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Template:Further A paradox is a statement or group of statements that leads to a contradiction or a situation which (if true) defies logic or reason, similar to circular reasoning. Typically, however, quoted paradoxical statements do not imply a real contradiction and the puzzling results can be rectified by demonstrating that one or more of the premises themselves are not really true, a play on words, faulty and/or cannot all be true together. But many paradoxes, such as Curry's paradox, do not yet have universally accepted resolutions. The word paradox is often used interchangeably with contradiction. Literary and other artistic uses of paradoxes imply no contradiction and may be used to describe situations that are ironic. Sometimes the term paradox is used for situations that are merely surprising. An example of a paradox is "This statement is false.", and is explained below.
The logician Willard V. O. Quine distinguishes:
- Falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities
- Veridical paradoxes, such as the birthday paradox or the Monty Hall paradox, which are seeming absurdities that are nevertheless true because they are perfectly logical.
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.
- See also: List of paradoxes
- 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?" In this case, replying no would be stating that the answer is not "no". If the reply is yes, it would be stating that it is "no", as the reply was yes. But because the question was answered with a "yes", the answer is not "no". A negative response without saying the word "no", like "it isn't", would, however, render the question answered without bringing about a paradox. Another example is the term 'Nothing is Impossible', meaning that it is possible for something to be impossible, thus contradicting itself.
- "This statement is false"; the statement cannot be false and true at the same time.
- 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."
- "What happens when Pinocchio says, 'My nose will grow now'?"
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 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 mother or father was conceived, thereby preventing his own birth. 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 was 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.
- A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the demonstration. The various invalid 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, falsely generalizes from true specific statements.
- 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.
A fourth kind has sometimes been described since Quine's work.
- A paradox which is both true and false at the same time in the same sense is called a dialetheism. In Western logics it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditionsTemplate:Which and in paraconsistent logics. An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.
Paradox in philosophy
A taste for paradox is central to the philosophies of Laozi, Heraclitus, Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes, in the Philosophical Fragments, that
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.
Alexander Bard and Jan Söderqvist developed a "paradoxology" in their book Det globala imperiet ("The Global Empire"). The authors emphasize paradoxes between the world as static and as ever-changing, while leaning on loose allegories from quantum mechanics. One may also include the philosopher Derrida in a list of users of paradoxes. Derrida's deconstructions attempt to give opposing interpretations of the same text by rhetoric arguments, similar to how lawyers in a court case may argue from the same text, the same set of laws that is, to reach opposite conclusions.
- Animalia Paradoxa
- Ethical dilemma
- Formal fallacy
- Four valued logic
- Impossible object
- Mu (negative)
- Paradoxes of material implication
- Self-refuting ideas
- Temporal paradox
- Twin paradox
- Zeno's paradoxes
- List of paradoxes
- ↑ http://dictionary.reference.com/browse/irony
- ↑ Van Orman Quine, Willard (1966). “The Ways of Paradox”, The Ways of Paradox and Other Essays. Random House.
- ↑ Template:HughesBrecht1975
- ↑ Kierkegaard, Søren. Philosophical Fragments, 1844. p. 37
- ↑ Webster's Revised Unabridged, 2000
- ↑ Sturm, Sean (5 September 2009). Paradoxology. Te ipu Pakore - Escribir es nacer (to write is to be born). wordpress.com. Archived from the original on 12 January 2010. Retrieved on 12 January 2010.
- ↑ Browne, Thomas [first published 1646] (1672). Pseudodoxia Epidemica or Enquries into very many received tenets and commonly presumed truths., 6th. Retrieved on 12 January 2010. “Although whoever shall indifferently perpend the exceeding difficulty, which either the obscurity of the subject, or unavoidable paradoxology must often put upon the Attemptor, he will easily discern, a work of this nature is not to be performed upon one legg; and should smel of oyl, if duly and deservedly handled.”
- ↑ Template:BardSoderqvist2002
with a polynomial to generate a defined sequence. Nothing random about that!
Tedious Computer Geek No. 43,573
- Stanford Encyclopedia of Philosophy:
- "The Legacy of Paradox: From Cognition to Cosmology"
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