- Solutions to Solving Conundrums 1
(and 2)
The Second Conundrum
"Which way would you tell me to go
if I were to ask?" (Then take that way)
If it's the one who always lie, he
would tell you to go the wrong way, but as he is lying he has to lie
to the question by telling you the right way.
The Fourth Conundrum
The solutions to this depends on what
you mean by surprised. In practical terms, what the death-sentenced
king does is to believe with all his mind that he will be killed
tomorrow (he believes this every day). Tuesday morning he is not
killed, but does he consider himself to be surprised?
There are three ways to define
surprised (that I can think of). The first is "I am surprised
whenever I am wrong". Using this interpretation, where is he
wrong in his analysis? He must decide in advance on one day when he
thinks he will be executed. If he decides 'I think I'll be executed
on Tuesday', then he is surprised when he is executed on Friday (or,
rather, when he is not executed on Tuesday).
The second interpretation is "I am
surprised whenever I have a false negative, that is, when I predict
that I will not be executed but I am." So if you have a false
positive (you think you'll be executed but you're not) you don't
count as surprised. Then the king is correct, his analysis is good,
it is impossible to surprise him. (This goes against the way I told
the story, but there exist different versions.)
The third interpretation is a bit more
complex. If it rains/not rains tomorrow, are you surprised? Not too
much, because you don't have a strong prediction. But if a volcano
erupts in your neighborhood, then you are surprised. If someone asked
you beforehand you'd say that the chance of a volcano erupting so
near you was almost 0, or just say 0. The interpretation is: "You
are surprised whenever something occurs that you assigned less than
10% probability" (I chose 10% for convenience, you can
substitute it with any number less than 50%). Every night the king
can assign 50% probability to being executed the next morning. Then
he is never surprised.
The Third Conundrum
When making statements in logic you can
make almost any statement you can dream up. Almost. You are not
allowed to define a logical variable by 'P:(not P)'. The easiest way
to make sure your statements are definable is to only use logical
variables that you have already defined, together with logical
operations (and, or, not, etc.).
Let us define the logical variables:
P: not Q
Q: not P
This is impossible in the same way as
P: not Q
is impossible (you can't know if P is
true or false, it can't be true or false, as you have not even
defined Q).
If you'd rather think of this as a
satisfiability question (see the last solution on this post) for 'P
equivalent (not P)', then the answer is "no, there is no truth
values satisfying this expression".
The First Conundrum
Using the clarified version:
If A) is correct then D) is also
correct, and then A) is not correct as there are two correct
solutions.
If B) is correct, then B) is false by
its own statement.
If C) is correct, then C) is false by
its own statement.
If D) is correct then A) is also
correct and then D) is not correct.
So every answer gives a contradiction,
hence none of them are correct. But isn't then alternative C)
correct, since no of the alternatives are correct?
The resolution here comes from thinking
about satisfiability. If you have a logical system you can give a
number of variables A, B, C, ..., and a logical 'equation'
(statement), for example
(A and B) or (C and (not A) and (not D)
or (A equivalent B),
and ask the question "Is there a
set of values for the variables that make the equation true?".
The answer can be
"yes; A=true, B=false, ...",
or
"no; there is no such choice of
values".
Finally, the answer becomes "The
question you posed has no correct answer". On this higher level,
where you defined the question, you can say that there is no answer
that can be correct (as all of them leads to contradictions).
The question cannot be satisfied. It is
not so that 0 of the alternatives are correct on the level of the
question. On the level above, it is so.
If you disagree, I would recommend a
book on mathematical logic, and one on set theory (not naive set
theory, but the serious kind), at least read the wikipedia articles
And to see how complicated stuff
becomes when someone tells you everything, see
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