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Formal Logic
3. The dog has a shiny coat and loves to bark. Consequently, the dog loves to bark.
4. If Paul is a good swimmer, then he is a good runner. If Paul is a good runner, then he is a good biker.
Therefore if Paul is a good swimmer, then he is a good biker.
For Exercises 5–8, decide what conclusion, if any, can be reached from the given hypotheses and justify your
answer.
5. If the car was involved in the hit-and-run, then the paint would be chipped. But the paint is not chipped.
6. Either the weather will turn bad or we will leave on time. If the weather turns bad, then the flight will be
canceled.
7. If the bill was sent today, then you will be paid tomorrow. You will be paid tomorrow.
8. The grass needs mowing and the trees need trimming. If the grass needs mowing, then we need to rake the
leaves.
9. Justify each step in the proof sequence of
A ` (B S C ) S (B S (A ` C ))
1. A
4. C
2. B S C
5. A ` C
3. B
10. Justify each step in the proof sequence of
B ` [(B ` C ) S A′] ` (B S C ) S A′
1. B
4. C
2. (B ` C ) S A′
5. B ` C
3. B S C
6. A′
11. Justify each step in the proof sequence of
[A S (B ~ C )] ` B′ ` C′ S A′
1. A S (B ~ C )
4. B′ ` C′
2. B′
5. (B ~ C )′
3. C′
6. A′
12. Justify each step in the proof sequence of
A′ ` B ` [B S (A ~ C )] S C
1. A′
5. (A′)′ ~ C
2. B
6. A′ S C
3. B S (A ~ C )
7. C
4. A ~ C
In Exercises 13–24, use propositional logic to prove that the argument is valid.
13. (A ~ B′)′ ` (B S C ) S (A′ ` C )
14. A′ ` (B S A) S B′
15. (A S B) ` [A S (B S C )] S (A S C )
16. [(C S D) S C] S [(C S D) S D]
17. A′ ` (A ~ B) S B