sm 3

| 5. U cm sm 3

| 6. D mp 1,4

| 7. S mp 2,5

| 8. S ∙ D cn 7,6

9. (E . U) > (S . D) cp 3-8

2. A doctor must disconnect a dying patient from a respirator only if the fact that patients are self-determining implies that the doctor must follow the patient’s orders. If a dying patient refuses treatment, the doctor must disconnect the patient from a respirator and the patient will die peacefully. Patients are self-determining, so if a dying patient refuses treatment, the doctor must follow the patient’s orders.

1. D > (S > F)

2. R > (D . P)

3. S / R > F

| 4. R acp

| 5. D . P mp 2,4

| 6. D sm 5

| 7. S > F mp 1,6

| 8. F mp 3,7

9. R > F cp 4-8

3. If astronauts attempt interplanetary space travel, heavy shielding will be required to protect them from solar radiation. If massive amounts of either fuel or water are carried, the spacecraft must be very large. Therefore if heavy shielding is required to protect them from solar radiation only if massive amounts of fuel are carried, then if astronauts attempt interplanetary space travel, the spacecraft must be very large.

1. A > H

2. (F v W) > L / (H > F) > (A > L)

| 3. H > F acp

| | 4. A acp

| | 5. H mp 1,4

| | 6. F mp 3,5

| | 7. F v W ad 6

| | 8. L mp 2,7

| 9. A > L cp 4-8

10. (H > F) > (A > L) cp 3-9