What is the liar paradox?

The liar paradox is a puzzle that arises from a sentence that says of itself that it is false, such as "This sentence is false." If the sentence is true, then what it says holds, so it is false. If it is false, then what it says is correct, so it is true. Either assumption leads to its opposite, and the sentence seems to be both true and false. Collins Dictionary describes it as a statement that is true only if it is false and false only if it is true. [1] [2]

The Stanford Encyclopedia of Philosophy sets out a sentence it calls FLiar, which says of itself that it is false. If FLiar is true, then given what it says, it is false. If it is false, then the sentence saying that it is false is true. So FLiar is false if and only if it is true. If every sentence must be true or false, FLiar is both, and that is a contradiction. [3]

The Internet Encyclopedia of Philosophy adds why this matters. In classical logic, anything follows from a contradiction, even 1 + 1 = 3. A contradiction therefore seems to threaten semantic coherence altogether. The Stanford entry calls the principle behind this EFQ, the rule that every sentence follows from a contradiction, and says that the real question the paradox raises is which step in the reasoning is wrong. [4] [3]

The puzzle is very old. The Stanford entry says it was discussed in classical times, notably by the Megarians, and was also mentioned by Aristotle and Cicero. Medieval logicians such as Buridan studied it as one of the insolubilia. Work on it has also been part of the development of modern mathematical logic. [3]

It is sometimes called the Epimenides paradox, because tradition attributes to Epimenides of Crete the claim that all Cretans are always liars. Oxford Reference notes that the New Testament letter to Titus mentions a Cretan saying this. The Cretan version is a looser relative of the pure self-referring sentence, and the Stanford entry stresses that the puzzle is not really about intentions or social norms around lying but about truth. [3] [5] [1]

The Internet Encyclopedia of Philosophy says experts in philosophical logic have never agreed on a way out despite about 2,300 years of attention. One obvious response is to deny bivalence, the principle that every sentence is true or false, and allow some sentences to be neither. The Stanford entry says descendants of this idea remain important in current work. [4] [3]

That response is not enough on its own. A variant sentence that says of itself that it is not true gives rise to the paradox even if some sentences are neither true nor false, because "not true" covers both false sentences and those that are neither. Giving up the Law of Excluded Middle also does not suffice, since there are versions of the reasoning that rely on Reductio instead. Blocking the paradox therefore requires further assumptions, for example about negation, and the sources present several competing ways of restoring coherence rather than one accepted solution. [3] [4]