The liar paradox was invented in the Ancient Greece and it amazed the people for 25 centuries. They claimed that it’s intractable. Some people are amazed with it still and they claim still that it’s intractable. They’re in error.

There are many versions of that paradox but the rule that works in all of these cases is always the same.

Here’s the example of the liar paradox:

Someone reads that quote and thinks: “The sentence above says that the sentence below is false. So the sentence below is false. The sentence below says falsely that the sentence above is true. So the sentence above is false. The sentence above says falsely that the sentence below is false. So the sentence below is true”.

At that point the contradiction appears: the assumption that the sentence below is false leads to the conclusion that the sentence below is true (and the same with the sentence above). It’s the paradox.

The same paradox appears in a shorter example:

For 2500 years people thought that the liar paradox is intractable. In the 20th century a Polish logician, mathematician, and philosopher – Alfred Tarski – solved that paradox (and many other as well). He invented so-called semantic theory of truth. The core of that theory is a concept of the metalanguages.

Lets imagine that John said: “It’s raining now”, Mary replied: “That’s true: ‘It’s raining now’”, and an Extraterrestrial commented: “Mary said truly that John said the truth”.

As we see John said the statement about the reality outside, Mary said the statement about John’s statement, and an Extraterrestrial said the statement about Mary’s statement about John’s statement. (We still don’t know whether it’s raining indeed because all of them may be the pathological liars but on the other hand we know now that Extraterrestrials exist because they can talk about the other beings talk.)

Lets see the following presentation (read it from the bottom to the top):

John’s statement talks about the reality outside and it belongs to the language. Mary’s statement talks about John’s statement and belongs to the first level metalanguage (it talks about the language which talks about the reality outside). Extraterrestrial’s statement talks about Mary’s statement about John’s statement and belongs to the second level metalanguage (it talks about the language which talks about the language which talks about the reality outside). And so on.

The above is the essence of the conception of the metalanguages. Each sensible sentence belongs either to the language or to the metalanguage of some level. The sensible sentence can’t belong to more than one level of the above stack. The sentence which belongs to a few levels at the same time is a nonsensical sentence and can’t be treated seriously.

Now let’s come back to the liar paradox. The sentence: “This sentence is false” talks about itself. So it’s the sentence from some language and – at the same time – the sentence from the metalanguage. Moreover while we think about that sentence more and more it changes its “truth value” many times back and forth. When we assume that it’s true the conclusion is: it’s false. When we assume that it’s false the conclusion is: it’s true. And so on. So during this considerations that sentence talks about itself more and more times passing as a result an infinite number of “the ordered metalanguages”.

As a result the sentence: “This sentence is false” is nonsensical and can’t be treated seriously.

The same with the other example. The sentence above talks about the sentence below and the sentence below talks about the sentence above. And so on. As a result each sentence talks about itself as well.

So that syllogism is nonsensical too.

That solves the liar paradox.

(I published the above tutorial because some future puzzles or jokes may concern the liar paradox or the pathological liars and the pathologically veracious people.)

What’s more probable as a result of a deal of a deck of the cards in a rubber bridge: 1) you and your partner have together all spades or 2) you both have none spade?

If you take three bananas from a platter with thirteen bananas how many bananas will you have?

A secretary prints six letters and addresses six envelopes to six recipients. A boss inserts these letters into these envelopes at random. Estimate the probability that exactly five letters hit the right envelopes.

zero

---------- Post added 04-07-13 at 03:38 PM ----------

three

...in addition to any you already had possession of prior to taking them. The puzzle doesn't explicitly state that you were devoid of 'narnas when you took them.

My answer would have been two, because I would have eaten one - I like 'narnas.

I agree.

Actually I would have had none before taking them (i.e., I would have already eaten them), and I would have taken all thirteen, which would still leave me with none, but with banana belly-ache!

EDIT

No I was wrong it is an impossible situation.

Let us go back to the ladies and the tigers (see: post #36).

The king decided to complicate the puzzle. So the servants prepared three rooms: one with the lady, one with the tiger, and one empty.

King said to the prisoner that the inscription on the door to the room with the lady is true, the inscription on the door to the room with the tiger is false, and the inscription on the door to the empty room may be true or false.

The prisoner sees the following plates:

Which room should he choose assuming that he is in love with the lady so he is not interested in an empty room?

