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Similar to Circular reasoning
Circular reasoning
Circular reasoning, or in other words, paradoxical thinking, is a type of formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises. For example:"Only an untrustworthy person would run for office...

, A paradox is a seemingly true statement
Statement
Statement may refer to:* A kind of expression in language *Statement , declarative sentence that is either true or false*Statement , the smallest standalone element of an imperative programming language...

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

or a situation which seems to defy logic or intuition
Intuition (knowledge)
Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...

. 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 premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...

s 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
Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules...

, do not yet have universally accepted resolutions.
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

. Literary and other artistic uses of paradoxes imply no contradiction and may be used to describe situations that are ironic
Irony
Irony is a rhetorical device, literary technique, or situation in which there is a sharp incongruity or discordance that goes beyond the simple and evident intention of words or actions...

. 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
Absurdism
In philosophy, "The Absurd" refers to the conflict between the human tendency to seek value and meaning in life and the human inability to find any...

, from
In probability theory, the birthday problem or birthday paradox pertains to the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability reaches 100% when the number of people reaches 366. However, 99%...

, which are seeming absurdities that are nevertheless true because they are perfectly logical.

Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

tend to be the veridical type, typically counterintuitive outcomes of economic theory, such as Simpson's paradox
In probability and statistics, Simpson's paradox is a paradox in which a correlation present in different groups is reversed when the groups are combined. This result is often encountered in social-science and medical-science statistics, and it occurs when frequencydata are hastily given causal...

. In literature
Literature
Literature is the art of written works, and is not bound to published sources...

In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

or obviously untrue statement, which resolves itself upon later inspection
Inspection
An inspection is, most generally, an organized examination or formal evaluation exercise. In engineering activities inspection involves the measurements, tests, and gauges applied to certain characteristics in regard to an object or activity...

.

Common themes in paradoxes include self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...

, infinite regress
Infinite regress
An infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, the truth of proposition P2 requires the support of proposition P3, .....

, circular definition
Circular definition
A circular definition is one that uses the term being defined as a part of the definition or assumes a prior understanding of the term being defined. Either the audience must already know the meaning of the key term, or the definition is deficient in including the term to be defined in the...

s, and confusion between different levels of abstraction
Abstraction
Abstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....

.

Patrick Hughes
Patrick Hughes (artist)
Patrick Hughes is a British artist working in London. He is the creator of "reverspective", an optical illusion on a 3-dimensional surface where the parts of the picture which seem farthest away are actually physically the nearest....

outlines three laws of the paradox:

Self reference: An example is "This statement is false", a form of the liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...

. The statement is referring to itself. Another example of self reference is the question of whether the barber shaves himself in the barber paradox
The Barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible.- The Paradox...

. One more example would be "Is the answer to this question no?" In this case, if you replied no, you would be stating that the answer is not no. If you reply yes, you are stating that it is no, because you said yes. But because you answered yes the answer is not no. However you could reply "It isn't." indicating a negative response without saying the word "no".
Contradiction: "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
Pinocchio
The Adventures of Pinocchio is a novel for children by Italian author Carlo Collodi, written in Florence. The first half was originally a serial between 1881 and 1883, and then later completed as a book for children in February 1883. It is about the mischievous adventures of Pinocchio , an...

says, 'My nose will grow now'?"

False statement
A false statement is a statement that is either willfully or unknowingly untrue. Though the word fallacy is often used as a synonym for false statement, this is not what is meant by "fallacy" in logic or most formal contexts....

s or half-truths
Half-truths
A half-truth is a deceptive statement that includes some element of truth. The statement might be partly true, the statement may be totally true but only part of the whole truth, or it may utilize some deceptive element, such as improper punctuation, or double meaning, especially if the intent is...

and the resulting biased
Cognitive bias
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations. Implicit in the concept of a "pattern of deviation" is a standard of comparison; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable...

assumptions. This form is common in howlers
Howler (error)
Howler, in the main sense this article deals with, is a glaring blunder, typically an amusing one.The word howler is variously used. Some usages are seen as correct English, others as slang...

.

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
Surgery
Surgery is an ancient medical specialty that uses operative manual and instrumental techniques on a patient to investigate and/or treat a pathological condition such as disease or injury, or to help improve bodily function or appearance.An act of performing surgery may be called a surgical...

. 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
Hasty generalization
Hasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence essentially making a hasty conclusion without considering all of the variables...

; 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
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

, 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 logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

ians and philosophers. This sentence is false is an example of the famous liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...

: 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
Four valued logic
In logic, a four-valued logic is used to model signal values in digital circuits: the four values are Z, X and the boolean values 1 and 0. Z stands for high impedance or open circuit, while X stands for "unknown"...

