Epimenides paradox

The Epimenides paradox is a problem in logic. This problem is named after the Cretan philosopher Epimenides of Knossos (flourished circa 600 BC), who stated Κρῆτες ἀεί ψεύσται, "Cretans, always liars". There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach (page 17), by Douglas R. Hofstadter.

Logical analysis

If we define "liar" to mean that every statement made by a liar is false (so that Epimenides' statement amounts to "Anything said by a Cretan is false"), then the statement "All Cretans are liars," if uttered by the Cretan Epimenides, cannot be consistently true. (And, as will be noted below, according to one interpretation it also cannot be consistently false, either.)

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The conjunction of "Epimenides said all Cretans are liars" and "Epimenides is a Cretan" would, if true, imply that a Cretan has truthfully asserted that no Cretan has ever spoken the truth; the truth of Epimenides' statement would be a counterexample (some Cretan has told the truth at least once) and would mean that not all Cretans are necessarily always liars, which would contradict Epimenides' statement and thus would render it false.

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Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truth-value "false" can be consistently assigned to the simple proposition that "All Cretans are liars," so that this statement by itself, when deemed false, is not, strictly speaking, paradoxical. Thus, if there ever existed a Cretan (not Epimenides in this instance) who even once spoke the truth, the categorical statement "All Cretans are (always) liars," would be false, and Epimenides might be simply regarded as having made a false statement himself. But if Epimenides' statement is understood as in essence asserting its own falsehood, then the statement cannot consistently be false, either, because its falsehood would imply the truth of its self-asserted falsehood.

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An interesting asymmetry is possible under one interpretation: the statement's truth clearly implies its falsehood, but, unless the statement is interpreted to refer specifically to itself (rather than referring categorically to all statements by Cretans), the statement could be contingently false without implying its own truth.

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Alternatively, if, by "liar", we were to mean someone whose statements are usually but not always false, the logical problem would dissolve: Epimenides might usually lie, but on this occasion it might be that he happened to speak the truth. This interpretation would not lead to an interesting logical problem.

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Another interpretation of the term "liar" is someone who is simply willing to lie (i.e., when it suits him). This is the most common interpretation of the phrase "X is a liar" in English and, indeed, in most languages. However, this also does not lead to an interesting logical problem as all Epimenides would essentially be claiming is that Cretans do not possess ethical qualms about lying. Thus, it is critical to caveat the Epimenides paradox by defining "liar" as someone who always and exclusively speaks falsehoods.

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Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides. (The Epimenides paradox is usually classified as a variation on the liar paradox, and sometimes the two are not distinguished.) The study of self-reference led to important developments in logic and mathematics in the twentieth century.

Related Topics:
Liar paradox - Russell's paradox - Burali-Forti paradox

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