IM – 10 | 16
5. ~(Q v S) 4, DEM
6. ~P 2, 5, MT
9. 1. P v (S & R) (Premise)
2. T → (~P & ~R) (Premise) /∴ ~T
Notice how simple item 10 becomes once we have the rule of conditional proof.
Exercise 10-17
Note: The level of formality you hold your students to is of course up to you. We allow
combining double negation with other steps, and in the items that follow, we often do such
combining, as indicated in the annotation. We probably don’t need to indicate that there is
usually a different, and equally correct, way of constructing a derivation.
1. ▲1. P → R (Premise)
2. R → Q (Premise) /∴ ~P v Q
3. P → Q 1, 2, CA
4. ~P v Q 3, IMPL
IM – 10 | 17
6. ~R v ~S 5, DEM
7. ~F v ~L 1, 2, 6, DD
4. ▲1. P v (Q & R) (Premise)
2. (P v Q) → S (Premise) /∴ S
3. (P v Q) & (P v R) 1, DIST
6. 1. ~L → (~P → M) (Premise)
2. ~(P v L) (Premise) /∴ M
3. ~P & ~L 2, DEM
4. ~L 3, SIM
8. 1. Q → L (Premise)
2. P → M (Premise)
3. R v P (Premise)
IM – 10 | 18
12. ~M → L 11, DN/IMPL
9. 1. Q → S (Premise)
2. P → (S & L) (Premise)
10. S 9, TAUT
11. R 4, 10, MP
12. R & S 10, 11 CONJ
10. ▲1. P v (R & Q) (Premise)
Exercise 10-18
Each of these has alternative symbolizations. We give the most obvious.
1. ▲D → ~B
2. ~C v B
Exercise 10-19
The fact that students have difficulty with this exercise is evidence that they are not as good at
determining just exactly what plain English sentences say as they believe.
IM – 10 | 19
1. ▲Equivalent to (b)
2. Equivalent to (c)
Exercise 10-20
Be careful with this exercise—some of these require a tricky step.
1. ▲1. P (Premise)
2. Q & R (Premise)
3. (Q & P) → S (Premise) /∴ S
2. 1. (P v Q) & R (Premise)
2. (R & P) → S (Premise)
3. (Q & R) → S (Premise) /∴ S
4. R & (P v Q) 1, COM
3. 1. P → (Q → ~R) (Premise)
2. (~R → S) v T (Premise)
3. ~T & P (Premise) /∴ Q → S
4. P 3, SIM
4. ▲1. P v Q (Premise)
2. (Q v U) → (P → T) (Premise)
3. ~P (Premise)
4. (~P v R) → (Q → S) (Premise) /∴ T v S
5. Q 1, 3, DA
IM – 10 | 20
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Education.
6. Q v U 5, ADD
7. P → T 2, 6, MP
8. ~P v R 3, ADD
9. Q → S 4, 8, MP
10. T v S 1, 7, 9, CD
5. 1. (P → Q) & R (Premise)
2. ~S (Premise)
3. S v (Q → S) (Premise) /∴ P → T
4. Q → S 2, 3, DA
5. ~Q 2, 4, MT
6. P → Q 1, SIM
7. ~P 5, 6, MT
8. ~P v T 7, ADD
9. P → T 8, IMPL
6. 1. P → (Q & R) (Premise)
2. R→ (Q → S) (Premise) /∴ P → S
3. (R & Q) → S 2, EXPORT
4. (Q & R) → S 3, COM
5. P → S 1, 4, CA
7. ▲1. P → Q (Premise) /∴ P → (Q v R)
2. ~P v Q 1, IMPL
3. (~P v Q) v R 2, ADD
4. ~P v (Q v R) 3, ASSOC
5. P → (Q v R) 4, IMPL
This next one could be shortened a little if you combine double negation with other steps.
We’ve laid it all out to make it as clear as possible.