In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...

, 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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

.

Thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

The grandfather paradox is a proposed paradox of time travel first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent . The paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler's...

, for example, would arise if a time travel
Time travel
Time travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...

er 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
The Pirates of Penzance
The Pirates of Penzance; or, The Slave of Duty is a comic opera in two acts, with music by Arthur Sullivan and libretto by W. S. Gilbert. The opera's official premiere was at the Fifth Avenue Theatre in New York City on 31 December 1879, where the show was well received by both audiences...

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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

demonstrates difficulties in mapping voting results to the will of the people.
• 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
Invalid proof
In mathematics, certain kinds of mistakes in proof, calculation, or derivation are often exhibited, and sometimes collected, as illustrations of the concept of mathematical fallacy...

(e.g., that 1 = 2) are classic examples, generally relying on a hidden division by zero
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...

. 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
Antinomy
Antinomy literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology....

, 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
Dialetheism
Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...

. In Western logics it is often assumed, following Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

, that no dialetheia exist, but they are sometimes accepted in Eastern traditions and in paraconsistent logic
Paraconsistent logic
A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...

s. 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.

The paradox as a literary device
Literary technique
A literary technique is any element or the entirety of elements a writer intentionally uses in the structure of their work...

has been assigned as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition
Exposition (literary technique)
At the beginning of a narrative, the exposition is the author's providing of some background information to the audience about the plot, characters' histories, setting, and theme. Exposition is considered one of four rhetorical modes of discourse, along with argumentation, description, and narration...

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
Oscar Wilde
Oscar Fingal O'Flahertie Wills Wilde was an Irish writer and poet. After writing in different forms throughout the 1880s, he became one of London's most popular playwrights in the early 1890s...

and G. K. Chesterton
G. K. Chesterton
Gilbert Keith Chesterton, KC*SG was an English writer. His prolific and diverse output included philosophy, ontology, poetry, plays, journalism, public lectures and debates, literary and art criticism, biography, Christian apologetics, and fiction, including fantasy and detective fiction....

; other literature deals with paradox of situation. Rabelais
François Rabelais
François Rabelais was a major French Renaissance writer, doctor, Renaissance humanist, monk and Greek scholar. He has historically been regarded as a writer of fantasy, satire, the grotesque, bawdy jokes and songs...

, Cervantes
Miguel de Cervantes
Miguel de Cervantes Saavedra was a Spanish novelist, poet, and playwright. His magnum opus, Don Quixote, considered the first modern novel, is a classic of Western literature, and is regarded amongst the best works of fiction ever written...

, Sterne
Laurence Sterne
Laurence Sterne was an Irish novelist and an Anglican clergyman. He is best known for his novels The Life and Opinions of Tristram Shandy, Gentleman, and A Sentimental Journey Through France and Italy; but he also published many sermons, wrote memoirs, and was involved in local politics...

, and Borges
Jorge Luis Borges
Jorge Francisco Isidoro Luis Borges Acevedo , known as Jorge Luis Borges , was an Argentine writer, essayist, poet and translator born in Buenos Aires. In 1914 his family moved to Switzerland where he attended school, receiving his baccalauréat from the Collège de Genève in 1918. The family...

, for instance, are all concerned with episodes and narratives designed around paradoxes. Two of literature's arguably most famous paradoxes is the Miltonic narrator's statement in Book One of 'Paradise Lost', that the fires of hell emit 'no light, but darkness visible.' Statements such as Wilde's "I can resist anything except temptation", Chesterton's "Spies do not look like spies" and Polonius
Polonius
Polonius is a character in William Shakespeare's Hamlet. He is King Claudius's chief counsellor, and the father of Ophelia and Laertes. Polonius connives with Claudius to spy on Hamlet...

' observation in Hamlet
Hamlet
The Tragical History of Hamlet, Prince of Denmark, or more simply Hamlet, is a tragedy by William Shakespeare, believed to have been written between 1599 and 1601...

that "though this be madness, yet there is method in't" are examples of rhetorical paradox.

A taste for paradox is central to the philosophies of Laozi
Laozi
Laozi was a mystic philosopher of ancient China, best known as the author of the Tao Te Ching . His association with the Tao Te Ching has led him to be traditionally considered the founder of Taoism...

, Heraclitus
Heraclitus
Heraclitus of Ephesus was a pre-Socratic Greek philosopher, a native of the Greek city Ephesus, Ionia, on the coast of Asia Minor. He was of distinguished parentage. Little is known about his early life and education, but he regarded himself as self-taught and a pioneer of wisdom...

