Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Paradox |
ParadoxNoun1. (logic) a self-contradiction; "I always lie". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "paradox" was first used: 1540. (references) |
Etymology: Paradox \Par`a*dox\, noun; plural Paradoxes. [French expression paradoxe, from Latin expression paradoxum, from the Greek expression beside, beyond, contrary to to think, suppose, imagine. See Para-, and Dogma.]. (references) |
| Domain | Definition |
Computing | Paradox |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
simple:ParadoxA paradox is an apparently true statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition. The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics.
In moral philosophy, paradox plays a particularly central role in debates on ethics. For instance, an ethical admonition to "love thy neighbour" is in (not just contrast but) contradiction with an armed neighbour actively trying to kill you: if he or she succeeds, then, you will not be able to love them. But to pre-emptively attack them or restrain them is not usually understood as very loving. This might be termed an ethical dilemma; another example is the conflict between an injuction not to steal and one to care for a family that you cannot afford to feed except with stolen money.
Common themes in paradoxes
Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning.
W. V. Quine [1] distinguished three classes of paradox.
- A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person may be more than N years old on his Nth birthday. Likewise, Arrow's paradox involves behavior of voting systems that is surprising but all too true.
- A falsidical paradox establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various proofs that 1 = 2 are classic examples, generally relying on a hidden division by zero. Another example would be the Horse paradox.
- A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. The Grelling-Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.
List of paradoxes
Not all paradoxes fit neatly into one category. Some paradoxes include:
Veridical paradoxes
These are unintuitive results of correct logical reasoning.Mathematical/Logical
- Apportionment paradox: Some systems of apportioning representation can have unituitive results
- Alabama paradox
- New states paradox
- Population paradox
- Averaging - the mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results - for example, it is possible that moving an entry from Wikipedia to Wiktionary would increase the average entry length on both sites
- Arrow's paradox/Voting paradox/Codorcet paradox: You can't have all the attributes of an ideal voting system at once
- Banach-Tarski paradox: Cut a ball into 5 pieces, re-assemble the pieces to get two balls, both of equal size to the first.
- Birthday paradox: What is the chance that two people in a room have the same birthday?
- Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number which is smaller than itself.
- Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured on the roof, and disassembled in the basement.
- Galileo's paradox: Though most numbers are not squares, there are no more numbers than squares.
- Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.
- Monty Hall problem: An unintuitive consequence of conditional probability.
- Monty Hell problem: Positive daily profits yield zero assets in the limit.
- Raven paradox (or Hempel's Ravens): Observing a red apple increases the likelihood of all ravens being black.
- Richard's paradox: A complete list of definitions of real numbers doesn't exist.
- Simpson's paradox: An association in sub-populations may be reversed in the population. It appears that two sets of data separately support a certain hypothesis, but, when considered together, they support the opposite hypothesis.
- Statistical paradox: It is quite possible to draw wrong conclusions from correlation. For example, towns with a larger number of churches generally have a higher crime rate - because both result from higher population. A professional organization once found that economists with a PhD actually had a lower average salary than those with a BS - but this was found to be due to the fact that those with a PhD worked in academia, where salaries are generally lower.
Psychological/Philosophical
- Abilene paradox: People take actions in contradiction to what they really want to do, and therefore defeat the very purposes of what they were trying to accomplish.
- Control paradox Man can never be free of control, for to be free of control is to be controlled by oneself.
Physical
- Braess' paradox: sometimes adding extra capacity to a network can reduce overall performance
- Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanics?
- Twin paradox: When the travelling twin returns, he's younger and older than his brother who stayed put.
- Zeno's paradoxes: When you reach the turtle's spot, it has already advanced a bit, so you can never catch it.
Falsidical paradoxes
These are incorrect results of subtly false reasoning.
- Epimenides paradox: A Cretan says "All Cretans are liars". (But see also the Liar paradox, an antinomy.)
