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Epimenides paradox

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Epimenides paradox

Epimenides from "Promptuarii Iconum Insigniorum"

The Epimenides paradox reveals a problem with self-reference in logic.

It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC) who is credited with the original statement.

A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter

Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."

A paradox of self-reference arises when one considers whether it is possible for Epimenides to have spoken the truth.

Logical paradox

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; saying 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."[1]

The Epimenides paradox in this form can however be solved, meaning that the statement can be assigned truth-value in a way that is not self-contradicting. Namely, if the statement "all Cretans are liars" (stated by Epimenides, himself a Cretan) is true, then Epimenides, being a Cretan, would be a liar; making the assumption that liars only make false statements, the statement should be false. So assuming the statement is true leads us to conclude that the statement is false and cannot be accepted. However, if we assume the statement is false, then its correct negation, "there exists a Cretan who is honest", is true. This does not lead to contradiction, since it is not required that this Cretan be Epimenides, meaning that Epimenides can tell false statements (honest people tell only true statements) and thus be a liar (Epimenides knows at least one honest Cretan and lies about it). Hence, from the assumption that the statement is false it does not follow that the statement is true. So we can avoid a paradox as seeing the statement "all Cretans are liars" as a false statement, which is made by a lying Cretan, Epimenides.[2][3] The mistake made by Thomas Fowler (and many other people) above is to think that the negation of "all Cretans are liars" is "all Cretans are honest", when in fact the negation is "there exists a Cretan who is honest". The Epimenides paradox can be slightly modified as to not allow the kind of solution described above, like it was in the first paradox of Eubulides but instead leading to a non-avoidable self-contradiction. Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Socratic 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.

Origin of the phrase

Epimenides was a 6th-century BC philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:

Denying the immortality of Zeus, then, was the lie of the Cretans.

The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides:

Emergence as a logical contradiction

The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus, who both used the phrase to emphasize their point, without irony.

Later, in the 1st century AD, the quote is given similar credence as it had in the past:

Nor does Clement of Alexandria, in the late 2nd century AD, indicate that the concept of logical paradox is an issue:

During the early 4th century, Saint Augustine restates the closely related liar paradox in Against the Academicians (III.13.29), but without mentioning Epimenides.

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.

Finally, in 1740, the second volume of Pierre Bayle's Dictionnaire Historique et Critique explicitly connects Epimenides with the paradox, though Bayle labels the paradox a "sophisme".[4]

Notes

  1. ^  
  2. ^ Liar paradox#History
  3. ^ http://mathworld.wolfram.com/EpimenidesParadox.html
  4. ^   Dictionnaire Historique et Critique at WorldHeritage.

References

All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the Cretica of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of The Expositor's Bible Commentary, Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976–1984), page 476. Longenecker in turn cites M.D. Gibson, Horae Semiticae X (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote:

The Syr. version of the quatrain comes to us from the Syr. church father Isho'dad of Merv (probably based on the work of Theodore of Mopsuestia), which J.R. Harris translated back into Gr. in Exp ["The Expositor"] 7 (1907), p 336.

