From Wikipedia, the free encyclopedia
For other uses, see
is a statement that apparently contradicts itself and yet might be
true (or wrong at the same time).
Some logical paradoxes are known to be invalid
arguments but are still valuable in promoting critical
Some paradoxes have
revealed errors in definitions assumed to be rigorous, and have
caused axioms of
mathematics and logic to be re-examined. One example is Russell's
paradox, which questions whether a "list of all lists that
do not contain themselves" would include itself, and showed
that attempts to found set
theory on the identification of sets with properties
Others, such as Curry's
paradox, are not yet resolved.
by M. C. Escher,
Examples outside logic include the Ship
of Theseus from philosophy (questioning whether a ship repaired
over time by replacing each of its wooden parts would remain the same
ship). Paradoxes can also take the form of images or other media. For
Escher featured perspective-based
paradoxes in many of his drawings, with walls that are regarded as
floors from other points of view, and staircases that appear to climb
In common usage, the word
"paradox" often refers to statements that are ironic
or unexpected, such as "the paradox that standing is more tiring
See also: List
Common themes in paradoxes include self-reference,
definitions, and confusion between different levels of
Hughes outlines three laws of the paradox:
An example is "This statement is false", a form of the
The statement is referring to itself. Another example of
self-reference is the question of whether the barber shaves himself
in the barber
paradox. One more example would be "Is the answer to this
"This statement is false"; the statement cannot be false
and true at the same time. Another example of contradiction is if a
man talking to a genie wishes that wishes couldn't come true. This
contradicts itself because if the genie grants his wish, he did not
grant his wish, and if he refuses to grant his wish, then he did
indeed grant his wish, therefore making it impossible to either
grant or not grant his wish because his wish contradicts itself.
Vicious circularity, or infinite regress
"This statement is false"; if the statement is true, then
the statement is false, thereby making the statement true. Another
example of vicious
circularity is the following group of statements:
"The following sentence is true."
"The previous sentence is false."
Other paradoxes involve false
statements ("impossible is not a word in my vocabulary",
a simple paradox) or half-truths
and the resulting biased
assumptions. This form is common in howlers.
For example, consider a situation in which a father and his son
are driving down the road. The car crashes into a tree and the father
is killed. The boy is rushed to the nearest hospital where he is
prepared for emergency surgery.
On entering the surgery suite, the surgeon says, "I can't
operate on this boy. He's my son."
The apparent paradox is caused by a hasty
generalization, for if the surgeon is the boy's father, the
statement cannot be true. The paradox is resolved if it is revealed
that the surgeon is a woman — the boy's mother.
Paradoxes which are not based on a hidden error generally occur at
the fringes of context or language,
and require extending the context or language in order to lose their
paradoxical quality. Paradoxes that arise from apparently
intelligible uses of language are often of interest to logicians
"This sentence is false" is an example of the well-known
it is a sentence which cannot be consistently interpreted as either
true or false, because if it is known to be false, then it is known
that it must be true, and if it is known to be true, then it is known
that it must be false. Russell's
paradox, which shows that the notion of the set
of all those sets that do not contain themselves leads to a
contradiction, was instrumental in the development of modern logic
and set theory.
experiments can also yield interesting paradoxes. The grandfather
paradox, for example, would arise if a time
traveller were to kill his own grandfather before his mother or
father had been conceived, thereby preventing his own birth. This is
a specific example of the more general observation of the butterfly
effect, or that a time-traveller's interaction with the past —
however slight — would entail making changes that would, in
turn, change the future in which the time-travel was yet to occur,
and would thus change the circumstances of the time-travel itself.
Often a seemingly paradoxical conclusion arises from an
inconsistent or inherently contradictory definition of the initial
premise. In the case of that apparent paradox of a time traveler
killing his own grandfather it is the inconsistency of defining the
past to which he returns as being somehow different from the one
which leads up to the future from which he begins his trip but also
insisting that he must have come to that past from the same future as
the one that it leads up to.
Quine's classification of paradoxes
W. V. Quine
(1962) distinguished between three classes of paradoxes:
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
twenty-one-year-old would have had only five birthdays, if he had
been born on a leap
day. Likewise, Arrow's
impossibility theorem demonstrates difficulties in mapping
voting results to the will of the people. The Monty
Hall paradox demonstrates that a decision which has an intuitive
50-50 chance in fact is heavily biased towards making a decision
which, given the intuitive conclusion, the player would be unlikely
to make. In 20th century science, Hilbert's
paradox of the Grand Hotel and Schrödinger's
cat are famously vivid examples of a theory being taken to a
logical but paradoxical end.
A falsidical paradox
establishes a result that not only appears false but actually
is false, due to a fallacy in the demonstration. The various
mathematical proofs (e.g., that 1 = 2) are classic examples,
generally relying on a hidden division
by zero. Another example is the inductive form of the horse
paradox, which falsely generalizes from true specific
paradoxes are falsidical, concluding for example that a flying
arrow never reaches its target or that a speedy runner cannot catch
up to a tortoise with a small head start.
A paradox that 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.
A fourth kind has sometimes been described since Quine's work.
A paradox that is both true and false at the same time and in the
same sense is called a dialetheia.
In Western logics it is often assumed, following Aristotle,
that no dialetheia exist, but they are sometimes accepted in
Eastern traditions (e.g. in the Mohists,
and in Zen)
and in paraconsistent
logics. It would be mere equivocation or a matter of degree, for
example, to both affirm and deny that "John is here" when
John is halfway through the door but it is self-contradictory to
simultaneously affirm and deny the event in some sense.
A taste for paradox is central to the philosophies of Laozi,
Chesterton, among many others. Søren Kierkegaard, for
example, writes, in the Philosophical
But one must not think ill of
the paradox, for the paradox is the passion of thought, and the
thinker without the paradox is like the lover without passion: a
mediocre fellow. But the ultimate potentiation of every passion is
always to will its own downfall, and so it is also the ultimate
passion of the understanding to will the collision, although in one
way or another the collision must become its downfall. This, then, is
the ultimate paradox of thought: to want to discover something that
thought itself cannot think.
reaction to a drug
is the opposite of what one would expect, such as becoming agitated
by a sedative or
sedated by a stimulant.
Some are common and are used regularly in medicine, such as the use
of stimulants such as Adderall
and Ritalin in
the treatment of attention
deficit disorder and attention
deficit hyperactivity disorder, while others are rare and can be
dangerous as they are not expected, such as severe agitation from a
(1985). Hong, Howard V.; Hong, Edna H., eds. Philosophical
Fragments. Princeton University Press. p. 37.
William Poundstone, 1989, Labyrinths of Reason: Paradox, Puzzles,
and the Frailty of Knowledge, Anchor
Mark Sainsbury, 1988, Paradoxes,
Cambridge: Cambridge University Press
Roy Sorensen, 2005, A Brief History of the Paradox:
Philosophy and the Labyrinths of the Mind, Oxford University Press
Wikiquote has quotations related to: Paradox
Look up paradox
in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Paradoxes.
Listen to this
audio file was created from a revision of the "Paradox"
article dated 2005-07-07, and does not reflect subsequent edits to
the article. (Audio