- Horse paradox: All horses are the same color.
- Unexpected hanging paradox: The day of the hanging will be a surprise, so it can't happen at all, so it will be a surprise. (Similar to the Liar paradox, an antinomy.)
Antinomies
Paradoxes that show flaws in accepted reasoning, axioms, or definitions. Note that many of these are special cases, or adaptations, of the Russell's paradox.
- Barber paradox: The barber who shaves all men who don't shave themselves, and no-one else.
- Berry paradox: What is "The first number not nameable in under ten words"?
- Curry's paradox: "If I'm not mistaken, the world will end in a week."
- Grelling-Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterological word?
- Liar paradox: "This sentence is false."
- Quine's Liar Paradox: "Yields a falsehood when appended to its own quotation."
- Russell's paradox: Is there a set of all those sets that do not contain themselves?
Antinomies of definition
These paradoxes rest simply on an ambiguous definition.
- Ship of Theseus/George Washington's axe: When every component of the ship has been replaced at least once, is it still the same ship?
- Sorites paradox: At what point does a heap stop being a heap as I take away grains of sand?
Conditional paradoxes
These are paradoxes only if certain special assumptions are made. Some of these show that those assumptions are false or incomplete, others are are other types of paradoxes.
- Fermi paradox: If there are many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?
- Grandfather paradox: You travel back in time and kill your grandfather before he meets your grandmother, resulting in you never being conceived.
- The GZK paradox: high-energy cosmic rays have been observed which seem to violate the Greisen-Zatsepin-Kuzmin limit which is a consequence of special relativity
- Jevons paradox: In economics, increases in efficiency lead to even larger increases in demand.
- Mere addition paradox: is a large population living barely tolerable lives better than a small happy population?
- Newcomb's paradox: How do you play a game against an omniscient opponent?
- Nihilist paradox: if truth does not exist, the statement "truth does not exist" is a truth, thereby proving itself incorrect.
- Olbers' paradox: If the universe is infinite, the sky should be entirely bright because there's a star in every direction.
- Omnipotence paradox: Can an omnipotent being create a rock too heavy to lift? Can an irresistible force move an unmovable object?
- Predestination paradox: A man travels back in time and impregnates his great-great-grandmother. The result is a line of offspring and descendants, including the man's parent(s) and the man himself. Therefore, unless he makes the time-travel trip at all, he will never exist.
Other paradoxes
- Giffen paradox: can increasing the price of bread make poor people eat more of it?
- Kavka's toxin puzzle: Can one intend to drink the nondeadly toxin, if the intention is the only thing needed to get the reward?
- Moore's paradox: "It's raining but I don't believe that it is."
See also
- Impossible objects
References
- [1] Quine, W. V: "Paradox", Scientific American, April 1962, pp. 84–96.
External links
- Google Directory category: Paradoxes
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Paradox."
(From Wikipedia, the free Encyclopedia)
Russell's paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A.
In Cantor's system, M is a well-defined set. Does M contain itself? If it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, then it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to contradictions.
In Frege's system, M corresponds to the concept does not fall under its defining concept. Frege's system also leads to a contradiction: that there is a class defined by this concept, which falls under its defining concept just in case it does not.
History
Exactly when Russell discovered the paradox is not clear. It seems to have been May or June 1901, probably as a result of his work on Cantor's theorem that that the number of entities in a certain domain is smaller than the number of subclasses of those entities. In Russell's Principles of Mathematics (not to be confused with the later Principia Mathematica) Chapter X, section 100, where he calls it "The Contradiction" he says that he was led to it by analyzing Cantor's proof that there can be no greatest cardinal. He also mentions it in a 1901 paper in the International Monthly, entitled "Recent work in the philosophy of mathematics" Russell mentioned Cantor's proof that there is no largest cardinal and stated that "the master" had been guilty of a subtle fallacy that he would discuss later.
