The paradox is an apparently true statement or class action of statements that seems to lead to the contradiction or to a situation that defies intuition. Generally, either a statements within wonder don't really indicate a contradiction, a puzzling effect is non really a contradiction, or even even the premiss themselves are non whole really avowedly (or, can't totally become avowedly together). A recognition of ambiguities, equivocations, and unsaid assumptions underlying known paradoxes has typically led to important advances within science, philosophy and mathematics.

A word paradox is typically utilized interchangeably using contradiction; however in which the contradiction by definition can't become avowedly, numbers of paradoxes wash allow guide, though numerous remain unsolved or even exclusively contentiously resolved (like Curry's paradox). However supplementary casually, the term is periodically utilized for situations that come but surprising (albeit inside a distinctly "logical" manner) like a Birthday Paradox. This is too a usage within economics, where the paradox is an unintuitive effect of economic theory.

A etymology of paradox can be traced back to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and a related Greek paradoxon. A word is composed of the preposition para which means "by way of", or "according to" conjoint to the noun stem doxa, meaning "what is received". Compare orthodox (literally, "straight teaching") and heterodox (literally, "different teaching"). A prevaricator paradox & more paradoxes were exposed inside mediaeval days under a running insolubilia.

Most common themes within paradoxes include directly & indirect self-information, eternity, round definitions, & confusion of levels of logical thinking. Paradoxes which are then non according to a hidden error usually happen at the fringes of context or language, and take extending a context (or even language) to lose their paradox quality.

Around moral philosophy, paradox plays a central role inside ethics debates. E.g., an moral admonition to "love thy neighbour" is non merely witharound counterpoint sustaining, however in contradiction to an armed neighbour actively trying to wipe out professional people: whenever he or even even she succeeds, smart shoppers may non exist as entity to love him or her. However to preemptively attack the two or even restrain the children is non unremarkably understood when caring. This will exist as termed an ethical dilemma. Another example is the conflict between an injunction does'nt to steal and one to care for the personal you just can not afford to feed forgoing purloined money.

Types of Paradoxes

W. V. Quine (1962) distinguished three classes of paradox: The real paradox produces the effect that appears absurd however is demonstrated to exist as confessedly still. So, a paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that the human can be to a higher degree Nine years old in his 9th birthday. Also, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but all too true. The falsidical paradox establishes the effect that non just appears faithlessly however actually is faithlessly; there is a false belief in the supposed demonstration. A various invalid proofs (e.g. that Unity = Deuce) come classic examples, usually relying in the hidden section by zero. Another example would become a Horse paradox. The paradox which is within neither class can be an antinomy, which reaches the self-contradictory effect by properly using accepted ways of abstract thought. For instance, a Grelling-Nelson paradox points out echt problems within my understanding of the ideas of truth and description.

List of paradoxes

Quine's classification, naturally, is utile exclusively it used to be that the paradox has the clear guide. That wonder is treated on a document for both single paradox; the ensuing come grouped thematically.

Logical (except mathematical)

Paradox of entailment: Inconsistent premises universally produce an argument valid. Raven paradox (or Hempel's Corvus corax): Researching a red apple increases the likeliness of 100% corvus corax existence melanise. Horse paradox: All horses come a equivalent colour. Unexpected hanging paradox: The day of the hanging is the surprise, then it can't happen in the least, thus it is the surprise. Russell's paradox: Does the set of all those sets that do not contain themselves contain itself? Barber paradox: The adult male barber world health organization shaves everthing men world health organization don't shave themselves, & there are no-a single else. (The close relative of Russell's paradox.) Richard's paradox: We appear to be able to use simple English to define a decimal expansion in a way which is self-contradictory. Löb's paradox: If this sentence is true, then Santa Claus exists.

Semantic Paradoxes

These form the easily-known (& easily-exposed) class with inside park that any allowable assignment of semantic value (truth, information) to an expression immediately implies the assignment of a different value.

Berry paradox: What is "The first number not nameable in under ten words"? (& has it non upright been known as inside nine?) Curry's paradox: "If I'm not mistaken, the world will end in a week." Epimenides paradox: A Cretan says "All Cretans are liars". Grelling-Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterologous word? Liar paradox: "This sentence is false." Quine's liar paradox: "Yields a falsehood when appended to its own quotation." A Y combinator in the lambda calculus and combinatory logic has been called a paradoxical combinator since it is related to the self-referential antinomies. Missing Dollar Paradox: Faulty logic makes it appear when whenever the dollar from either the eating house bill has never again missing. Non in a equivalent class when the others. Hegel's paradox: "Man learns from history that man learns nothing from history." Petronius' paradox: "Moderation in all things. Including moderation."

