Epimenides paradox

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The Greek philosopher Epimenides relates in the sixth century BC that "All Cretans are liars... One of their own poets has said so." The poet's statement is sometimes wrongly considered to be paradoxical because the poet himself is a Cretan; it is not.

Common usage defines a "liar" as someone who occasionally produces answers that differ from the known truth. This presents 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 lies always. 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" (see DeMorgan's laws), in other words: some Cretans sometimes speak the truth. This does not contradict the fact that our Cretan poet just lied.

Therefore, the statement "All Cretans are liars", if uttered by a Cretan, is 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.

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