I'm having some trouble with the logic here. I wonder if the parable got a bit garbled.
"You see," the jester said, "let us hypothesize that the first inscription is the true one."
The first inscription says, "Either this box contains an angry frog, or the box with a false inscription contains an angry frog, but not both." Now we are hypothesizing that this is the true one. Therefore "the box with a false inscription" means "the second box". So, "Either the 1st box contains an angry frog, or the 2nd box contains an angry frog, but not both".
The jester goes on, "Then suppose the first box contains an angry frog."
So we know (by assumption) that the 1st clause of the inscription is true, the 1st box contains an angry frog. Since "not both" clauses are true, it means the 2nd clause is false, and so the 2nd box does not contain an angry frog - it must contain gold.
But the jester claims that this is a contradiction: "Then the other box would contain gold and this would contradict the first inscription which we hypothesized to be true." For this to be a contradiction, the 1st inscription would have had to say that the 2nd box should contain an angry frog, but we just saw that it doesn't say that.
I can't make much progress with the 2nd inscription either. I'm getting pretty confused now!
Bx is true if box x has gold, false if frog. one contains frog, other gold -> B1 == ~B2. only one inscription is true -> Bf == ~Bt
We know:
B2 && Bf || Bt && B1 (I1)
B2 && Bt || B1 && Bt (I2)
Bt == B1 && Bf == B2 && I1 && ~I2 || Bf == B1 && Bt == B2 && ~I1 && I2 # only one inscription is true
From this:
((B2 && B2 || B1 && B1) && ~(B2 && B1 || B1 && B1)) || (~(B2 && B1 || B2 && B1) && (B2 && B2 || B1 &...
Once upon a time, there was a court jester who dabbled in logic.
The jester presented the king with two boxes. Upon the first box was inscribed:
On the second box was inscribed:
And the jester said to the king: "One box contains an angry frog, the other box gold; and one, and only one, of the inscriptions is true."
The king opened the wrong box, and was savaged by an angry frog.
"You see," the jester said, "let us hypothesize that the first inscription is the true one. Then suppose the first box contains gold. Then the other box would have an angry frog, while the box with a true inscription would contain gold, which would make the second statement true as well. Now hypothesize that the first inscription is false, and that the first box contains gold. Then the second inscription would be—"
The king ordered the jester thrown in the dungeons.
A day later, the jester was brought before the king in chains, and shown two boxes.
"One box contains a key," said the king, "to unlock your chains; and if you find the key you are free. But the other box contains a dagger for your heart, if you fail."
And the first box was inscribed:
And the second box was inscribed:
The jester reasoned thusly: "Suppose the first inscription is true. Then the second inscription must also be true. Now suppose the first inscription is false. Then again the second inscription must be true. So the second box must contain the key, if the first inscription is true, and also if the first inscription is false. Therefore, the second box must logically contain the key."
The jester opened the second box, and found a dagger.
"How?!" cried the jester in horror, as he was dragged away. "It's logically impossible!"
"It is entirely possible," replied the king. "I merely wrote those inscriptions on two boxes, and then I put the dagger in the second one."
(Adapted from Raymond Smullyan.)