Paradoxes

[Flannel Jesus]

I once presented you forum members with a logic puzzle I had found on the site lesswrong.com.
Here is the puzzle I presented

I present to you two baskets, with lids closed. Both of the baskets have notes attached to them.

On basket 1 is written,
"Either this basket contains nothing, or the basket with the false note contains nothing, but not both."

On basket 2 is written,
"Either this basket contains $1million and the basket with the false note contains nothing, or this box contains nothing and the box with the true note contains $1million."

One note is true, and one note is false. One basket contains $1million, and one contains nothing. I will give a picture of $1million to the the first one to both answer correctly and tell me how they proved it. Let the race begin!

I must also assure you that there's no word game going on here, and the answer is deducable.

Here's where I got it from

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:

"Either this box contains an angry frog, or the box with a false inscription contains an angry frog, but not both."

On the second box was inscribed:

"Either this box contains gold and the box with a false inscription contains an angry frog, or this box contains an angry frog and the box with a true inscription contains gold."

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."

As you can see, I changed the details -- boxes for baskets, gold for $1million, and an angry frog for nothing, but I kept the syntax essentially the same, preserving the puzzle.

What I didn't do in this thread, however, was post the second part of this article, which is perhaps a bit more interesting, if you're into this kinda stuff.

The paradox:

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:

"Either both inscriptions are true, or both inscriptions are false."

And the second box was inscribed:

"This box contains the key."

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."

I will end this post here. My next post will explain its paradoxical nature, and why I find it so interesting and so clever.

[Flannel Jesus]

so, 2 boxes, 1 contains a key, one a dagger

box1: "Either both inscriptions are true, or both are false."

box2: "This box contains the key."

The Jester's logic was sound. To avoid paradox, the knife has to be in box2. If the knife is in box1, this is the analysis of the status of the inscriptions:

box2 is false for sure, since obviously the key was in box1
if box1 is true...well, essentially what box1 is saying is that the truth-value of both boxes has to be the same, so if box1 is true, and box2 is false, box1 must also be false, because it has to have the same truth-value as box 2. so "box1 is true" results in a contradiction.
if box1 is false on the other hand...well, that's logically the same as saying that the truth value of the inscriptions are different. one is true and the other is false. but if both box1 and box2 are false, the "or both are false" part of the inscription on box1 is true, making box 1 true. So, box1 is false also results in a contradiction.

Both possible truth-values for box1 result in contradiction if the key is in box1...and yet it was. When I read that for the first time, it hit me really hard. The King's response is absolutely brilliant: "I merely wrote those inscriptions on two boxes, and then I put the dagger in the second one."

You can try it for yourself, as well. Take two envelopes and a ring, write on one envelope "Either both inscriptions are true or both are false," put the ring in that envelope, and write on the other envelope "This envelope contains the ring." You'll find, I think, that reality doesn't collapse in on itself. A black hole doesn't suddenly form where the envelopes were. The world remains existent, you continue on living, pigs don't fly, etc.

But something seems off, ya know? There's a writing on an envelope, and what's written on it is a truth statement. Normally, we'd think that a truth statement is either true or it's false. But this thing is written, and it's a truth statement, and if we assume it's true...it's false, and if we assume it's false...it's true! So what's going on here? Right? Something weird is happening here. Logic's getting turned on it's head.

That's what I thought until I realized that nothing actually contradictory had happened in the real world. You see, in our minds, we see a paradox, but in reality, the configuration of atoms that you arranged...are just a configuration of atoms. Nothing paradoxical happened. You made a series of pen strokes, and put a ring in an envelope. There's nothing in there that implies that there is an actual logical paradox in reality. Despite our inability to evaluate the truth-value of the inscriptions on the box...well, just let what the king said sink in. "I merely wrote those inscriptions on two boxes, and then I put the dagger in the second one."

[phyllo]

Is it possible to logically deduce the location of the key from the given information?

[Flannel Jesus]

Is it possible to logically deduce the location of the key from the given information?

The jester did deduce it. It just so happened that the king put the key in the other box.

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."

[phyllo]

or both inscriptions are false

Is one of those nonsense phrases that you can create by stringing a bunch of words together. Basically the same as "This statement is false." It doesn't give you any more information about the state of the boxes. The correct answer would be "This puzzle can't be solved because it's not a logical puzzle.'

"One fine day in the middle of the night" (Journal Versions)

One fine day in the middle of the night,
Two dead boys* got up to fight, [*or men]
Back to back they faced each other,
Drew their swords and shot each other,

One was blind and the other couldn't, see
So they chose a dummy for a referee.
A blind man went to see fair play,
A dumb man went to shout "hooray!"

A paralyzed donkey passing by,
Kicked the blind man in the eye,
Knocked him through a nine inch wall,
Into a dry ditch and drowned them all,

A deaf policeman heard the noise,
And came to arrest the two dead boys,
If you don't believe this story’s true,
Ask the blind man he saw it too!

[Flannel Jesus]

no need to troll buddy. i mean no harm.

[phyllo]

Trolling? Hardly.

Don't you agree that the phrase is a self-referential negation? Therefore nonsensical.

[Flannel Jesus]

Nonsensical is one possibility. I do see its resemblance to "This statement is false," and even thought about that in my thoughts of this one.
There are other approaches to self-referential negation as well though, which don't all result in "that statement is nonsensical." I'm not sure which one I accept.

