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The Monty Hall paradox: which door do you choose?
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 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
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 believed that there are 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.
- Paradox of entailment: Inconsistent premises always
make an argument valid.
- Apportionment paradox: Some systems of apportioning
representation can have unintuitive 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 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: The exists a countable subset C of the
sphere S such that S\C is equidecomposable with 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.
- 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.
- Epicurian paradox: The existence of evil is incompatible with
the existence of an omnipotent and caring God.
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.
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?
- 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, 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.
- 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
1 See Quine, W. V: "Paradox," Scientific American, April 1962, pp. 84–96.
See also
External links
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