'This sentence is false.' Five words that logicians still can't tame.
If the sentence is true, then what it says holds, so it's false. If it's false, then it's true. Thinkers in ancient Greece, fifth-century India and the medieval Islamic world all wrestled with this tiny loop. Every neat fix seems to spawn a nastier version of the same problem.
One early version is credited to Eubulides of Miletus in the fourth century BC, who asked whether a man who says he is lying speaks truly or falsely. The better-known line from the Cretan seer Epimenides, 'All Cretans are liars', is often cited as the original, but it is not a true paradox: it can simply be false, provided at least one Cretan tells the truth.
The puzzle was not only a Greek preoccupation. The Indian grammarian and philosopher Bhartrhari, in the late fifth century AD, analysed the statement 'everything I am saying is false'. Scholars in the Islamic world discussed the liar for at least five centuries from the late ninth century, apparently independently of other traditions, and Nasir al-Din al-Tusi may have been the first logician to pin the trouble on self-reference.
The obvious escape is to say the liar sentence is neither true nor false. But then consider 'This sentence is not true.' If it is neither true nor false, it is certainly not true, which is exactly what it claims, so it is true after all. The philosopher Graham Priest takes a bolder line, accepting that some statements are both true and false, but critics have built liar sentences aimed at that view too. The paradox can also be spread across two sentences that each describe the other.
In the 20th century Alfred Tarski argued the problem arises only in languages that can talk about their own truth. His solution was a hierarchy: a sentence can only call sentences at a lower level true or false. Saul Kripke later showed that whether a sentence is paradoxical can depend on ordinary facts about the world, and developed an influential theory of truth in response. The liar matters because truth is one of logic's most basic ideas, and this little loop suggests we still don't fully understand it.
Source: Wikipedia — Liar paradox · Text summarised from Wikipedia (CC BY-SA 4.0)
