'This statement is false.' What do we make of this statement (call it S)? If S is true, then S is false. On the other hand, if S is false, then it is true to say S is false; but, because the Liar sentence is saying precisely that (namely that it is false), S is true. So S is true if and only if it is false. Since S is one or the other, it is both! Debate about sentences like S has been going on among philosophers and logicians for more than 2,000 years without any clear resolution.
The roots of the Liar paradox stretch back to the philosopher Epimenides in the 6th century BC. Epimenides said: "All Cretans are liars.... One of their own poets has said so." Another version of this can be found in the Bible, Titus 1, verse 12-13: "Even one of their own prophets has said, 'Cretans are always liars, evil brutes, lazy gluttons.' This testimony is true." The poet's (or prophet's) statement is sometimes wrongly considered to be paradoxical because he himself is a Cretan. But actually there is no paradox here. A 'liar,' in everyday language, is someone who on occasions knowingly gives false answers. This leads to no problem at all: the poet, while lying occasionally, this time spoke the truth. However, most formulations of logic define a 'liar' as an entity that always produces the negation of the true answer, that is, someone who does nothing but lie. Thus, the poet's statement cannot be true: if it were, then he himself would be a liar who just spoke the truth, but liars don't do that. However, no contradiction arises if the poet's statement is taken to be false: the negation of 'All Cretans are liars' is 'Some Cretans aren't liars', in other words: some Cretans sometimes speak the truth. This doesn't contradict the fact that our Cretan poet just lied. Therefore, the statement 'All Cretans are liars', if uttered by a Cretan, is necessarily false, but not paradoxical. Even the statement 'I am a liar' is not paradoxical; depending on the definition of 'liar' it may be true or false. However, the statement 'I am lying now,' first attributed to Eubulides of Miletus in the 4th century BC, definitely is paradoxical. It is exactly equivalent to the sentence, we started with: 'This statement is false.'
Various elaborations of the basic Eubulides Liar paradox have appeared over the ages. In the fourteenth century, the French philosopher Jean Buridan applied it in his argument for the existence of God. In 1913, the English mathematician Philip Jourdain (1879-1921) offered a version that is sometimes referred to as 'Jourdain's Card Paradox.' On one side of a card is written:
THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS TRUE.
On the other side is written:
THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS FALSE.
Yet another popular version of the Liar paradox, guaranteed to perplex, is given by the following three sentences written on a card:
(1) THIS SENTENCE CONTAINS FIVE WORDS.
(2) THIS SENTENCE CONTAINS EIGHT WORDS.
(3) EXACTLY ONE SENTENCE ON THIS CARD IS TRUE.