Famously, Russell wrote to Frege about the paradox in June 1902, just as Frege was preparing the second volume of his Grundgesetze. Frege was forced to prepare an appendix in response to the paradox, but this later proved unsatisfactory. It is commonly supposed that this led Frege completely to abandon his work on the logic of classes*.
[*Some revisionist historians have argued against this, can someone supply references?]
While Zermelo was working on his version of set theory, he also noticed the paradox, but thought it too obvious and never published anything about it! Zermelo's system avoids the difficulty through the famous Axiom of separation (Aussonderung).
Russell, with Alfred North Whitehead, undertook to accomplish Frege's task, this time using a more restricted version of set theory that, they thought, would not admit Russell's Paradox, but would still produce arithmetic. Kurt Gödel later showed that, even if it was consistent, it did not succeed in reducing all mathematics to logic -indeed this could not be done: arithmetic is "incomplete."
Easy-to-understand version of the Paradox
There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: For example, the Barber paradox which considers a barber who shaves everyone who does not shave himself, and no one else. When you start to think about whether he should shave himself or not you will get puzzled...
Similarly, Russell's paradox proves that, on Wikipedia, if we had an entry on list of all lists which do not contain themselves, then that list must be either incomplete (if it does not list itself) or incorrect (if it does).
After this paradox was described, set theory had to be reformulated axiomatically as axiomatic set theory in a way that avoided this and other related problems. Russell himself, together with Alfred North Whitehead, developed a comprehensive system of types in his work Principia Mathematica. This system does indeed avoid the known paradoxes and allows for the formulation of all of mathematics, but it has not been widely accepted. The most common version of axiomatic set theory in use today is Zermelo-Fraenkel set theory, which avoids the notion of types and restricts the universe of sets to those which can be constructed from given sets using certain axioms. The object M discussed above cannot be constructed like that and is therefore not a set in this theory; it is a proper class.
The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success: Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of the Halting problem (and with that the Entscheidungsproblem) by using the same trick.
Russell's Paradox is closely related to the Liar paradox.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Russell's paradox."
| The following table is compiled from various sources, across various languages. When English abbreviations or acronyms come from a non-English source, this is noted. | |||
| Entry | Source | Expression | Field |
| PAL | English | Paradox Application Language | Computer - (Borland, DB) |
Source: compiled by the editor, based on several corpora (additional references). | |||
| Context | Synonyms within Context (source: adapted from Roget's Thesaurus). |
Absurdity | Noun: absurdity, absurdness; Adjective: imbecility; alogy, nonsense, utter nonsense; paradox, inconsistency; stultiloquy, stultiloquence; nugacity. |
Difficulty | Nice point, delicate point, subtle point, knotty point; vexed question, vexata quaestio, poser; puzzle; (riddle); paradox; hard nut to crack, nut to crack; bone to pick, |
Secret | Problem; (question); paradox; (difficulty); unintelligibility; terra incognita; (ignorance). |
Unintelligibility | Paradox, oxymoron; riddle, enigma, puzzle; (secret); diagnus vindice nodus; sealed book; steganography, freemasonry. |
| Source: adapted from Roget's Thesaurus. | |
Crosswords: Paradox |
| English words defined with "paradox": Para-, Paradoxes, Paradoxist, Paradoxy. (references) |