8. 1. ~P v ~Q (Premise)
2. (Q → S) → R (Premise) /∴ P → R
3. ~R → ~(Q → S) 2, CONTR
4. ~~R v ~(Q → S) 3, IMPL
5. R v ~(Q → S) 4, DN
6. R v ~(~Q v S) 5, IMPL
7. R v (~~Q & ~S) 6, DEM
8. R v (Q & ~S) 7, DN
9. (R v Q) & (R v ~S) 8, DIST
10. R v Q 9, SIM
11. Q v R 10, COM
12. ~Q → R 11, IMPL
13. P → ~Q 1, IMPL
14. P → R 12, 13, CA
IM – 10 | 21
9. 1. S (Premise)
2. P → (Q & R) (Premise)
3. Q → ~S (Premise) /∴ ~P
4. ~~S 1, DN
5. ~Q 3, 4, MT
10. ▲1. (S → Q) → ~R (Premise)
2. (P → Q) → R (Premise)/∴ ~Q
3. ~R → ~(P → Q) 2, CONTR
4. (S → Q) → ~(P → Q) 1, 3, CA
5. ~(S → Q) v ~(P → Q) 4, IMPL
Exercise 10-21
3. 1. P → (Q → R) (Premise) /∴ (P → Q) → (P → R)
2. P → Q CP Premise
3. P CP Premise
4. Q → R 1, 3, MP
5. Q 2, 3, MP
6. R 4, 5, MP
IM – 10 | 23
5. Q → ~R 3, 4, MP
6. ~Q v ~R 5, IMPL
7. ~(Q & R) 6, DEM
8. P 1, 7, DA
9. P v U 8, ADD
10. ▲1. (P & Q) v R (Premise)
2. ~R v Q (Premise) /∴ P → Q
3. P CP Premise
4. ~Q CP Premise
IM – 10 | 24
Exercise 10-22
1. ▲Valid, as per the following deduction:
1. C → ~S Premise
2. ~L → S Premise /∴ C → L
3. ~S → L 2, DN/CONTR
4. C → L 1, 3, CA
3. Valid, as per the following:
1. ~R v A (Premise)
2. A → E (Premise)
3. M → ~E (Premise) /∴ ~R v ~M
4. ▲~M v C
~M → ~K
C v H
T → ~H
T → K
IM – 10 | 25
Invalid:
C N T E P
T F T F F
7 ▲Valid, as per the following deduction:
1. C v S Premise
2. S → E Premise
3. C → R Premise /∴ R v E
9. Here’s the symbolization, with an assignment of claim letters:
C = The creation story in Genesis is compatible with the theory of evolution.
L = The creation story is taken literally.
E = There is plenty of evidence for the theory of evolution.
T = The creation story is true.
C → ~L
E → (~C → ~T) /∴ L → ~T
Invalid:
10. ▲Valid, as per the following deduction:
11. Valid, as per this deduction:
1. ~M → H (Premise)
2. (H → I) & (I → C) (Premise) /∴ ~M → C
IM – 10 | 26
3. H → I 2, SIM
12. Valid:
1. ~T v N (Premise)
2. ~T → W (Premise)
7. ~D → ~T 5, 6, CA
8. ~D → W 2, 7, CA
9. D v W 8, IMPL
13. ▲Valid, as per the following:
1. S →~F Premise
2. ~S → ~T Premise
14. Valid, as per the following deduction. The symbolization can be confusing; here’s how we
did it:
B = Jane goes on eating binges for no apparent reason.
F = Jane looks forward to times when she can eat alone.
S = Jane eats sensibly in front of others.
A = Jane makes up for it when she’s alone.
D = Jane has an eating disorder.
In the first premise, the parentheses look more confusing than they actually are. You
can help students by building the symbolization up from its parts.
15. Valid.
1. I & (I → M) (Premise)
IM – 10 | 27
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Education.
5. I 1, SIM
6. I → M 1, SIM
7. M 5, 6 MP
8. G 3, 7, MP
9. ~C 4, 8, MP