Vagueness

Ship of Theseus/George Washington's axe: When every component of the ship has been replaced at least once, is it still the same ship? Sorites paradox: At what point does the heap prevent existence a heap when I personally choose away grains of sand? Alternately, at what point does somebody be bald?

Mathematical and statistical

Apportionment paradox: Some systems of apportioning representation could keep close at hand unintuitive results Alabama paradox New states paradox Population paradox Will Rogers phenomenon: the mathematical conception of an average, whether defined as a mean or median, leads to apparently paradoxical resolutions - for instance, these are conceivable that moving an entry from either Wikipedia to Wiktionary would increase the typical entry length in each sites Berkson's paradox Bertrand's paradox (probability): Different common-sense definitions of randomness give quite different results. Birthday paradox: What is a chance that deuce humans inside the room keep around the equivalent birthday? Borel's paradox: Conditional probability density functions are not invariant under coordinate transformations. Elevator paradox: Elevators can seem to become mostly enter a single counsel, when whenever it were existence made midmost of the building & existence disassembled on the roof & basement. Hodgson's paradox: the ratio of two Gaussian random variables has neither mean nor variance. Richard's paradox: A complete list of definitions of real numbers do not exist. Monty Hall problem: An unintuitive consequence of contingent probability. Simpson's paradox: An association in sub-populations may be reversed in the population. It appears that ii sets of information singly trend lines a certain hypothesis, however, whilst considered together, it trend lines the paired hypothesis. Sleeping beauty paradox: a single half or even one third? news://rec.puzzles can't agree in the probability. Technical indicator paradox: These are quite imaginable to draw incorrectly conclusions from either correlation. E.g., towns sustaining the pack of churches usually have a higher crime rate — because two symptom from either higher people. The professional organization another time encountered that economists with a Ph.D. actually had a lower average salary than those with a BS — but this was found to be due to the fact that those with a Ph.D. worked in academia, where salaries are generally lower. This is likewise known as the spurious relationship. Low birth weight paradox: moo birth weight new borns stand a higher death rate, children of smoking moms have moo typical birth weight, infants of smoking moms have a higher deathrate, however moo birth weight infants of smoking moms have a moo death rate than more low birth weight toddlers. Two-envelope paradox: Given two envelopes, a single of which contains twice when much money as a more, a advantage seems universally to lie inside switching from either one to the more, & universally sticking by using your computers original selection.

Infinity

Burali-Forti paradox: If a ordinal numbers formed a set, it would be an ordinal total which is little than itself. Galileo's paradox: Though most numbers are not squares, there are no more numbers than squares. (Look at too Cantor, Diagonal Argument) Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests. Monty Hell problem: Positive day-to-day successfully yield zero plus in the (infinite) set boundaries. Skolem's paradox: Countably infinite models of set theory contain uncountably infinite sets.

Geometry and Topology

Banach-Tarski paradox: Cut a ball into Quintuplet pieces, re-assemble a pieces for deuce balls, each of equal size to the number 1. Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Likewise, a Mandelbrot set and various other fractals have finite locality, however infinite perimeter (as a matter of fact, there is no 2 distinct points on a border of the Mandelbrot placed that may be reached from either 1 an additional by moving a finite few feet away along the border, which as well implies that in a way that you last no more if you hike "the wrong way" in the placed to email a nearby point). Hausdorff paradox: There exists the numerable subset One hundred of the sphere S such that S\C is equidecomposable sustaining deuce copies of itself. Smale's paradox: A sphere can, topologically, be turned inside out.

