One-Step Proofs. Complete the following proofs. They can be done in one step using the rules from 7.1 and 7.2 of your text. 14 points.

Example:

1. A

2. A>B / B

3. B 1,2 MP ßYou add this line

1. G > H

2. ~H / ~G

3.

1. J > K

2. K > (L * M) / J > (L * M)

3.

1. O v P

2. ~O / P

3.

1. R * S / S

2.

1. R

2. T / R * T

3.

1. W / W v X

2.

1. (X > Y) * (Y > Z)

2. X v Y / Y v Z

3.

1. (C * D) > (E v F)

2. C * D / E v F

3.

1. (G > H) > (I * J)

2. ~(I * J) / ~(G > H)

3.

1. (J v K) > ~(L > M)

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

3.

1. O v (P v Q)

2. ~(P v Q) / O

3.

1. R * (S v T) / R

2.

1. R > S

2. S > (T v U) / (R > S) * (S > (T v U))

3.

1. W * X / (W * X) v (X * Y)

2.

Two-Step Proofs. Complete the following proofs. They can be done in two steps using the rules from 7.1 and 7.2 of your text. There may be more than one way to do some of the proofs. 14 points.

Example:

1. A

2. A > B

3. B > C / C

4. B 1,2 MP ß You add either version à 4. A > C 2,3 HS

5. C 3,4 MP of lines 4 and 5 5. C 1,4 MP

1. ~C

2. A > B

3. B > C / ~A

1. D > E

2. F > G

3. D v F / E v G

1. A * B / A v C

1. (H v I) > J

2. H / J

1. (H * I) > J

2. H

3. I / J

1. H > (I * J)

2. H / J

1. (K * L) v (M * N)

2. ~(K * L) / N

Proofs with Translation. Translate the following proofs into symbols, using the letters suggested. Each letter should stand for a simple sentence. Then do the proofs. 12 points.

Example:

If we are morally responsible for our actions, we have free will. If we are not morally responsible for our actions, jails should be emptied. We don’t have free will. Therefore, jails should be emptied. (M, F, J)

You write everything that follows:

M = We are morally responsible for our actions

F = We have free will

J = Jails should be emptied.

1. M > F

2. ~M > J

3. ~F / J

4. ~M 1,3 MT

5. J 2,4 MP

1. Either the mind is identical to the brain, or it is immaterial. If the mind is immaterial, it doesn’t make any difference. The mind makes a difference. Therefore, the mind is identical to the brain. (B, I, D)

2. If there is not an afterlife, then all is hopeless. If all is hopeless, despair is the right attitude. Despair is not the right attitude. Therefore, there is an afterlife. (A, H, D)