Shantila's
Inside logic #7+
More
Exercises:
The
ping pong competition
Having
shown Joe the way to the bathroom you sit back
down. The television is still on ESPN. The hockey
game is over and now there is a ping pong
tournament being shown. The announcer is making a
number of claims about what is going on. For
example, he says that if Kim wins in the first
round, Dmitri will give up the match. That sounds
odd. But, anyway, what do you care about ping
pong?
*
As
mentioned earler in chapter 7 we will use
parentheses in obvious ways to prevent confusion
in 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.
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, of course, we might want our
logic to differentiate between the tenses --
past, present, and future.)
2. If
Dmitri gives up and Kim wins the first round,
then Clark will give up.
(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
7.5 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.
7.6 For each of the following, (a)
symbolize the reasoning, and (b) construct a
proof to show that the reasoning is valid.
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.
7.7 Use the rules A, MP, DN, MT,
&E, and &I to construct proofs for each
of the following sequents.
31 P>R, P>Q, ~~P
} R&Q
32 P>(P>Q), P } Q
33 Q>(P>R), ~R, Q
} ~P
34 P>(Q>R),
P&Q } R
35 P>(R&Q),
~(R&Q), Q>P } ~Q
36 (P&Q)>R, ~S,
R>S } ~(P&Q)
7.8 Back to the ping pong
tournament. For each of the following, (a)
symbolize the reasoning, and (b) construct a
proof to show that the reasoning is valid.
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.
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