An oblique reference to Epimenides in the context of logic anchor_year disambiguator if any ]] function check_date (date_string) local year; -- assume that year2, months, and days are not used; local year2=0; -- second year in a year range local month=0; local month2=0; -- second month in a month range local day=0; local day2=0; -- second day in a day range local anchor_year; local coins_date; if date_string:match("^%d%d%d%d%-%d%d%-%d%d$") then -- year-initial numerical year month day format year, month, day=string.match(date_string, "(%d%d%d%d)%-(%d%d)%-(%d%d)"); month=tonumber(month); if 12 < month or 1 > month or 1583 > tonumber(year) then return false; end -- month number not valid or not Gregorian calendar anchor_year = year; elseif date_string:match("^%a+ +[1-9]%d?, +[1-9]%d%d%d%a?$") then -- month-initial: month day, year month, day, anchor_year, year=string.match(date_string, "(%a+)%s*(%d%d?),%s*((%d%d%d%d)%a?)"); month = get_month_number (month); if 0 == month then return false; end -- return false if month text isn't one of the twelve months elseif date_string:match("^%a+ +[1-9]%d?–[1-9]%d?, +[1-9]%d%d%d%a?$") then -- month-initial day range: month day–day, year; days are separated by endash month, day, day2, anchor_year, year=string.match(date_string, "(%a+) +(%d%d?)–(%d%d?), +((%d%d%d%d)%a?)"); if tonumber(day) >= tonumber(day2) then return false; end -- date range order is left to right: earlier to later; dates may not be the same; month = get_month_number (month); if 0 == month then return false; end -- return false if month text isn't one of the twelve months elseif date_string:match("^[1-9]%d? +%a+ +[1-9]%d%d%d%a?$") then -- day-initial: day month year day, month, anchor_year, year=string.match(date_string, "(%d%d*)%s*(%a+)%s*((%d%d%d%d)%a?)"); month = get_month_number (month); if 0 == month then return false; end -- return false if month text isn't one of the twelve months elseif date_string:match("^[1-9]%d?–[1-9]%d? +%a+ +[1-9]%d%d%d%a?$") then -- day-range-initial: day–day month year; days are separated by endash day, day2, month, anchor_year, year=string.match(date_string, "(%d%d?)–(%d%d?) +(%a+) +((%d%d%d%d)%a?)"); if tonumber(day) >= tonumber(day2) then return false; end -- date range order is left to right: earlier to later; dates may not be the same; month = get_month_number (month); if 0 == month then return false; end -- return false if month text isn't one of the twelve months elseif date_string:match("^[1-9]%d? +%a+ – [1-9]%d? +%a+ +[1-9]%d%d%d%a?$") then -- day initial month-day-range: day month - day month year; uses spaced endash day, month, day2, month2, anchor_year, year=date_string:match("(%d%d?) +(%a+) – (%d%d?) +(%a+) +((%d%d%d%d)%a?)"); if (not is_valid_month_season_range(month, month2)) or not is_valid_year(year) then return false; end -- date range order is left to right: earlier to later; month = get_month_number (month); month2 = get_month_number (month2); elseif date_string:match("^%a+ +[1-9]%d? – %a+ +[1-9]%d?, +[1-9]%d%d%d?%a?$") then -- month initial month-day-range: month day – month day, year; uses spaced endash month, day, month2, day2, anchor_year, year=date_string:match("(%a+) +(%d%d?) – (%a+) +(%d%d?), +((%d%d%d%d)%a?)"); if (not is_valid_month_season_range(month, month2)) or not is_valid_year(year) then return false; end month = get_month_number (month); month2 = get_month_number (month2); elseif date_string:match("^[1-9]%d? +%a+ +[1-9]%d%d%d – [1-9]%d? +%a+ +[1-9]%d%d%d%a?$") then -- day initial month-day-year-range: day month year - day month year; uses spaced endash day, month, year, day2, month2, anchor_year, year2=date_string:match("(%d%d?) +(%a+) +(%d%d%d%d?) – (%d%d?) +(%a+) +((%d%d%d%d?)%a?)"); if tonumber(year2) <= tonumber(year) then return false; end -- must be sequential years, left to right, earlier to later if not is_valid_year(year2) then return false; end -- year2 no more than one year in the future month = get_month_number (month); month2 = get_month_number (month2); elseif date_string:match("^%a+ +[1-9]%d?, +[1-9]%d%d%d – %a+ +[1-9]%d?, +[1-9]%d%d%d%a?$") then -- month initial month-day-year-range: month day, year – month day, year; uses spaced endash month, day, year, month2, day2, anchor_year, year2=date_string:match("(%a+) +(%d%d?), +(%d%d%d%d) – (%a+) +(%d%d?), +((%d%d%d%d)%a?)"); if tonumber(year2) <= tonumber(year) then return false; end -- must be sequential years, left to right, earlier to later if not is_valid_year(year2) then return false; end -- year2 no more than one year in the future month = get_month_number (month); month2 = get_month_number (month2); elseif date_string:match("^%a+ +[1-9]%d%d%d–%d%d%a?