The Paradoxe of the liar is derived from the paradox of Crétois (or paradox of Épiménide). In its most concise form, it is stated:

I lie!
One can see two interpretations there:
* as a statement, this sentence known as: “This sentence is false. ”
* as matter, it is necessary to include/understand: “I lie now. ”

History

The oldest trace of this paradox is reported in the Bible:
“Somebody of them, their own prophet, said: Crétois are always lying, of malicious animals, the lazy bellies. ”
the epistle with Tite, chapter 1 verse 12, Paul de Tarse.
This prophet, who lived at seventh century BC, would be Épiménide ''' Crétois '''. However, this first formulation of the paradox of the liar appeared paradoxical only well later; when at the fourth century BC, Eubulide of Millet stated
“A man said that it was lying. What the man did he say is true or false? ”
Many formulations transfer the day, thereafter. Among most recent
“Épiménide, thinker crétois, emitted an immortal assertion: All Crétois are liars. ”
Gödel, Escher, Bach, Douglas Hofstadter, page 19.

Attempts at explanations

Let us allot to Épiménide Crétois the matter “All Crétois are liars.” Épiménide being Crétois itself, if this assertion is true, then Épiménide is a liar, therefore its assertion is false: contradiction! In fact, there is no really paradox: all that one can deduce from the quotation of Épiménide, it is that it is false; in particular all Crétois are not liars, but Épiménide, is to him one. One thus solves the paradox by spreading out it in space.

In fact, the negation of “All Crétois are liars. ” is not: “All Crétois say the truth”, but: “There exists at least Crétois which says the truth” (and it would even be necessary to say, in the direction where lying is used up to now, “There exists at least Crétois which says sometimes the truth”). Therefore, it can exist one or more Crétois liars.

In a similar way, the paradoxical sentence: “I always lie” cease to be it when one spreads out it in time: at the time when I say “I always lie”, I necessarily lie (if not, there is the same problem as with Épiménide), which implies that I always do not lie. There is no contradiction: it sometimes happens to me to lie, but not always!

The paradox of the liar becomes more interesting when one considers the following version of it: “I even lie in this moment”. If the sentence is true, then it is that it is false. But if it is false, then it becomes true!

That indicates that when a sentence can be caught itself for statement, that can lead to an unstable situation (see autodescriptif Pangramme).

Approaches by mathematics

That one considers that the sentence “This sentence is false” is neither true, nor false or that it is about a Non-sens, she refutes in all the cases the Principe of the third excluded. Traditional mathematics, which is based on the excluded third, cannot thus make it possible to build such a statement formally. However, even with the restriction of the excluded third, the paradoxical statements can reappear via a coding of formal logic in a sufficiently rich theory for that, like the arithmetic one or the majority of the theories intended to found mathematics.

Thus Kurt Gödel explicitly refers to the paradox of the liar in the article of 1931 on its two famous theorems of incomplétude: to establish the proof of the first theorem of incomplétude, he manages to code a certain form of this paradox of the liar (where however demonstrability replaces the truth). There is no more paradox, but it is shown that the statement thus built is not provable (and for other reasons, its negation is not it either). The Théorème of Tarski illustrates this step even more clearly: this time it is a question of showing that one cannot express the truth in the arithmetic one, because if not the paradox could be expressed and would provide a contradiction.

One can try to clear up the bond between the paradox of the liar and the incomplétude of certain mathematical theories. Do somebody known as “I lie”, it lies? If he lies it is that he does not lie. If he does not lie it is that he lies. What he says affirms its own falseness. It was seen that this paradox can be presented in another form: the present sentence which starts with “the present sentence” and finishes by “is false” is false, or more simply, this sentence is false.

A theory is a whole of sentences. One can regard it as a kind of teller of truths. The theory says that all its sentences are true. The paradox of the liar proves that there are restrictions on the capacities of the tellers of truth when those are able to formulate statements in connection with what they say. Let us suppose that a teller of truth is able to answer by advance all the questions about what it will answer. Him then the question “with this question you ask will answer not? ”. That he answers yes or not, in both cases he known as forgery. He cannot thus answer without being mistaken.

It is about an essential incomplétude for the theories and the tellers of truth. They cannot say all the truth on all that they say as from the moment when their means of expression are sufficiently rich to make it possible to put questions such as that which has just been quoted. In short, as soon as one can pose with a teller of truth questions such as “to this question you will answer not? ” it cannot at the same time always answer and always say the truth.

See too

Related articles

External bonds

  • Liar Paradox - At the Internet Encyclopedia off Philosophy

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