Psychological, action, and practical reason

Abilene paradox: population might produce decisions depending nin in what it actually obviously run, however on what it believe that more people obviously clean, by having a symptom that everybody decides to wash something that cipher really wants to wash, but only what it thought that everybody else wanted to wash. Buridan's ass: How can a rational choice be made between two outcomes of equal value? Control paradox: Man potty never become unhampered control, for to exist as unhampered control is to become controlled by oneself. Paradox of hedonism: while a single pursues happiness itself, a single is miserable; however, when 1 pursues something else, a single achieves happiness. Newcomb's paradox: How do you play a game against an omniscient opponent? Kavka's toxin puzzle: Can one intend to drink the deadly toxin, if the intention is the only thing needed to get the reward? Socratic Paradox: One might never wittingly select a lesser effective or even volitionally clean a incorrectly tool. Cursed Friend: If a single is cursed into hurting a single’s friends by upright existence by owning a children, potty 1 however exist as the friend & search the more's company? Paradox of Denial: How can a single assume oneself when a single is, in case 1 is non a way 1 thinks one should exist as? Eventually, how else does 1 profits vary oneself forgoing accepting oneself when 1 is, & working from either that? Hypocritical paradox: I personally hate pretender, however 1 tool I hate possibly supplementary is illiberal dissimulator.

Physical

Braess' paradox: Sometimes adding additional capacity to the network could reduce overall performance. Carroll's paradox: the angular momentum of a stick should be zero, but is not D'Alembert's paradox: An inviscid liquid produces no drag. Denny's paradox: Surface-dwelling arthropods (such as the water strider) should not be able to propel themselves horizontally. Einstein-Podolsky-Rosen paradox: Can far away cases influence both more inside quantum mechanics? Gibbs paradox: In an ideal gas, is entropy an extensive variable? A GZK paradox: High-energy cosmic rays have been observed which seem to violate a Greisen-Zatsepin-Kuzmin limit which is a symptom of special relativity. Loschmidt's paradox: Why is there an inevitable increase in entropy when the laws of physics are invariant under time reversal? A instance reversal symmetry of physical laws appears to allow a 2nd law of thermodynamics to become broken. Mpemba paradox: Hot a river might under certain conditions freeze sooner than cold water, possibly though it must pass a frigidness en route to freezing. Olbers' paradox Why is the night sky black whenever there exists an eternity of stars? Twin paradox: When a travel twin is restored, he is immature & older than his brother world health organization stick around. Black hole information paradox Supplee's paradox: the buoyancy of a relativistic object (such as a bullet) appears to change when the reference frame is changed from one in which the bullet is at rest to one in which the fluid is at rest

Philosophy

Fermi paradox: If there come numbers of more animate coinage in the Universe, so in which are it? Shouldn't their presence exist as conspicuous? Grandfather paradox: You travel back in time & kill the granddaddy prior to he meets your systems computers granny which precludes your have conception and, so, wise shoppers couldn't last back eventually & wipe out your computers granddaddy. Mere addition paradox: Is the big people dwelling barely passable resides better than a little happy people? Nihilist paradox: If truth doesn't survive, a statement "truth does not exist" is a truth, thereby proving itselfWrong. Omnipotence paradox: Can an omnipotent being create the rock as well heavily to lift? Could an resistless click move an immovable object? Predestination paradox: A human travels back eventually & impregnates his dandy-neat-grandmother. A effect occurs as line of offspring & descendent, including a human's parent(s) & a human himself. So, unless he makes a instance-travel hike the least bit, he may never survive. Epicurean paradox, or even Problem of evil: a being of evil is incompatible by using the being of an all-powerful & positive God. Moore's paradox: "It's raining but I don't believe that it is." Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on..."

Economics

Abilene paradox: A class action of population typically has to decide against its have interests. Arrow's paradox/Voting paradox/Condorcet paradox: You can't have all the attributes of an ideal voting system at once. Bertrand paradox: Two players reaching the state of Nash equilibrium both find themselves using there is no successfully. Bird in the bush paradox: Why come population therefore chance-averse? Diamond-water paradox Why is a lake inexpensive than diamonds, after human being want water to live, non diamonds? Edgeworth paradox: With capacity constraints, there might not become an equilibrium. Ellsberg paradox: A self-contradictory symptom inside experimental guide theory. Gibson's paradox: Why were interest rates and prices correlated? Giffen paradox: Can increasing a price of bread produce poor eat additional of it? Jevons paradox: Increases in efficiency lead to potentially big increases coveted. St. Petersburg paradox: People will only offer a modest fee for a reward of infinite value. Easterlin Paradox: Wealthier nations don't necessarily own higher proportions of "happy" population (equally revealed by survey information).