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Wikipedia: Paradox
Paradox
From Wikipedia, the free encyclopedia.

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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 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 except with stolen money.

Types of paradox

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 [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.

Mathematical/Logical

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.
  • St. Petersburg paradox: People will only offer a modest fee for a reward of infinite value.

Physical

Falsidical paradoxes

These are incorrect results of subtly false reasoning.

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.

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, resulting in your 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."

Links and references

References

See also

External links


  

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