The Liar Paradox – La Libre

Take as an example a dishonest person, a liar. Imagine that the liar suddenly says to you: “I’m a liar”. Should you believe the liar who tells you that he is a liar? Famous dilemma.

If you believe him, he’s not a liar. You’re done. If you don’t believe him, he is a liar, but he isn’t because he says he is. You are overcooked. Whichever way you look at it, you will find a paradox.

This famous paradox is called the liar’s paradox. It dates back to 700 BC and is attributed to Epimenides the Cretan, who is said to have said: ““All Cretans are liars.”. We find it in the epistle to Titus attributed to Saint Paul: “Someone among them, their own prophet, said: The Cretans are always liars, evil beasts, lazy bellies.”

A trick of reasoning

This antinomy is very powerful because it is based on a contradictory self-reference. The paradox is insoluble, unless one leaves the box. Consider the written version of the liar’s paradox: “This sentence is false.”. Should we believe the sentence that says it is false? We got out of there by a trick of reasoning. “The phrase ‘This phrase is false’ is false.”. We have moved the verification of truth to another level of language. We create a kind of metalanguage.

On a lighter note, here is Pinocchio’s version of the liar’s paradox: “My nose is getting longer! Should we believe it? If he tells the truth, his nose doesn’t grow any more. If he doesn’t go to bed, he is lying since she said she would go to bed. On the other hand, if he lies, his nose grows, but in this case he would have told the truth. It is insoluble.

Some logicians get away with prohibiting contradictory statements. In other words, it is not possible for Pinocchio to say: ““My nose is getting longer.”. You can say anything but that. It is not possible for a Cretan to affirm that “All Cretans are liars.” and we cannot write “this sentence is false”.

Wrestling between Plato and Socrates

In practice, we often use metalanguage to get by. It happens that a politician or an expert (it is fashionable) says more or less “Don’t believe the experts”. We quickly understand that the expert tells us: “Don’t believe other experts.” or the politician: ““Don’t believe other politicians.”

The contradictory statement can also be developed with two characters passing the buck, in this case Plato and Socrates.

– Plato says: “Socrates’ next proposition will be a lie.”.

– Socrates answers: “Plato tells the truth”.

Our two boys are “pale blue chocolate.” They fell into the logical trap that they themselves set. The first says that the following proposition will be false and the second says that it is true. Create the paradox of the first. If it is true, it is false, therefore it is not true (and vice versa in the opposite sense).

Even AI is chocolate

Before the development of artificial intelligence, we had imagined theoretical machines capable of analyzing a proposition. If it is true, the machine turns on a green light, if it is false, a red light. How to create a self-reference that crashes the machine? We would have to predict the button that will turn on the machine, preventing it from doing so: “The next illuminated button will be red”. Ready. The machine trips and turns on again. If it’s red, it’s wrong but it’s true. If it is green, it is true but it is false. What could the machine do? Don’t turn on anything at all. She would abstain, because we are entering the realm of prohibited propositions.

There is a slightly more sophisticated version of the liar paradox, that of Protagoras. He once again stages self-reference in a subtle game of nonsense. Protagoras was a sophist. He had a penniless disciple whom he taught argumentation.

– “It doesn’t matter, Protagoras told him: You will pay me with the money from your first test.”

Therefore, the agreement stipulated that when the disciple won his first trial, he would pay the teacher for his teaching. So far, so good. The matter was this: the trained and well-trained disciple did not touch any. He was left lying on the beach counting stones.

Protagoras’ mustard ran up his nose. So he himself sued his own disciple to force him to pay him. Protagoras’ reasoning was astute. If Protagoras won the case, the disciple would have been forced to pay him. Or Protagoras lost the trial, the disciple would also have been forced to pay because he would have finally won a first trial!

The clever trick is that the disciple had made a similar reasoning, but in the other direction. If the disciple won his case, he did not have to pay Protagoras because the case had been won. If the disciple lost the case, he also did not have to pay Protagoras because he had not yet won the case.

Physicists would not be happy, for them the problem is poorly posed. It is unknown whether the lawsuit that Protagoras files against his disciple to finally get paid is part of the agreement or not. Confusion creates the trap we fall into: it is paradox.

The so-called Jewish humor likes clever jokes with logical paradoxes. This represents the paradox of the liar with a person who perhaps does not lie! This is the height of the height. Two friends meet on a station platform.

– “Where are you going ?”

–“I’m going to Krakow.”

– “You tell me that you are going to Krakow, so that I believe that you are going to Lodz, when I know that you are going to Krakow. So why do you lie?

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