paradox

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time.

In

logic

, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were 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 or predicates were flawed. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.

Examples outside logic include the ship of Theseus from philosophy, a paradox that questions whether a ship repaired over time by replacing each and all of its wooden parts, one at a time, would remain the same ship. Paradoxes can also take the form of images or other media. For example, M.C. 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 endlessly.

In common usage, the word "paradox" often refers to statements that are

ironic

or unexpected, such as "the paradox that standing is more tiring than walking".

Introduction

Common themes in paradoxes include self-reference,

infinite regress

, circular definitions, and confusion or equivocation between different levels of

abstraction

.

Patrick Hughes outlines three laws of the paradox:
;Self-reference:An example is the statement "This statement is false", a form of the

liar paradox

. The statement is referring to itself. Another example of self-reference is the question of whether the

barber

shaves himself in the barber paradox. Yet another example involves the question "Is the answer to this question 'No'?"
;Contradiction:"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 either to grant or not grant his wish without leading to a contradiction.
;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 and half-truths ("''impossible'' is not in my vocabulary") or rely on a hasty assumption. (A father and his son are in a car crash; the father is killed and the boy is rushed to the hospital. The doctor says, "I can't operate on this boy. He's my son." There is no paradox if the boy's mother is a surgeon.)

Paradoxes that 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

logic

ians and

philosopher

s. "This sentence is false" is an example of the well-known

liar paradox

: it is a sentence that cannot be consistently interpreted as either true or false, because if it is known to be false, then it can be inferred that it must be true, and if it is known to be true, then it can be inferred 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.

Thought-experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a

time-travel

er 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 that 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

W. V. O. Quine (1962) distinguished between three classes of paradoxes:

According to Quine's classification of paradoxes:
* A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless. 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 had he been born on a leap day. Likewise, Arrow's impossibility theorem

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