[phyllo]

There are other approaches to self-referential negation as well though, which don't all result in "that statement is nonsensical."

The alternate approaches make me think that the phrase does not add enough to make a unique logical choice. So although not nonsensical, it's not very informative.

[gib]

That's what I thought until I realized that nothing actually contradictory had happened in the real world. You see, in our minds, we see a paradox, but in reality, the configuration of atoms that you arranged...are just a configuration of atoms. Nothing paradoxical happened. You made a series of pen strokes, and put a ring in an envelope. There's nothing in there that implies that there is an actual logical paradox in reality. Despite our inability to evaluate the truth-value of the inscriptions on the box...well, just let what the king said sink in. "I merely wrote those inscriptions on two boxes, and then I put the dagger in the second one."

You're absolutely right! The nagging feeling that "something is off" remains only because we want to sort out the logic of the statements in our heads, not because we need to set reality straight. I mean, if you think about it, the fact that the king put the dagger in the second box is itself an element of fantasy (i.e. only part of the thought experiment) and so even there what's nagging at us is not something about the state of reality but the state of an imaginary scenario.

My solution to these kinds of problems has always been to reject the premise that statements must be either true or false, and to replace it with the idea that some statements can have an "undefined" truth value. You see, when it comes to statements, psychologically speaking, they can represent either concepts or beliefs. Before you attach a truth value to them, they are just concepts (i.e. understandings of the meaning of the statement without deciding whether it is true or false), but once you do attach truth values to them, they become beliefs (an attachment of 'false' results in a belief against the statement). We can understand what a statement like "this statement is false" means to convey (easily), thus furnishing us with the concept thereof, but we cannot decide on whether or not to believe it until we can somehow be certain about what truth value to attach to it, and this proves difficult, maybe impossible, with these kinds of statements.

[James S Saint]

Is it possible to logically deduce the location of the key from the given information?

The jester did deduce it. It just so happened that the king put the key in the other box.

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."

No. The jester did not deduce it (properly).
If the first inscription is false, then you know nothing of either box.

The inscription was, "Both inscriptions are true or both are false"
At first glance the possible states are;
A) 1 is true AND 2 is true
B) 1 is true BUT 2 is false
C) 1 is false AND 2 is false
D) 1 is false BUT 2 is true

If the inscription 1 is assumed to be true , then only A and B qualify
A) 1 is true AND 2 is true
B) 1 is true BUT 2 is false

Although 2 cannot be yet deduced, since the inscription 1 is assumed true, the inscription disqualifies B, leaving;
A) 1 is true AND 2 is true
B) 1 is true BUT 2 is false

So IF 1 is true, then 2 can be deduced to also be true.

If the inscription 1 is assumed to be false, then only C and D qualify;
C) 1 is false AND 2 is false
D) 1 is false BUT 2 is true

But option C is self contradicting if false and is thus not a logical statement. Option C cannot be said to be false. Such statements cannot be used for deduction at all, neither for nor against any conclusion. Thus in reality if 1 is assumed false you have;
A) 1 is true AND 2 is true
B) 1 is true BUT 2 is false
D) 1 is unknown BUT 2 is false
C) 1 is unknown AND 2 is true

The second inscription, 2, cannot be deduced from those possible states.
So you merely have 1 equation and 2 unknowns = insufficient information to resolve the puzzle.

[Flannel Jesus]

Thus in reality if 1 is assumed false you have;
A) 1 is true AND 2 is true
B) 1 is true BUT 2 is false
D) 1 is unknown BUT 2 is false
C) 1 is unknown AND 2 is true

You actually just did a great job of defending the jester's logic, ironically.

"in reality if 1 is assumed false, you have...1 is unknown."
How is it assumed false and unknown at the same time? You just proved that "1 is false" is not an option. Leaving only 1 is true, which implies 2 is true.

[James S Saint]

"in reality if 1 is assumed false, you have...1 is unknown."
How is it assumed false and unknown at the same time? [that was the point.. it cannot be both false and known because it contradicts itself]
You just proved that "1 is false" is not an option. [it is not an option for use in deduction. One must remove that option as meaningful]
Leaving only 1 is true, which implies 2 is true. [leaving only that IF 1 IS TRUE, then 2 is true. You are left with merely a contingent truth and nothing else]

I guess it would have been better worded as, "If 1 is assumed to not be true, then.."

In all things there are 3 options, not 2
1) true
2) false
3) not applicable

[phyllo]

[Flannel Jesus]

sounds like you're now saying the same thing phyllo was saying. is that it? that self-contradiction makes a statement nonsensical, and therefore neither true nor false? i thought you were making a new point, hence my confusion.

[James S Saint]

sounds like you're now saying the same thing phyllo was saying. is that it? that self-contradiction makes a statement nonsensical, and therefore neither true nor false? i thought you were making a new point, hence my confusion.

You didn't seem to believe him so I thought that I would go through the details and reasoning.

[Flannel Jesus]

Your approach was less detailed though (despite having more words). You didn't ever say why what he said was true, you just posted a whole bunch more words and repeated that it was true. He was actually more detailed in his explanation, even with less words.

"It's not applicable because it's self-contradiction" is more detail than "it's not applicable," which ended up being the point of your post. We're still no closer to agreement about the best approach to self-contradiction than we were before your post.