| Specialty definitions using "paradox": ANCESTORS, Axiom of Choice, Axiom of Comprehension ♦ Banach-Tarski paradox ♦ Inprise Corporation ♦ kentsmithite ♦ liar paradox ♦ ObjectPAL, Open DataBase Connectivity, oxygen paradox ♦ Paradox Application Language ♦ Russell's Paradox ♦ salt anticline, Seeming paradox ♦ Zermelo set theory. (references) |
| Non-English Usage: "Paradox" is also a word in the following languages with English translations in parentheses. Czech (paradox), Dutch (paradox), German (ironic, paradox, paradoxical, paradoxically), Hungarian (paradoxical), Romanian (antinomy, paradox), Swedish (irish bull, paradox). |
| Domain | Usage | |
Screenplays | History abhors a paradox. (Legacy of Kain: Soul Reaver II; writing credit: Amy Hennig) That is a paradox, sir, I hate paradoxes (Ideal Husband, An; writing credit: Oscar Wilde; Oliver Parker) | |
Movie/TV Titles | Paradox Lake (2002) Welcome to Paradox (1998) Hell's Paradox (1996) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title | ||
Books | |||
Periodicals |
| ||
Theater & Movies | |||
Music |
| ||
High Tech |
| ||
Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit |  | Chesapeake Docks, a geographic paradox, advertising sailfish fishing prowess. Credit: Fisheries. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Author | Quotation |
Charles Caleb Colton | There is this paradox in pride -- it makes some men ridiculous, but prevents others from becoming so. |
Friedrich Schlegel | Irony is the form of paradox. Paradox is what is good and great at the same time. |
Source: compiled by the editor from various references. | |
| Title | Author | Quote |
Les Miserables | Hugo, Victor | There are men still on earth who know how to open and shut pleasantly the surprise boxes of paradox. |
Source: compiled by the editor from various references. | ||
| Subject | Topic | Quote |
Health | The reason for this paradox is unclear and may reflect the current state of the research. (references) | |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Paradox" is generally used as a noun (singular) -- approximately 98.49% of the time. "Paradox" is used about 595 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 98.49% | 586 | 10,818 |
| Noun (proper) | 1.51% | 9 | 117,287 |
| Total | 100.00% | 595 | N/A |
Source: compiled by the editor from several corpora; see credits.
1. Paradox, NY |
Expressions using "paradox": d'Alembert's paradox ♦ French paradox ♦ Hydrostatic paradox ♦ liar paradox ♦ oxygen paradox ♦ paradox Application Language ♦ paradox control ♦ paradox gate ♦ posthypoxia paradox ♦ russells paradox ♦ Russell's Paradox ♦ Solow paradox ♦ twin paradox. Additional references. | |
| Source: compiled by the editor from various references; see credits. |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "paradox"; alternative meanings/domain in parentheses. | |
Albanian | paradoks (anomaly), thënie e gabuar, njeri kontradiktor. (various references) | |
Arabic | العبارة الموهمة للصحة, ظاهر متناقض. (various references) | |
Bulgarian | парадокс (antinomy). (various references) | |
Chinese | 自相矛盾的人 (paradoxes). (various references) | |
Czech | protismyslnost, paradox. (various references) | |
Danish | post-hypoxia paradoks (oxygen paradox, posthypoxia paradox), Piper's adaptations-paradoks (Piper paradox), tvillingeparadokset (twin paradox), tætningsringventil (paradox gate, ring-follower gate, ring-seal gate), saerlig type af afbalanceret naaleventil (paradox control), Opitz'paradox (Opitz paradox), Opie's paradox (Opie paradox), hypoxisk paradox (hypoxic paradox). (various references) | |
Dutch | paradox. (various references) | |
Esperanto | paradokso. (various references) | |
Faeroese | tvørsøgn, ósamsvar. (various references) | |
Farsi | مهمل نما, قیاس ضدونقیض , اضداد, بیان مغایر. (various references) | |
Finnish | Paradox-ohjaus (paradox control), rengasluukku (paradox gate, ring-follower gate, ring-seal gate), happivajausparadoksi (oxygen paradox, posthypoxia paradox). (various references) | |
