Paradox: Meaning (information, definition, explanation, facts)

A paradox is an apparently true statement or group of statements that seems to lead to a contradiction or to a situation that defies intuition. The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has often led to significant advances in science, philosophy and mathematics.

The etymology of paradox can be traced back to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek paradoxon. The word is composed of the prefix para- which means "contrary to", "altered" or "opposite of" conjoined to the noun stem doxa, meaning "opinion." Compare orthodox and heterodox. The liar paradox and other paradoxes were studied in medieval times under the heading insolubilia.

Not all paradoxes are equal. For example, the Birthday paradox is more of a surprise than a paradox, while the resolution of Curry's paradox is still a matter of contention. Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning.

In moral philosophy, paradox plays a particularly central role in debates on ethics. For instance, an ethical admonition to "love thy neighbor" is not just in contrast with, but in contradiction to an armed neighbor actively trying to kill you: if he or she succeeds, then, you will not be able to love them. But to preemptively 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 injunction not to steal and one to care for a family that you cannot afford to feed without stolen money.

Types of paradoxes

W. V. Quine (1962) 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 impossibility theorem 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 invalid proofs (e.g. 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.

Not all paradoxes fit neatly into one category. Some paradoxes include:

Veridical paradoxes

These are unintuitive results of correct logical reasoning.

Mathematical/Logical

• Paradox of entailment: Inconsistent premises always make an argument valid.
• Apportionment paradox: Some systems of apportioning representation can have unintuitive 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 - Will Rogers phenomenon
• Arrow's paradox/Voting paradox/Condorcet 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?
• Borel's paradox: Conditional probability density functions are not invariant under coordinate transformations.
• 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.
• Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals have finite area, but infinite perimeter.
• Hausdorff paradox: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.
• 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.
• Sleeping beauty paradox: One half or one third? news://rec.puzzles cannot agree on a probability.
• 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.
• Buridan's ass: How can a rational choice be made between two outcomes of equal value?
• Control paradox: Man can never be free of control, for to be free of control is to be controlled by oneself.
• Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.
• Epicurean paradox: The existence of evil is incompatible with the existence of an omnipotent and caring God.
• Mere addition paradox

Physical

• Braess' paradox: sometimes adding extra capacity to a network can reduce overall performance
• Cosmic ray paradox: physical theory predicts an upper limit to the possible energy of cosmic rays, but cosmic rays with energies above the theoretical limit have been observed.
• D'Alembert's paradox: An inviscid liquid produces no drag.
• Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanics?
• Gibbs paradox: In an ideal gas, is entropy an extensive variable?
• Loschmidt's paradox: Why is there an inevitable increase in entropy when the laws of physics are invariant under time reversal?
• Mpemba paradox: hot water can under certain conditions freeze faster than cold water, even though it must pass the lower temperature on the way to freezing.
• Olbers' paradox Why is the night sky black, if there is an infinity of stars ?
• Twin paradox: When the travelling twin returns, he's younger and older than his brother who stayed put.
• Black hole information paradox

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.)
• Zeno's paradoxes: When you reach the turtle's spot, it has already advanced a bit, so you can never catch it.

Antinomies

Paradoxes that show flaws in accepted reasoning, axioms, or definitions. Note that many of these are special cases, or adaptations, of 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?
• Richard's paradox

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 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 which precludes your own conception.
• 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, with infinitely many luminous stars uniformly distributed, 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.
• St. Petersburg paradox: People will only offer a modest fee for a reward of infinite value.
• Galloway's Paradox: What would happen in a battle between Star Trek redshirts (well known for dying before the first commercial break) and Star Wars Imperial Stormtroopers (who never hit their targets)?

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."
• Low birth weight paradox: Low birth weight babies have a higher mortality rate, babies of smoking mothers have lower average birth weight, babies of smoking mothers have a higher mortality rate, but low birth weight babies of smoking mothers have a lower mortality rate than other low birth weight babies.
• Loschmidt's paradox

Quine, W. V. (1962) "Paradox". Scientific American, April 1962, pp. 84–96.

Michael Clarke. Paradoxes from A to Z. London: Routledge, 2002.