Epimenides paradox
Encyclopedia
The Epimenides paradox is a problem in logic
. It is named after the Cretan
philosopher Epimenides
of Knossos
(alive circa 600 BC), There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach
, by Douglas Hofstadter
:
The Epistle to Titus
makes apparent reference to Epimenides: the author says of Cretans that "they are always liars, as one of their own has said."
A paradox
of self-reference
is commonly supposed to arise when one considers whether Epimenides spoke the truth. However, if Epimenides knew of at least one Cretan (other than himself) who was not a liar, then his statement is a non-paradoxical lie in that it does not lead to a logical contradiction. (The contradictory of the statement, "All Cretans are liars" is the statement, "Some Cretans are not liars", which might be true at the same time as the statement, "Some Cretans are liars.")
was immortal, as in the following poem:
Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus
in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted in the Epistle to Titus
:
The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus
. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of; accusing Cretans (other than himself) denying the immortality of Zeus while he did not deny it himself. It is also quite natural to understand the Cretan poet as possibly having employed the figure of speech known as hyperbole (deliberate exaggeration) rather than having advanced a vigorous logical claim. It is not clear when Epimenides became associated with the Epimenides paradox. Epimenides himself does not appear to have intended any irony or paradox in his statement, "Cretans, always liars", nor did Callimachus, nor the author of Titus, nor Clement of Alexandria
:
Saint Augustine
restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). In the Middle Ages
, many forms of the liar paradox were studied under the heading of insolubilia
, but these were not explicitly associated with Epimenides. The second volume of Pierre Bayle
's Dictionnaire Historique et Critique
of 1740 explicitly connects Epimenides with the paradox, though Bayle labels the paradox a "sophisme".
In his epistle to Titus, Apostle Paul wants to warn Titus that Cretans don't believe in the one truth of Christianity, because "Cretans are always liars". To justify his claim, Apostle Paul cites Epimenides. This forms a contradiction because he cites the words of a Cretan (always a liar) in order to prove that Cretans are always liars, also concluding that Epimenides (a Cretan) had surely told the truth (that Cretans are always liars).
Define a "liar" as someone who is never truthful. Then the statement, "All Cretans are liars", if uttered by a Cretan, in this case Epimenides, implies that the speaker's statement is not true—that is, some Cretans are not liars. Some logicians have treated the Epimenides paradox as identical to the Liar paradox
.
For example, Thomas Fowler (1869) states the paradox as follows: "Epimenides the Cretan says, 'that all the Cretans are liars,' but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; hence the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful." However, the inference from "not all Cretans are liars", to "the Cretans are veracious" is not valid.
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 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.
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.
Naturally, any truly logical idea of a paradox with the statement falls flat if one understands that while "all Cretans" may be "liars," such a statement in realistic terms does not necessarily mean that all Cretans lie all the time or that they lie only. Even if it is said that "Cretans are always liars," this does not produce a paradox if one understands the various meanings of the term always—as in "John always says No!" does not mean that "No" is all—or the only thing—John ever says. Certainly even the most prolific liars in history told the truth at least some of the time, so an idea that anyone lies in every single sentence they speak is merely simpleminded at best. The word "always" can also be interpreted to mean "in all instances", as in, "when you find a Cretan, you will always have found a liar," where "liar," again, need not mean more than a person who is known to lie.
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. Indeed, 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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
. It is named after the Cretan
Crete
Crete is the largest and most populous of the Greek islands, the fifth largest island in the Mediterranean Sea, and one of the thirteen administrative regions of Greece. It forms a significant part of the economy and cultural heritage of Greece while retaining its own local cultural traits...
philosopher Epimenides
Epimenides
Epimenides of Knossos was a semi-mythical 6th century BC Greek seer and philosopher-poet. While tending his father's sheep, he is said to have fallen asleep for fifty-seven years in a Cretan cave sacred to Zeus, after which he reportedly awoke with the gift of prophecy...
of Knossos
Knossos
Knossos , also known as Labyrinth, or Knossos Palace, is the largest Bronze Age archaeological site on Crete and probably the ceremonial and political centre of the Minoan civilization and culture. The palace appears as a maze of workrooms, living spaces, and store rooms close to a central square...
(alive circa 600 BC), There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach
Gödel, Escher, Bach
Gödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
, by Douglas Hofstadter
Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...
:
- Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."
The Epistle to Titus
Epistle to Titus
The Epistle of Paul to Titus, usually referred to simply as Titus, is one of the three Pastoral Epistles , traditionally attributed to Saint Paul, and is part of the New Testament...
makes apparent reference to Epimenides: the author says of Cretans that "they are always liars, as one of their own has said."
A paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
of self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
is commonly supposed to arise when one considers whether Epimenides spoke the truth. However, if Epimenides knew of at least one Cretan (other than himself) who was not a liar, then his statement is a non-paradoxical lie in that it does not lead to a logical contradiction. (The contradictory of the statement, "All Cretans are liars" is the statement, "Some Cretans are not liars", which might be true at the same time as the statement, "Some Cretans are liars.")