French | paradoxe. (various references) | |
German | Paradoxon, paradoxie (paradoxicalness), paradoxen, paradox (ironic, paradoxical, paradoxically). (various references) | |
Greek | παράδοξος (paradoxical, peculiar), παράδοξο, παραδοξολογία (paradoxology, strange talk). (various references) | |
Hebrew | "בר ו"פוכו. (various references) | |
Hungarian | paradoxon, ellentmondás (antagonism, antilogy, caveat, clash, conflict, contradiction, discrepancy, repugnance). (various references) | |
Indonesian | lawan asas. (various references) | |
Italian | paradosso (paradoxical), paradossale (paradoxical, strange). (various references) | |
Japanese Kanji | 逆説 , 逆説 , パラチフス菌 (clattering, falling indrops, flipping through a book, paradoxical, paraffin, paranoia, paratyphoid bacillus, pattering, sprinkle). (various references) | |
Japanese Katakana | ぎゃくせつ (contradictory conjunction), パラドックス . (various references) | |
Korean | 역설 (paradoxes, Paradoxical). (various references) | |
Pig Latin | aradoxpay.(various references) | |
Portuguese | paradoxo (crank, paradoxical). (various references) | |
Romanian | paradox (antinomy), pãrere (belief, conceit, conviction, dictum, estimate, estimation, hint, idea, judgement, mind, notion, opinion, say, thinking, verdict, view, voice), fapt (affair, case, circumstance, deed, event, fact, incident, phenomenon, reality), care contrazice ceva în general acceptat. (various references) | |
Russian | парадокс (antinomy). (various references) | |
Serbo-Croatian | paradoks. (various references) | |
Spanish | paradoja. (various references) | |
Swedish | paradox (irish bull). (various references) | |
Turkish | paradoks, mantıkla çelişen ama doğru olan söz, çelişki (antinomy, cleavage, contradiction, contradictoriness, contrast, discrepancy, excursion, variable). (various references) | |
Ukrainian | парадокс (antinomy). (various references) | |
Vietnamese | vật ngược đời. (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | paradoxum. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "paradox": paradoxes, paradoxical, paradoxicalities, paradoxicality, paradoxically, paradoxicalness, paradoxicalnesses. (additional references) | |
Words containing "paradox": ultraparadoxical. (additional references) | |
| |
"Paradox" is suggested in spellcheckers for the following: Apricocks, Arado, Caradog, Jardox, pagado, parad, parado, paradoi, paradol, paradom, paradon, parador, paradoxe, paradoy, paradoz, Parakou, Paramo, paratex, pardox, paredo, Parejo, Parodos, parodox, Parralo, perado, Prados, pradox. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "paradox" (pronounced pe"rudÄ'ks) |
| 5 | -u d Ä' k s | heterodox, orthodox, unorthodox. |
| 4 | -d Ä' k s | boondocks. |
| 3 | -Ä' k s | aftershocks, boombox, breadbox, deadlocks, detox, equinox, feedstocks, flintlocks, gearbox, hollyhocks, icebox, jukebox, mailbox, matchbox, matchlocks, padlocks, peacocks, roadblocks, saltbox, Sandbox, skybox, smallpox, soapbox, spacewalks, tinderbox. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-a-d-o-p-r-x" | |
-3 letters: apod, dopa, dorp, drop, orad, para, pard, prao, proa, prod, road. | |
-4 letters: ado, dap, dor, oar, ora, pad, par, pax, pod, pox, pro, rad, rap, rax, rod. | |
-5 letters: aa, ad, ar, ax, do, od, op, or, ox, pa. | |
| Words containing the letters "a-a-d-o-p-r-x" | |
+2 letters: paradoxes. | |
+4 letters: paradoxical. | |
+5 letters: approximated, extrapolated. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Crosswords 3. Usage: Modern 4. Usage: Commercial | 5. Images: Photo Album 6. Quotations: Familiar 7. Quotations: Fiction 8. Quotations: Non-fiction | 9. Usage Frequency 10. Cities 11. Expressions 12. Expressions: Internet | 13. Translations: Modern 14. Translations: Ancient 15. Abbreviations 16. Acronyms | 17. Derivations 18. Rhymes 19. Anagrams 20. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
