Paradoxo

Arquivo:180px-Boyle'sSelfFlowingFlask.png

O frasco com auto-fluxo de Robert Boyle preenche a si próprio neste diagrama, mas máquinas de moto contínuo não existem.

Um paradoxo é uma declaração aparentemente verdadeira que leva a uma contradição lógica, ou a uma situação que contradiz a intuição comum. Em termos simples, um paradoxo é 'o oposto do que alguém pensa ser a verdade'. A identificação de um paradoxo baseado em conceitos aparentemente simples e racionais tem, por vezes, auxiliado significativamente o progresso da ciência, filosofia e matemática.

A etimologia da palavra paradoxo pode ser traçada a textos que remontam à aurora da Renascença, um período de acelerado pensamento científico na Europa e Ásia que começou por volta do ano de 1500. As primeiras formas da palavra tiveram por base a palavra latina paradoxum, mas também são encontradas em textos em grego como paradoxon (entretanto, o Latim é fortemente derivado do alfabeto grego e, além do mais, o Português é também derivado do Latim romano, com a adição das letras "J" e "U"). A palavra é composta do prefixo para-, que quer dizer "contrário a", "alterado" ou "oposto de", conjungada com o sufixo nominal doxa, que quer dizer "opinião". Compare com ortodoxia e heterodoxo.

Na filosofia moral, o paradoxo tem um papel central nos debates sobre ética. Por exemplo, a admoestação ética para "amar o seu próximo" não apenas contrasta, mas está em contradição com um "próximo" armado tentando ativamente matar você: se ele é bem sucedido, você não será capaz de amá-lo. Mas atacá-lo preemptivamente ou restringi-lo não é usualmente entendido como algo amoroso. Isso pode ser considerado um dilema ético. Outro exemplo é o conflito entre a injunção contra roubar e o cuidado para com a família que depende do roubo para sobreviver.

Deve ser notado que muitos paradoxos dependem de uma suposição essencial: que a linguagem (falada, visual ou matemática) modela de forma acurada a realidade que descreve. Em física quântica, muitos comportamentos paradoxais podem ser observados (o princípio da incerteza de Heisenberg, por exemplo) e alguns já foram atribuídos ocasionalmente às limitações inerentes da linguagem e dos modelos científicos. Alfred Korzybski, que fundou o estudo da Semântica Geral, resume o conceito simplesmente declarando que, "O mapa não é o território". Um exemplo comum das limitações da linguagem são algumas formas do verbo "ser". "Ser" não é definido claramente (a área de estudos filosóficos chamada ontologia ainda não produziu um significado concreto) e assim se uma declaração incluir "ser" com um elemento essencial, ela pode estar sujeita a paradoxos.

Tipos de paradoxos

Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning.

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.

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.¹

List of paradoxes

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

Veridical paradoxes

These are unintuitive results of correct logical reasoning.

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The Monty Hall paradox: which door do you choose?

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.

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.
• 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.

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.

References

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

See also

• Impossible objects

External links

• Google Directory: Paradoxes
• Definability paradoxes