Stefan Waner and Steven R. Costenoble

Arguments and Proofs

** Answers** To see an answer to any odd-numbered exercise, just click on the exercise number.

Prove each of the following valid arguments:

1. |
(pr)~q | 2. |
~p (qs) |
3. |
~p(r~t) |

pr | ~p | ~(r~t) | |||

~q | qs | p | |||

4. |
(~pr) ~q |
5. |
~p(qr) | 6. |
p (rs) |

q | ~ps | ~r | |||

~(~pr) | r | ~p | |||

7. |
pq | 8. |
p (qr) |
9. |
(pr) q |

~(qr) | rs | sp | |||

p | s | ||||

~p | |||||

s | q | ||||

10. |
(pq) r |
11. |
(p~q)r | 12. |
(p~q)r |

~r | s(tu) | sp | |||

t q | sp | q~u | |||

us | |||||

~t | ru | ||||

r | |||||

13. |
(pq)r | 14. |
(pq)(pr) |
15. |
p(qr) |

~(qr) | qp | q | |||

p | r | pr | |||

16. |
p(qr) |
17. |
- | 18. |
- |

~r | |||||

(pq)(pq) | p~(q~p) | ||||

p(~q) | |||||

19. |
- | 20. |
- |
21. |
- |

~(p~p) | (p~p)q | (p~p)~p | |||

22. |
- | ||||

~p(p~p) |

Give counterexamples to each of the following fallacies by finding truth values for the variables making the premises true and the conclusion false.

23. |
pq | 24. |
pq |
25. |
(pq)r | 26. |
(pq)r |

~p | pr | p | ~r | ||||

~q | qr | r | ~p |

Decide whether each of the following is a valid argument. If it is valid, give a proof. If it is invalid, give a counterexample. In any case, supply verbal statements that make all the premises true; if the argument is invalid make sure that they also make the conclusion false.

27. |
p(qr) | 28. |
p(qr) |
29. |
pr | 30. |
~(pq) |

~q | ~q | ~q~r | p | ||||

~p | ~p | pq | q |

In the remaining exercises, convert each argument into symbolic form and then decide whether or not it is valid. If invalid, give a counterexample by supplying truth values for the various atomic statements.

**31.** If I were a cow, then I would eat grass. However, I am not a cow, so I don't eat grass.

**32.** If we were meant to fly, we would have wings. Since we do indeed fly, and since we would not fly unless we were meant to fly, it follows that we have wings.

**33.** I am wise if you are a fool; either you are a fool or the sun sets in the east; but the sun cannot both set and rise in the east; observe that the sun rises in the east. Therefore I am wise and you are a fool.

**34. **If factories in the west pollute the air, then acid rain will lead to damage in the east. Indeed, acid rain is leading to considerable damage in the east, so the factories in the west must be polluting the air.

**35.** If factories in the west pollute the air, then acid rain will lead to damage in the east. The factories in the west do not pollute the air, so acid rain will not be a problem in the east.

**36.** If interest rates go down, inflation will rise. If interest rates go down, the stock market will also rise. Therefore, if inflation rises so will the stock market.

**37.** If interest rates go down, inflation will rise. If inflation rises, so will the bond market. Therefore, if interest rates go down, the bond market will rise.

**38.** If mortgage rates go down, or prices fall, then housing starts will rise. Mortgage rates are falling, therefore housing starts will rise.

**39.** If mortgage rates go down or prices fall, then housing starts will rise. Prices are rising, therefore housing starts will not rise.

**40.** When it rains on the great plains of the Nile, Saggitarius is in the shadow of Jupiter, and as you know, it always rains if Mercury is ascending. On the other hand, Saggitarius falls in Jupiter's shadow only when either the moon is full or Mercury is ascending. I have noticed a disturbing pattern to the weather predictions of Desmorelda, so-called "Mistress of the Zodiac." It seems that she always predicts that it will rain on the plains of the Nile when Saggitarius ventures into the Shadow of Jupiter and the moon is full. Should I replace her as royal meterologist?

**41.** My stereo system is faulty: there is no sound coming out of the left speaker. Switching the speaker leads will not bring sound to the left speaker if and only if the left speaker is faulty. If switching the speaker leads causes the right speaker to fail, then there is a fault with either the amplifier or the CD player. Switching the leads from the CD player has no effect if and only if there is no problem with the CD player. I discovered the following: switching the leads to the speakers resulted in both channels failing, and switching the leads from the CD player reversed the problem from the left to the right speaker. Therefore replacing the CD player and the left speaker will solve the problem.

**42.** You are chairing an important committee at the UN, and are faced with the following predicament. Upper Volta refuses to sign your new peace accord unless both Costa Rica and Bosnia sign as well. Since Bosnia has a lucrative trade agreement with Iraq, Iraq's signing the peace accord is a sufficient condition for Bosnia to sign the accord. On the other hand, Bosnia, fearful of Upper Volta's recent military buildup, refuses to sign the accord unless Upper Volta also signs. You conclude that Iraq won't sign unless Costa Rica also signs.

**43.** Continuation of Exercise 42 Just as you are about to arrange details for the signing ceremony, Bosnia's representative informs your office that due to a recent scandal involving highly placed Costa Rican officials, the Bosnians refuse to sign any accord with Costa Rica. In retaliation, the Costa Rican government announces a hard-line position: they will not sign the accord unless Iraq also signs. After thinking things over, you come to the depressing conclusion that it will be impossible to have anyone sign the accord. [To make your analysis less time consuming, you may assume the following tautology (which we proved earlier): (p~p)~p. You may also feel free to quote the result of Exercise 42.]

If no violets are red and some roses are blue,

Then nobody loves you; that is certainly true.

But Lucille does love you,

And Susan and Joy.

Some roses are blue

Or you're just a boy.

You assure me you're grown up

And that I believe;

(The brew in your cup

Is so strong, I perceive.)

Now this surely implies

Violets red as your eyes.

**45.** (Dedicated to the Memory of Dr. Seuss)

If you glue, then Wu glues

And Golly glues too.

If Golly glues, Molly glues

And Solly, you too!

But Holly, not Solly glues

With green gooey glue.

Thus Dolly or Holly glues

But not you nor Wu!