Shantila's Inside logic #8+

More Exercises:

The ping pong competition

As mentioned earlier in chapter 7 we will use parentheses in obvious ways to make clear what we are saying. To illustrate further the use of parentheses, here is a list of some English sentences that could be made about a ping pong tournament, together with the representations of these sentences in our simple symbolic language. You should examine these 13 examples to make sure you understand the symbolization before tackling the Practice exercises in this section.

We will use these letters as abbreviations. K: Kim wins the first round. E: Esther wins the first round. D: Dmitri gives up. C: Clark gives up.

1. If Kim wins the first round, then both Dmitri will give up and so will Clark.

This sentence can be represented as: K>(D&C)

Note: For present purposes, we treat future tense statements like "Dmitri will give up" or "Dmitri is going to give up" as meaning the same thing as the simple present tense statement "Dmitri gives up" -- so here we would represent both of these just using D. Likewise we represent the past tense statements "Dmitri did give up" and "Dmitri gave up" in the same way --simply using D. (In more advanced logical investigations, we might want our logic to differentiate between the past, present, and future tenses.)

2. If Dmitri gives up and Kim wins the first round, then Clark will give up.

This sentence is represented as: (D&K)>C

3. Kim will not win the first round and Esther will not win the first round.

~K & ~E

4. It is not true that both Kim and Esther will win the first round. (Notice how this one differs from the preceding statement 3.)

~(K&E)

5. If Kim wins the first round, then Esther will not win the first round.

K>~E

6. Esther will not win the first round if Kim wins the first round. (Notice this means the very same thing as the preceding statement 5.)

K>~E

7. If Kim does not win the first round and Clark gives up, then Esther will win the first round.

(~K&C)>E

8. Kim will win the first round; and if Dmitri gives up, then so will Clark.

K & (D>C)

9. Esther will not win the first round; and Kim will win the first round if Dmitri gives up.

~E & (D>K)

10. If Dmitri gives up, then Esther will win the first round if Clark gives up.

D > (C>E)

11. Dmitri will not give up and Clark will not give up.

~D & ~C

12. Neither Dmitri nor Clark will give up. (Notice this means just the same thing as 11.)

~D & ~C

13. Neither Kim nor Esther will win the first round. (Notice this means just the same thing as 3.)

~K & ~E

*Practice

8.4 Symbolize each of the following sentences, using the same letters as above.

a. Dmitri will not give up and Kim will not win the first round.

b. Kim will not win the first round if Esther does.

c. If Kim wins the first round and Dmitri gives up, then Clark will not give up.

d. If Esther wins the first round, then Dmitri will give up and Kim will not win the first round.

e. If Kim wins the first round and Clark does not give up, then Dmitri will give up.

f. If Esther wins the first round and Dmitri does not give up, then Kim will not win the first round and Clark will give up.

g. If Esther wins the first round then both Dmitri will give up and Kim will not win the first round

h. It is not the case that both Dmitri and Clark will give up.

i. Dmitri will not give up and neither will Clark.

j. If neither Dmitri nor Clark gives up, then Kim will not win the first round.

k. If Dmitri does not give up, then not both Kim and Esther will win the first round.

8.5 For each of the following, (a) symbolize the reasoning, and (b) construct a proof to show that the reasoning is valid. Use the same symbols as in the preceding exercise.

a. Kim will not win the first round if Esther does. Dmitri will give up and Esther will win the first round. So Kim won't.

b. If Esther wins the first round, then Dmitri will give up and Kim will not win the first round. Yet Esther will win the first round. So Kim won't.

c. If Kim wins the first round and Clark does not give up, then Dmitri will give up. Clark will not give up. Kim will win the first round. So Dmitri will give up.

d. Kim wins the first round and Dmitri will give up. So Dmitri will give up and Kim wins the first round.

e. Kim will win the first round; and if Dmitri gives up, then so will Clark. Dmitri will give up. So Clark will give up as well.

f. If Kim wins the first round, then Dmitri will give up and so will Clark. Kim will win the first round. So Dmitri will give up.

g. If Kim wins the first round, then Esther will not win the first round. If Dmitri will give up, then Esther will win the first round. Kim will win the first round. Therefore, Dmitri will not give up.

h. If Dmitri gives up then Kim will win the first round. Clark will give up and so will Dmitri. If Kim wins the first round and Clark gives up, then Esther will not win the first round. So Esther will not win the first round and Dmitri will give up.

i. Kim will not win the first round and Esther will not win the first round. If Dmitri gives up, then Esther will win the first round. So Dmitri does not give up and Esther will not win the first round.

j. If Kim wins the first round and Dmitri gives up, then Clark will not give up. But Clark will give up. So it is not the case that both Kim will win the first round and Dmitri gives up.

k. If Dmitri gives up, then Esther will win the first round if Clark gives up. Esther will not win the first round and Dmitri does give up. So Clark does not give up.

l. If Esther wins the first round, then not both Clark and Dmitri will give up. But Clark will give up. Dmitri will give up if Kim wins the first round, and Kim does win the first round. So Esther does not win the first round.

m. If Dmitri gives up and Esther wins the first round, then I'll be a monkey's uncle. If Kim wins, Dmitri gives up. My nephew is a monkey if I am a monkey's uncle. But my nephew is not a monkey (although he acts like it most of the time). Therefore, it is not the case that both Dmitri gives up and Esther wins the first round.