247 lines
10 KiB
Typst
247 lines
10 KiB
Typst
|
#import "@local/handout:0.1.0": *
|
||
|
|
|
||
|
|
= Gods, Demons, and Mortals
|
||
|
|
|
||
|
|
#definition()
|
||
|
|
Shortly after Inspector Craig returned to London from his
|
||
|
|
strange experience in Subterranea, he had a curious dream.
|
||
|
|
He had been browsing that day in a library specializing in rare
|
||
|
|
books on mythology, another of his many interests. His head
|
||
|
|
was filled with gods and demons, and so his dream was perhaps not so surprising.
|
||
|
|
Time sometimes passes in unusual ways in the course of a
|
||
|
|
dream. Craig dreamed that he spent nine days in a region in
|
||
|
|
which dwelled gods, demons, and mortals. The gods, of
|
||
|
|
course, always told the truth, and the demons always lied. As
|
||
|
|
to the mortals, half were knights and half were knaves. As
|
||
|
|
usual, the knights told the truth and the knaves lied.
|
||
|
|
|
||
|
|
#problem("The First Day")
|
||
|
|
Craig dreamed that on the first day he met a dweller of the
|
||
|
|
region who looked as ifhe might be a god, though Craig could
|
||
|
|
not be sure. The dweller evidently guessed Craig's thoughts,
|
||
|
|
smiled, and made a statement to reassure him. From this state-
|
||
|
|
ment, Craig knew that he was in the presence of a god.
|
||
|
|
Can you supply such a statement?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
One statement that works is: "I am not a knight." If the
|
||
|
|
speaker were a knave or a demon, then it would be true that
|
||
|
|
he was not a knight, but knaves and demons don't make true
|
||
|
|
statements. Therefore the speaker was neither a knavtr nor a
|
||
|
|
demon, hence he was a knight or a god and his statement was
|
||
|
|
true. Since it was true, then he really is not a knight; hence
|
||
|
|
he must be a god.
|
||
|
|
]
|
||
|
|
|
||
|
|
|
||
|
|
#problem("The Second Day")
|
||
|
|
In this episode of the dream, Craig met a terrifying-looking
|
||
|
|
being who had every appearance of being a demon.
|
||
|
|
"What sort of being are'you?" asked Craig, in some alarm.
|
||
|
|
The being answered, and Craig then realized that he was confronting not a demon, but a knave. What could the being have
|
||
|
|
answered?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
A statement that works is: "I am a demon." Obviously
|
||
|
|
no demon can claim he is a demon, so the speaker is not a
|
||
|
|
demon. Therefore his statement was false and since he is not
|
||
|
|
a demon, he must be a knave.
|
||
|
|
Incidentally, this and the last puzzle are essentially the same
|
||
|
|
as Problems 4 and 5 of Chapter 1, the puzzles about the prizes.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("The Third Day")
|
||
|
|
In this episode, Craig met a totally nondescript-looking being
|
||
|
|
who from appearances could have been anything at all. The
|
||
|
|
being then made a statement from which Craig could deduce
|
||
|
|
that he was either a god or a demon, but Craig could not tell
|
||
|
|
which.
|
||
|
|
Can you supply such a statement?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
This is a bit more tricky: A statement that works is: "I
|
||
|
|
am either a god or a knave." That could be said by a god,
|
||
|
|
since a god is either a god or a knave; it could also be falsely
|
||
|
|
said by a demon. It couldn't be said by a knight, because a
|
||
|
|
knight would never lie and claim that he is either a god or a
|
||
|
|
knave, and it couldn't be said by a knave, because a knave
|
||
|
|
would never admit to the true fact that he is either a god or
|
||
|
|
a knave. And so the speaker must be either a god or a demon,
|
||
|
|
but there is no way to tell which.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("The Fourth Day")
|
||
|
|
Craig next met a being who made the following two statements:
|
||
|
|
1. A god once claimed that I am a demon.
|
||
|
|
2. No knight has ever claimed that I am a knave.
|
||
|
|
What sort of being was he?