History of the phrase
Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that ZeusZeus
In the ancient Greek religion, Zeus was the "Father of Gods and men" who ruled the Olympians of Mount Olympus as a father ruled the family. He was the god of sky and thunder in Greek mythology. His Roman counterpart is Jupiter and his Etruscan counterpart is Tinia.Zeus was the child of Cronus...
was immortal, as in the following poem:
Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus
Callimachus
Callimachus was a native of the Greek colony of Cyrene, Libya. He was a noted poet, critic and scholar at the Library of Alexandria and enjoyed the patronage of the Egyptian–Greek Pharaohs Ptolemy II Philadelphus and Ptolemy III Euergetes...
in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted in the Epistle to Titus
Epistle to Titus
The Epistle of Paul to Titus, usually referred to simply as Titus, is one of the three Pastoral Epistles , traditionally attributed to Saint Paul, and is part of the New Testament...
:
The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus
Callimachus
Callimachus was a native of the Greek colony of Cyrene, Libya. He was a noted poet, critic and scholar at the Library of Alexandria and enjoyed the patronage of the Egyptian–Greek Pharaohs Ptolemy II Philadelphus and Ptolemy III Euergetes...
. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of; accusing Cretans (other than himself) denying the immortality of Zeus while he did not deny it himself. It is also quite natural to understand the Cretan poet as possibly having employed the figure of speech known as hyperbole (deliberate exaggeration) rather than having advanced a vigorous logical claim. It is not clear when Epimenides became associated with the Epimenides paradox. Epimenides himself does not appear to have intended any irony or paradox in his statement, "Cretans, always liars", nor did Callimachus, nor the author of Titus, nor Clement of Alexandria
Clement of Alexandria
Titus Flavius Clemens , known as Clement of Alexandria , was a Christian theologian and the head of the noted Catechetical School of Alexandria. Clement is best remembered as the teacher of Origen...
:
Saint Augustine
Augustine of Hippo
Augustine of Hippo , also known as Augustine, St. Augustine, St. Austin, St. Augoustinos, Blessed Augustine, or St. Augustine the Blessed, was Bishop of Hippo Regius . He was a Latin-speaking philosopher and theologian who lived in the Roman Africa Province...
restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). In the Middle Ages
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...
, many forms of the liar paradox were studied under the heading of insolubilia
Insolubilia
In the Middle Ages, variations on the liar paradox were studied under the name of insolubilia .Although the liar paradox was well known in antiquity, interest seems to have lapsed until the twelfth century, when it appears to have been reinvented independently of ancient authors...
, but these were not explicitly associated with Epimenides. The second volume of Pierre Bayle
Pierre Bayle
Pierre Bayle was a French philosopher and writer best known for his seminal work the Historical and Critical Dictionary, published beginning in 1695....
's Dictionnaire Historique et Critique
Dictionnaire Historique et Critique
The Dictionnaire Historique et Critique is a biographical dictionary written by Pierre Bayle , a Huguenot who lived and published in Holland after fleeing his native France due to religious persecution. The dictionary was first published in 1697, and enlarged in the second edition of 1702...
of 1740 explicitly connects Epimenides with the paradox, though Bayle labels the paradox a "sophisme".
Logical analysis
The logical contradiction exists on Saint Paul's epistle rather than the poem of Epimenides:In his epistle to Titus, Apostle Paul wants to warn Titus that Cretans don't believe in the one truth of Christianity, because "Cretans are always liars". To justify his claim, Apostle Paul cites Epimenides. This forms a contradiction because he cites the words of a Cretan (always a liar) in order to prove that Cretans are always liars, also concluding that Epimenides (a Cretan) had surely told the truth (that Cretans are always liars).
Define a "liar" as someone who is never truthful. Then the statement, "All Cretans are liars", if uttered by a Cretan, in this case Epimenides, implies that the speaker's statement is not true—that is, some Cretans are not liars. Some logicians have treated the Epimenides paradox as identical to the Liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
.
For example, Thomas Fowler (1869) states the paradox as follows: "Epimenides the Cretan says, 'that all the Cretans are liars,' but Epimenides is himself a Cretan; therefore he is himself a liar. But if he be a liar, what he says is untrue, and consequently the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; hence the Cretans are liars, Epimenides is himself a liar, and what he says is untrue. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful." However, the inference from "not all Cretans are liars", to "the Cretans are veracious" is not valid.
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 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.
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.
Naturally, any truly logical idea of a paradox with the statement falls flat if one understands that while "all Cretans" may be "liars," such a statement in realistic terms does not necessarily mean that all Cretans lie all the time or that they lie only. Even if it is said that "Cretans are always liars," this does not produce a paradox if one understands the various meanings of the term always—as in "John always says No!" does not mean that "No" is all—or the only thing—John ever says. Certainly even the most prolific liars in history told the truth at least some of the time, so an idea that anyone lies in every single sentence they speak is merely simpleminded at best. The word "always" can also be interpreted to mean "in all instances", as in, "when you find a Cretan, you will always have found a liar," where "liar," again, need not mean more than a person who is known to lie.
Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
, Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
, and the Burali-Forti paradox
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...
, all of which have self-reference in common with Epimenides. Indeed, 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.