Paradox
A 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. For example, 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.
- Apportionment paradox: Some systems of apportioning representation can have unituitive results
- 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 - Will Roger's phenomenon
- 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.
- 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.
- 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.)
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
References
- [1] Quine, W. V: "Paradox", Scientific American, April 1962, pp. 84–96.