Paradox

For other uses, see Paradox (disambiguation).

A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion.[1][2] Some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.[3]

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 or predicates were flawed.[4] Others, such as Curry's paradox, are not yet resolved.

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 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.[5]

In common usage, the word "paradox" often refers to statements that may be both true and false i.e. ironic or unexpected, such as "the paradox that standing is more tiring than walking".[6]

Logical paradox

Common themes in paradoxes include self-reference, infinite regress, circular definitions, and confusion between different levels of abstraction.

Patrick Hughes outlines three laws of the paradox:[7]

Self-reference
An example is "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. One more example would be "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 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 and philosophers. "This sentence is false" is an example of the well-known liar paradox: 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.

Thought 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

See also: Veridicality

W. V. Quine (1962) distinguished between three classes of paradoxes:[8]

A fourth kind has sometimes been described since Quine's work.

In philosophy

A taste for paradox is central to the philosophies of Laozi, Zhuangzi, Heraclitus, Bhartrhari, Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes, in the Philosophical Fragments, that

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.[12]

In medicine

A paradoxical 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 hyperactivity disorder (also known as ADHD or ADD,) while others are rare and can be dangerous as they are not expected, such as severe agitation from a benzodiazepine.

See also

References

Notes
  1. ↑ "paradox". Oxford Dictionary. Oxford University Press. Retrieved 21 June 2016.
  2. ↑ Bolander, Thomas (2013). "Self-Reference". The Metaphysics Research Lab, Stanford University. Retrieved 21 June 2016.
  3. ↑ Eliason, James L. (March–April 1996). "Using Paradoxes to Teach Critical Thinking in Science". Journal of College Science Teaching. 15 (5): 341–44. (subscription required (help)).
  4. ↑ Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York: Oxford University Press. pp. 59–60. ISBN 0-19-888087-1. Zbl 0251.02001.
  5. ↑ Skomorowska, Amira (ed.). "The Mathematical Art of M.C. Escher". Lapidarium notes. Retrieved 2013-01-22.
  6. ↑ "Paradox". Free Online Dictionary, Thesaurus and Encyclopedia. Retrieved 2013-01-22.
  7. ↑ Hughes, Patrick; Brecht, George (1975). Vicious Circles and Infinity - A Panoply of Paradoxes. Garden City, New York: Doubleday. pp. 1–8. ISBN 0-385-09917-7. LCCN 74-17611.
  8. ↑ Quine, W.V. (1966). "The ways of paradox". The Ways of Paradox, and other essays. New York: Random House.
  9. ↑ The Logicians (Warring States period),"Miscellaneous paradoxes" Stanford Encyclopedia of Philosophy
  10. ↑ Graham, Angus Charles. (1990). Studies in Chinese Philosophy and Philosophical Literature, p. 334., p. 334, at Google Books
  11. ↑ Chung-ying Cheng (1973) "On Zen (Ch’an) Language and Zen Paradoxes" Journal of Chinese Philosophy, V. 1 (1973) pp. 77-102
  12. ↑ Kierkegaard, Søren (1844). Hong, Howard V.; Hong, Edna H., eds. Philosophical Fragments. Princeton University Press (published 1985). p. 37. ISBN 9780691020365.
Bibliography
  • 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
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