|
||
|
|
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
The speaker's first statement was obviously false, for if
|
||
|
|
it were true, a god would have once claimed that the speaker
|
||
|
|
was a demon, which would mean that the speaker really was
|
||
|
|
a demon, but no one who tells the truth can be a demon. Since
|
||
|
|
the first statement was false, so was the second statement, since
|
||
|
|
it was made by the same speaker. Therefore a knight did once
|
||
|
|
claim that the speaker was a knave, hence the speaker really
|
||
|
|
is a knave.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("The Fifth Day")
|
||
|
|
A being made the following two statements to Craig:
|
||
|
|
1. I never claim to be a knave.
|
||
|
|
2. I sometimes claim that I am a demon.
|
||
|
|
What sort of being are we now dealing with?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
The speaker's second statement was obviously a lie, be-
|
||
|
|
cause no truth-teller would ever say that he sometimes claims
|
||
|
|
to be a demon. Therefore the first statement was also a lie,
|
||
|
|
hence the speaker does sometimes claim to be a knave, hence
|
||
|
|
he must be a demon.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("The Sixth Day")
|
||
|
|
In this episode, Craig came across two beings, each of whom
|
||
|
|
made a statement. Craig could then infer that at least one of
|
||
|
|
them must be a god, but he could not tell which one. From
|
||
|
|
neither statement alone could Craig have deduced this.
|
||
|
|
What statements could the beings have made?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
Many solutions are possible; here is one. Let us call the
|
||
|
|
two beings A and B. Now, suppose A and B make the fol-
|
||
|
|
lowing two statements:
|
||
|
|
A: B is a knight.
|
||
|
|
B: A is not a knight.
|
||
|
|
A is either telling the truth or lying.
|
||
|
|
Case 1-A is telling the truth: Then B really is a knight,
|
||
|
|
hence his statement is true, hence A is not a knight, therefore
|
||
|
|
A must be a god, since he is telling the truth.
|
||
|
|
Case 2-A is lying: Then B is not a knight, since A says
|
||
|
|
he is. Also, since A is lying, then A is certainly not a knight,
|
||
|
|
hence B's statement is true. Therefore B is telling the truth,
|
||
|
|
but is not a knight, hence B is a god.
|
||
|
|
So if Case 1 is true, A is a god; if Case 2 is true, then B is
|
||
|
|
a god. There is no way to tell whether A is telling the truth
|
||
|
|
or lying.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("The Seventh Day")
|
||
|
|
On the next day, Craig again met two beings each of whom
|
||
|
|
made a statement. Craig could then infer that one of them was
|
||
|
|
a knave and the other a demon, though he could not tell which
|
||
|
|
was which. Again, from neither statement alone could Craig
|
||
|
|
have inferred this. Can you supply two such statements?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
gain let us call the two beings A and B. The following
|
||
|
|
statements would work:
|
||
|
|
A: Both of us are knaves.
|
||
|
|
B: Both of us are demons.
|
||
|
|
It is obvious that both are lying. Since A is lying, they are
|
||
|
|
not both knaves. Since B is lying, they are not both demons.
|
||
|
|
Therefore one is a knave and one is a demon, but there is no
|
||
|
|
way to tell which one is which.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("Introducing Thor")
|
||
|
|
On the eighth day, Craig met a being who had every ap-
|
||
|
|
pearance of being the god Thor. The being made a statement,
|
||
|
|
and Craig then knew he must be Thor.
|
||
|
|
What statement could Thor have made?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
A statement that works is: "I am either a knave or a demon
|
||
|
|
or the god Thor."
|
||
|
|
If the speaker were either a knave or a demon, then it would
|
||
|
|
be true that he is either a knave or a demon or the god Thor.
|
||
|
|
This would mean that a knave or a demon made a true state-
|
||
|
|
ment, which is not possible. Therefore the speaker is neither
|
||
|
|
a knave nor a demon, hence his statement is true. Hence he
|
||
|
|
must be the god Thor.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("A Perplexity Resolved")
|
||
|
|
Craig and Thor became fast friends. In fact, on the evening of
|
||
|
|
the ninth day, Thor gave a magnificent banquet in Craig's
|
||
|
|
honor. "I propose a toast to our illustrious guest!" said Thor,
|
||
|
|
as he raised his glass of nectar.
|
||
|
|
After a round of cheers, Craig was asked to speak.
|
||
|
|
"I am very perplexed!" said Craig as he rose. "I wonder
|
||
|
|
if this may not all be a dream!"
|
||
|
|
"Why do you think you may be dreaming?" asked Thor.