, Meister Eckhart
Meister Eckhart
Eckhart von Hochheim O.P. , commonly known as Meister Eckhart, was a German theologian, philosopher and mystic, born near Gotha, in the Landgraviate of Thuringia in the Holy Roman Empire. Meister is German for "Master", referring to the academic title Magister in theologia he obtained in Paris...

, Hegel
Georg Wilhelm Friedrich Hegel
Georg Wilhelm Friedrich Hegel was a German philosopher, one of the creators of German Idealism. His historicist and idealist account of reality as a whole revolutionized European philosophy and was an important precursor to Continental philosophy and Marxism.Hegel developed a comprehensive...

, Kierkegaard
Søren Kierkegaard
Søren Aabye Kierkegaard was a Danish Christian philosopher, theologian and religious author. He was a critic of idealist intellectuals and philosophers of his time, such as Georg Wilhelm Friedrich Hegel, Friedrich Wilhelm Joseph Schelling and Karl Wilhelm Friedrich Schlegel...

, and Nietzsche
Friedrich Nietzsche
Friedrich Wilhelm Nietzsche was a 19th-century German philosopher, poet, composer and classical philologist...

, among many others. Søren Kierkegaard, for example, writes, in the Philosophical Fragments
Philosophical Fragments
Philosophical Fragments was a Christian philosophic work written by Danish philosopher Søren Kierkegaard in 1844. It was the first of three works written under the pseudonym Johannes Climacus, the other two were Johannes Climacus, 1841 and Concluding Unscientific Postscript to Philosophical...

, 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.

In moral philosophy, paradox in a loose sense plays a role in ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

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
Ethical dilemma
An Ethical dilemma is a complex situation that will often involve an apparent mental conflict between moral imperatives, in which to obey one would result in transgressing another....

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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

; 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.

Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...

es". "As a word it originates from Thomas Browne
Thomas Browne
Sir Thomas Browne was an English author of varied works which reveal his wide learning in diverse fields including medicine, religion, science and the esoteric....

in his book Pseudodoxia Epidemica
Pseudodoxia Epidemica
Pseudodoxia Epidemica or Enquries into very many received tenets and commonly presumed truths, also known simply as Pseudodoxia Epidemica or Vulgar Errors, is a work by Thomas Browne refuting the common errors and superstitions of his age. It first appeared in 1646 and went through five subsequent...

."

Alexander Bard
Alexander Bard
Alexander Bengt Magnus Bard is a Swedish artist, music producer and writer.-Background and education:Bard was born Alexander Bengt Magnus Bard in Medevi, Motala Municipality, Sweden. After he had completed his upper secondary education, Bard studied in the United States and in Amsterdam, Netherlands...

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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

.
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.

are the mythical, magical or otherwise suspect animals included in early editions of Carl Linnaeus' seminal work under the cryptid wastebasket taxon of "Paradoxa". It includes fantastic creatures found in medieval bestiaries as well as those reported by explorers from abroad...

• Dilemma
Dilemma
A dilemma |proposition]]") is a problem offering two possibilities, neither of which is practically acceptable. One in this position has been traditionally described as "being on the horns of a dilemma", neither horn being comfortable...

• Irony
Irony
Irony is a rhetorical device, literary technique, or situation in which there is a sharp incongruity or discordance that goes beyond the simple and evident intention of words or actions...

• Oxymoron
Oxymoron
An oxymoron is a figure of speech that combines contradictory terms...

• Ethical dilemma
Ethical dilemma
An Ethical dilemma is a complex situation that will often involve an apparent mental conflict between moral imperatives, in which to obey one would result in transgressing another....

• Formal fallacy
Formal fallacy
In philosophy, a formal fallacy is a pattern of reasoning that is always wrong. This is due to a flaw in the logical structure of the argument which renders the argument invalid...

• Four valued logic
Four valued logic
In logic, a four-valued logic is used to model signal values in digital circuits: the four values are Z, X and the boolean values 1 and 0. Z stands for high impedance or open circuit, while X stands for "unknown"...

• Impossible object
Impossible object
An impossible object is a type of optical illusion consisting of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object although it is not actually possible for such an object to exist An impossible...

• Mu (negative)
Mu (negative)
or Wu , is a word which has been translated variously as "not", "nothing", "without", "nothingness", "non existent", "non being", or evocatively simply as "no thing"...

The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment....

• Self-refuting idea
Self-refuting idea
Self-refuting ideas are ideas or statements whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are accused by their detractors of being self-refuting, and such accusations are therefore almost always controversial, with defenders claiming that...

s