A Test


Challenge #1

An out-of-control train is hurtling towards five people on the track, who face certain death. You are nearby a switch and, by turning it, you could send the train onto a spur and save their lives. But one man is chained to the spur and would be killed if the train is diverted. Do you flick the switch? Yes or no?

Don’t think too long because the train is coming fast.

Don’t go to the next challenge before you solved the first.


Challenge #2

The train is about to kill five people. You are on a footbridge overlooking the track, next to a fat man. If you were to push him off the bridge onto the track his bulk would stop the train and save the lives of those five people – but kill the fat man. Do you push him? Yes or no?

Don’t think too long because the train is coming fast.

Don’t speculate about the possibility to stop the train using a fat man – just answer the question.


The Diagnoses

To see the diagnoses select the text below.

1. If you decided to do nothing in both cases (ten people killed – two people saved) your emotions play a crucial role in your decisions and you are a rare case.

2. If you decided to flick the switch but refused to push the fat man (six people killed – six people saved) in the first case your reason made the decision and in the latter case – your emotions made it. This behavior is typical for most of the people.

3. If you decided to flick the switch and to push the fat man (two people killed – ten people saved) you use your reason without admitting the emotions. Such a pattern is typical for psychopaths but also for politicians, assassins, chairmen, spies, military men, robbers, surgeons, etc.

4. If you decided to not flick the switch but to push the fat man it seems that you did not understand these challenges well. Maybe you should read them once again?


The Comment

To see my comment select the text below.

The above diagnoses are prepared by the psychologist (with the exception of the fourth one which I prepared in order to complete the set). In my opinion the third diagnosis is not valid because a real psychopath does exactly the same what does an extremely emotional man. However the difference between an emotional one and a psychopath is crucial. An emotional man refuses to do anything because he or she does not want to have something in common with that situation – he or she prefers to pretend that is not there and can not save someone. A psychopath – true psychopath as I understand that – decides to do nothing because he likes when people die so the more people will die the better for the psychopath.


The Exceptions

Because the above is a psychological test it has not one valid solution (each answer – at least from 1. to 3. – is valid). So you can publish here your result if you would like to do that. On the other hand it is not necessary at all so you do not have to do that.

As for the answer 4. if you chose it that not necessary means that you misunderstood the question. Maybe you have very strange sense of humor or you are a loony (from the other people’s point of view it is the same). Both these circumstances justify such an answer so do not worry. Let the other people worry contacting with you.

I am not intelligent enough to be using Slackware, so I hope my puzzle will not be considered offensive by the smart people. In fairness, however, the rules do not explicitly forbid us intellectual inferiors.

A small island far out in the ocean is populated by two tribes: the Good tribe and the Bad tribe. Members of the Good tribe are so honest that they never tell a lie. They also welcome visitors with feasting, singing and dancing. Members of the bad tribe are so dishonest that they never tell the truth. Visitors are attacked, captured and taken to the village to be cooked alive and eaten.

Your ship arrives offshore and you row a small boat ashore. When you reach the beach, you see two men running toward you. One of the men says, "I am Good and so is my friend."

If you do not row back to the ship, will you be feasting or feasted upon?

feasted upon!

A wanderer arrives to the island inhabited by two tribes: the vegetarians which always tell the truth and the cannibals which always lie. At the fork of the road the wanderer meets some native. According to the guidebook one way leads to the cannibals’ village and the other to the vegetarians’ village. Unfortunately the guidebook does not tell which road is which. So the wanderer has to ask the question to the native to learn about these roads. He can ask just one question because after the second question the vegetarians welcome the strangers but the cannibals eat them. What question is the right one in these circumstances?

It may seem so to you, but isn't so in reality. I know "average computer users", and even some people without computers, who are smarter than me. Of course, that might mean I'm not very smart.

P.S.
"Which is the road to the vegetarian village?"

Wrong question (and wrong answer at the same time). The vegetarian will show the way to the vegetarians’ village but the cannibal will show the way to the cannibals’ village so that question does not allow the wanderer to choose the right way.

I asked the people to not publish the answers here. Fortunately some people publish answers which are correct while the other ones publish the answers which are wrong so the next person has still the chance to solve the problem oneself despite the previously published answer which may be right or wrong (and you do not know that before you solve the problem).