$") then -- special case Winter/Summer year-year (YYYY-YY); year separated with unspaced endash if nil == date_string:match("^Winter") and nil == date_string:match("^Summer") then return false end; -- 'month' can only be Winter or Summer local century; year, century, anchor_year, year2=date_string:match("%a+ +((%d%d)%d%d)–((%d%d)%a?)"); anchor_year=year..'–'..anchor_year; -- assemble anchor_year from both years year2 = century..year2; -- add the century to year2 for comparisons if 1 ~= tonumber(year2) - tonumber(year) then return false; end -- must be sequential years, left to right, earlier to later if not is_valid_year(year2) then return false; end -- no year farther in the future than next year elseif date_string:match("^%a+ +[1-9]%d%d%d–[1-9]%d%d%d%a?$") then -- special case Winter/Summer year-year; year separated with unspaced endash if nil == date_string:match("^Winter") and nil == date_string:match("^Summer") then return false end; -- 'month' can only be Winter or Summer year, anchor_year, year2=date_string:match("%a+ +(%d%d%d%d)–((%d%d%d%d)%a?)"); anchor_year=year..'–'..anchor_year; -- assemble anchor_year from both years if 1 ~= tonumber(year2) - tonumber(year) then return false; end -- must be sequential years, left to right, earlier to later if not is_valid_year(year2) then return false; end -- no year farther in the future than next year elseif date_string:match("^%a+ +[1-9]%d%d%d% – %a+ +[1-9]%d%d%d%a?$") then -- month/season year - month/season year; separated by spaced endash month, year, month2, anchor_year, year2=date_string:match("(%a+) +(%d%d%d%d) – (%a+) +((%d%d%d%d)%a?)"); anchor_year=year..'–'..anchor_year; -- assemble anchor_year from both years if tonumber(year) >= tonumber(year2) then return false; end -- left to right, earlier to later, not the same if not is_valid_year(year2) then return false; end -- no year farther in the future than next year if not((0 ~= get_month_number(month) and 0 ~= get_month_number(month2)) or -- both must be month year or season year, not mixed (0 ~= get_season_number(month) and 0 ~= get_season_number(month2))) then return false; end elseif date_string:match ("^%a+–%a+ +[1-9]%d%d%d%a?$") then -- month/season range year; months separated by endash month, month2, anchor_year, year=date_string:match ("(%a+)–(%a+)%s*((%d%d%d%d)%a?)"); if (not is_valid_month_season_range(month, month2)) or (not is_valid_year(year)) then return false; end elseif date_string:match("^%a+ +%d%d%d%d%a?$") then -- month/season year month, anchor_year, year=date_string:match("(%a+)%s*((%d%d%d%d)%a?)"); if not is_valid_year(year) then return false; end if not is_valid_month_or_season (month) then return false; end elseif date_string:match("^[1-9]%d%d%d?–[1-9]%d%d%d?%a?$") then -- Year range: YYY-YYY or YYY-YYYY or YYYY–YYYY; separated by unspaced endash; 100-9999 year, anchor_year, year2=date_string:match("(%d%d%d%d?)–((%d%d%d%d?)%a?)"); anchor_year=year..'–'..anchor_year; -- assemble anchor year from both years if tonumber(year) >= tonumber(year2) then return false; end -- left to right, earlier to later, not the same if not is_valid_year(year2) then return false; end -- no year farther in the future than next year elseif date_string:match("^[1-9]%d%d%d–%d%d%a?$") then -- Year range: YYYY–YY; separated by unspaced endash local century; year, century, anchor_year, year2=date_string:match("((%d%d)%d%d)–((%d%d)%a?)"); anchor_year=year..'–'..anchor_year; -- assemble anchor year from both years if 13ppears in "The Logical Calculus" by W. E. Johnson, Mind (New Series), volume 1, number 2 (April, 1892), pages 235–250. Johnson writes in a footnote,

Compare, for example, such occasions for fallacy as are supplied by "Epimenides is a liar" or "That surface is red," which may be resolved into "All or some statements of Epimenides are false," "All or some of the surface is red."

The Epimenides paradox appears explicitly in "Mathematical Logic as Based on the Theory of Types", by Bertrand Russell, in the American Journal of Mathematics, volume 30, number 3 (July, 1908), pages 222–262, which opens with the following:

The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?

In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox. Since Russell, the Epimenides paradox has been referenced repeatedly in logic. Typical of these references is Gödel, Escher, Bach by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.

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