The liar paradox
"This sentence is false." Is it true or false?
Eubulides of Miletus, 4th century BCE
The parrot that only told lies
Meera's grandmother had a parrot named Mithu who, the family joked, had never said a true thing in his life. When it rained, Mithu said "Sunny!" When the milkman came, he said "Nobody's here!"
One evening Meera's little cousin taught Mithu a new sentence. The parrot puffed up his green feathers and announced: "Everything Mithu says is a lie!"
The whole family went quiet. If Mithu always lies, then that sentence must be a lie too, so he does sometimes tell the truth. But if he does tell the truth sometimes, then the sentence was false, which is exactly what a lying parrot would say. Meera tried a simpler version on a slip of paper: This sentence is false. If it is true, it is false. If it is false, it is true.
Meera's grandmother laughed and said the parrot had just invented philosophy.
The puzzle
A sentence that says of itself that it is false seems to be true exactly when it is false.
See it for yourself
What's really going on
Most sentences point at something outside themselves, such as the weather or a number. The liar sentence points only at itself, and its truth depends on its own truth, so it chases its own tail.
Logicians have tried several fixes. Alfred Tarski suggested that a language cannot talk about its own truth; you need a higher 'meta-language' to say whether sentences in the first one are true. Others allow some sentences to be neither true nor false, or to be both. Kurt Gödel turned a similar self-reference into a famous theorem in 1931: any rich enough system of maths contains true statements that it cannot prove.
Still argued about. There is no single accepted solution. Each fix seems to create a new 'revenge' version of the paradox.
Think about it
Is the sentence "This sentence is true" true or false? Can you ever tell?