A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth (cf. kōan, Catuskoti). Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction. It is also used to describe situations that are ironic.
But many paradoxes, such as Curry's paradox, do not yet have universally accepted resolutions.
Sometimes the term paradox is used for situations that are merely surprising. The birthday paradox, for instance, is unexpected but perfectly logical. The logician Willard V. O. Quine distinguishes falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities, from veridical paradoxes, such as the birthday paradox, which are seeming absurdities that are nevertheless true. Paradoxes in economics tend to be the veridical type, typically counterintuitive outcomes of economic theory, such as Simpson's paradox. In literature a paradox can be any contradictory or obviously untrue statement, which resolves itself upon later inspection.
"The statement below is false." "The statement above is true."
For example, consider a situation in which a father and his son are driving down the road. The car collides with 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; 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 happen at the fringes of context or language, and require extending the context or language to lose their paradoxical quality. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. This sentence is false is an example of the famous liar paradox: it is a sentence which cannot be consistently interpreted as 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. Therefore, it can be concluded that it is unknowable. 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 traveler were to kill his own grandfather before his father was conceived, thereby preventing his own birth. This paradox can be resolved by postulating that time travel leads to parallel or bifurcating universes, or that only contradiction-free timelines are stable.
W. V. Quine (1962) distinguished between three classes of paradoxes:
A fourth kind has sometimes been described since Quine's work.
The paradox as a literary device has been defined as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unorthodox insight. It functions as a method of literary analysis which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.
Literary or rhetorical paradoxes abound in the works of Oscar Wilde and G. K. Chesterton; other literature deals with paradox of situation. Rabelais, Cervantes, Sterne, Borges, and Chesterton are all concerned with episodes and narratives designed around paradoxes. Statements such as Wilde's "I can resist anything except temptation" and Chesterton's "spies do not look like spies" are examples of rhetorical paradox. Further back, Polonius' observation in Hamlet that "though this be madness, yet there is method in't" is a memorable third.
In moral philosophy, paradox in a loose sense plays a role in ethics debates. For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to armed neighbors actively intending murder. If the hostile neighbors succeed, it is impossible to follow the dictum. On the other hand, to attack, fight back, or restrain them is also not usually considered loving. This might be better termed an ethical dilemma rather than a paradox in the strict sense. However, for this to be a true example of a moral paradox, it must be assumed that "loving" and restraint cannot co-exist. In reality, this situation occurs often, notably when parents punish children out of love.
Another example is the conflict between a moral injunction and a duty that cannot be fulfilled without violating that injunction. For example, take the situation of a parent with children who must be fed (the duty), but cannot afford to do so without stealing, which would be wrong (the injunction). Such a conflict between two maxims is normally resolved through weakening one or the other of them: the need for survival is greater than the need to abide by the law. However, as maxims are added for consideration, the questions of which to weaken in the general case and by how much pose issues related to Arrow's impossibility theorem; it may not be possible to formulate a consistent system of ethics rules with a definite order of preference in the general case, a so-called "ethical calculus".
Paradoxes in a more strict sense have been relatively neglected in philosophical discussion within ethics, as compared to their role in other philosophical fields such as logic, epistemology, metaphysics, or even the philosophy of science. Important book-length discussions appear in Derek Parfit's Reasons and Persons and in Saul Smilansky's 10 Moral Paradoxes.
I. Tis true the beauteous Starre To which I first did bow Burnt quicker, brighter far Then that which leads me now; Which shines with more delight: For gazing on that light So long, neere lost my sight. II. Through foule, we follow faire, For had the World one face And Earth been bright as Ayre, We had knowne neither place; Indians smell not their Neast: A Swisse or Finne tastes best, The Spices of the East. III. So from the glorious Sunne, Who to his height hath got, With what delight we runne To some black Cave, or Grot! And Heav'nly Sydney you Twice read, had rather view Some odde Romance, so new. IV. The God that constant keepes Unto his Dieties, Is poore in Joyes, and sleepes Imprison'd in the skies: This knew the wisest, who From Juno stole, below To love a Beare, or Cow.
|This work published before January 1, 1923 is in the public domain worldwide because the author died at least 100 years ago.|
PARADOX (Gr. irapa, beyond, contrary to, S6Ea, opinion), a proposition or statement which appears to be at variance with generally-received opinion, or which apparently is self-contradictory, absurd or untrue, but either contains a concealed truth or may on examination be proved to be true. A "paradox" has been compared with a "paralogism" (7rapa, X6 yos, reason), as that which is contrary to opinion only and not contrary to reason, but it is frequently used in the sense of that which is really absurd or untrue.
's self-flowing flask fills itself in this picture, but perpetual motion machines cannot exist.]] A paradox is a sentence in logic that cannot be true but also cannot be false. Many famous problems of this kind exist.
One of most famous paradoxes is called the liar's paradox. It is the simple sentence "This sentence is a lie."
If the sentence is true, then it is a lie, as it says. But if it is a lie, how can it be true? A lie cannot also be the truth. So the sentence being true makes it a lie.
If the sentence is a lie, then it is not as it says, it is true. But that is just what the sentence says. So that makes it true. So the sentence being a lie makes it true.
This paradox is not just true in English but in any language powerful enough for a sentence to make a claim about itself. This is true of mathematics as well. Paradox can never be removed from any symbol system that makes claims about itself.
Another example is the statement that there is no cabal. Only a cabal can know if there is no cabal, so this is either a guess, or, it is a cabal trying to pretend it does not exist.
Other famous examples:
A paradox can also arise in ethics. Taking power over others is often also required to protect them, but also, one of the things being protected is their ability to do as they please, which this power interferes with. There is another article on ethical dilemma which means "a paradox arising in ethics".
Because a paradox forces us to think "out of the box", about possibilities other than true or false in logic, right or wrong in morality, it is considered very important in education. People who do not see a paradox where others do, are likely to be too certain they are right.