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A paradox is an argument that produces an inconsistency, typically within logic or common sense.[1] Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.[2] However, some have revealed errors in definitions assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined (e.g., Russell's paradox).[3] Still others, such as Curry's paradox, are not yet resolved. In common usage, the word "paradox" often refers to irony or contradiction. Examples outside logic include the Grandfather paradox from physics, and the Ship of Theseus from philosophy. 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.[4]

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:[5]

Self-reference
"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'?"

Other paradoxes involve false statements 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. 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 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 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.

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 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, which falsely generalizes from true specific statements.
• 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 dialetheism. In Western logics it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern traditions[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 into the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.

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

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.[6]

Footnotes

1. ^ "paradox - definition of paradox by the Free Online Dictionary, Thesaurus and Encyclopedia". Thefreedictionary.com. Unknown parameter `|access date=` ignored (help)
2. ^ "Using Paradoxes to Teach Critical Thinking in Science". Eric.ed.gov. Unknown parameter `|access date=` ignored (help)
3. ^ http://scimath.unl.edu/MIM/files/MATExamFiles/LaFleurK_MATPaperFinal_LA.pdf
4. ^ "The Mathematical Art of M.C. Escher “For me it... - Lapidarium notes". Ami notes.tumble.com. Unknown parameter `|access date=` ignored (help)
5. ^ 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.
6. ^ Kierkegaard, Søren. Philosophical Fragments, 1844. p. 37
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