|
||
|
|
"Because," said Craig, "two incidents have occurred today
|
||
|
|
that seem totally inexplicable. This morning I met someone
|
||
|
|
who made a statement which no knight, knave, god, or demon
|
||
|
|
could possibly make. Then this afternoon I met someone else
|
||
|
|
who also made a statement which no dweller of this region
|
||
|
|
could possibly make. That is why I suspect that I may be
|
||
|
|
dreaming. "
|
||
|
|
"Oh!" said Thor. "Be reassured; you are not dreaming.
|
||
|
|
The two incidents have a perfectly rational explanation. You
|
||
|
|
see, we have had two visitors here from another realm. Both
|
||
|
|
of them are mortal. One is Cyrus, who always tells the truth,
|
||
|
|
although he is not called a knight since he is not from this
|
||
|
|
region. The other is Alexander, who sometimes tells the truth
|
||
|
|
and sometimes lies. It must have been those two whom you
|
||
|
|
met today. What statements did they make?"
|
||
|
|
Craig then told the company what each had said.
|
||
|
|
"That explains it perfectly!" said Thor. "Moreover, it fol-
|
||
|
|
lows from their having said what they did that Cyrus was the
|
||
|
|
one you met in the morning. And interestingly enough, if you
|
||
|
|
hadn't met Alexander in the afternoon, you could never have
|
||
|
|
known whether the one you met in the morning was Cyrus
|
||
|
|
or Alexander."
|
||
|
|
Craig thought the matter over and realized that Thor was
|
||
|
|
right.
|
||
|
|
What statements could these two outsiders have made
|
||
|
|
which fulfill all of the above conditions?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
Here is one possible solution.
|
||
|
|
MORNING SPEAKER: "I am neither a knight nor a god."
|
||
|
|
AFTERNOON SPEAKER: "I am either a knave or a demon."
|
||
|
|
No inhabitant of the region could make either of those
|
||
|
|
statements. No knight or god could claim that he is neither a
|
||
|
|
knight nor a god; no knave or demon could make the true
|
||
|
|
statement that he is neither a knight nor a god. As for the
|
||
|
|
second statement, obviously no knight or god would claim to
|
||
|
|
be either a knave or a demon and no knave or demon would
|
||
|
|
admit to being a knave or a demon. Therefore both were out-
|
||
|
|
siders; namely, Cyrus and Alexander. The statement of the
|
||
|
|
morning speaker was true and the statement of the afternoon
|
||
|
|
sp,eaker was false. Since Cyrus never makes false statements,
|
||
|
|
he couldn't have been the afternoon speaker. Thus he was the
|
||
|
|
morning speaker.
|
||
|
|
]
|
||
|
|
|
||
|
|
#problem("A Philosophical Puzzle")
|
||
|
|
The next morning when Craig was wide awake and recalling
|
||
|
|
his dream, he wondered whether he had been logically inconsistent in his sleep. "The trouble is this," thought Craig: "In
|
||
|
|
my dream I believed that Thor was a god and that gods always
|
||
|
|
tell the truth. Yet Thor told me that I wasn't dreaming. Now
|
||
|
|
how could Thor, who tells the truth, say that I wasn't dreaming when in fact I was? Wasn't this an inconsistency on my
|
||
|
|
part?"
|
||
|
|
Would you say that Craig's dream was logically inconsistent?
|
||
|
|
|
||
|
|
#solution[
|
||
|
|
As I see it, Craig's dream was not necessarily inconsistent. If
|
||
|
|
Craig had actually believed in the dream that he was dreaming,
|
||
|
|
then the set of his beliefs during his dream would have been
|
||
|
|
inconsistent, since the following propositions are indeed log-
|
||
|
|
ically contradictory: (1) Thor is a god; (2) Gods make only
|
||
|
|
true statements; (3) Thor stated that Craig was not dreaming;
|
||
|
|
(4) Craig was dreaming.
|
||
|
|
The contradiction is obvious. However, there is no evi-
|
||
|
|
dence that Craig at any time of his dream believed that he was
|
||
|
|
dreaming, although at one point he wondered whether he
|
||
|
|
might be dreaming. Craig presumably believed that he was
|
||
|
|
awake, and this belief, though false, was perfectly consistent
|
||
|
|
with the other beliefs of his dream.
|
||
|
|
Curiously enough, if Craig had formulated the belief that
|
||
|
|
he was dreaming, then this belief, though correct, would have
|
||
|
|
created a logical inconsistency!
|
||
|
|
]
|