This is a list of paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and have their own article.
Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. Informally, the term is often used to describe a counter-intuitive result.
Barbershop paradox The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. Not to be confused with the Barber paradox.
What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...", also known as Carroll's paradox, not to be confused with the physical paradox of the same name.
Catch-22 A situation in which someone is in need of something that can only be had by not being in need of it.
Drinker paradox In any pub there is a customer of whom it is true to say: if that customer drinks, everybody in the pub drinks.
Lottery paradox If there is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not reasonable to believe that no lottery ticket will win.
Raven paradox (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.
Ross's paradox Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred.
Berry paradox The phrase "the first number not nameable in under ten words" appears to name it in nine words.
Crocodile dilemma If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father correctly guesses that the child will not be returned?
Paradox of the Court A law student agrees to pay his teacher after winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
Exception paradox "If there is an exception to every rule, then every rule must have at least one exception; the exception to this one being that it has no exception." "There's always an exception to the rule, except to the exception of the rule—which is, in of itself, an accepted exception of the rule." "In a world with no rules, there should be at least one rule - a rule against rules."
Hypergame A finite game is a game which always ends after a finite number of moves. The hypergame is a game where the first move is that player 1 names a finite game, and then player 2 makes the first move. If the hypergame is a finite game, then when playing the hypergame player 1 can name the hypergame as the game to be played, and then player 2 can name the hypergame, etc., ad infinitum. But if the hypergame is not a finite game, then there is a series of moves that never ends.
Quine's paradox "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.
Yablo's paradox An ordered infinite sequence of sentences, each of which says that all following sentences are false. Uses neither self-reference nor circular reference.
Opposite Day "It is opposite day today." Therefore it is not opposite day, but if you say it is a normal day it would be considered a normal day.
Petronius' paradox "Moderation in all things, including moderation" (unsourced quotation sometimes attributed to Petronius).
Richard's paradox We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.
Russell's paradox Does the set of all those sets that do not contain themselves contain itself?
Bhartrhari's paradox Hans and Radhika Herzbergers (1981) argue that the Indian grammarian-philosopher Bhartrhari (late fifth century CE) held a thesis which the authors call the Unnameability Thesis (the thesis that there are some things which are unnameable), and that Bhartrhari was well aware that it leads to a problematic situation which they call 'Bhartrhari's paradox'.
Unnameability paradox "There are some things which are unnameable: but they become nameable by calling them unnameable." See: Bhartrhari's paradox.
Unsignifiability paradox "There are some things which are unsignifiable: but they become signifiable by calling them unsignifiable." See: Bhartrhari's paradox.
Ship of Theseus (a.k.a. George Washington's axe or Grandfather's old axe or Trigger's Broom in the UK): It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with.
Sorites paradox (also known as the paradox of the heap): If you remove a single grain of sand from a heap, you still have a heap. Keep removing single grains, and the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?
Lindley's paradox Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.
Low birth weight paradox Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a special case of Simpson's paradox.
Simpson's paradox, or the Yule–Simpson effect: A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.
Will Rogers phenomenon The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books.
Two-envelope paradox You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.
Coin rotation paradox a coin rotating along the edge of an identical coin will make a full revolution after traversing only half of the stationary coin's circumference.
Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong way" around the set to reach a nearby point). This can be represented by a Klein bottle.
Hausdorff paradox There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.
Missing square puzzle Two similar-looking figures appear to have different areas while built from the same pieces.
Nikodym set A set contained in and with the same Lebesgue measure as the unit square, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point.
Abilene paradox People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do.
Apportionment paradox Some systems of apportioning representation can have unintuitive results due to rounding
Alabama paradox Increasing the total number of seats might shrink one block's seats.
New states paradox Adding a new state or voting block might increase the number of votes of another.
Cool tropics paradox A contradiction between modelled estimates of tropical temperatures during warm, ice-free periods of the Cretaceous and Eocene, and the lower temperatures that proxies suggest were present.
Holographic principle The amount of information that can be stored in a given volume is not proportional to the volume but to the area that bounds that volume.
Algol paradox In some binaries the partners seem to have different ages, even though they are thought to have formed at the same time.
Faint young Sun paradox The contradiction between existence of liquid water early in the Earth's history and the expectation that the output of the young Sun would have been insufficient to melt ice on Earth.
Bentley's paradox In a Newtonian universe, gravitation should pull all matter into a single point.
Boltzmann brain If the universe we observe resulted from a random thermodynamic fluctuation, it would be vastly more likely to be a simple one than the complex one we observe. The simplest case would be just a brain floating in vacuum, having the thoughts and sensations you have.
Fermi paradox If there are, as various arguments suggest, many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?
Heat death paradox If the universe was infinitely old, it would be in thermodynamical equilibrium, which contradicts what we observe.
Olbers' paradox Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?
Extinction paradox In the small wavelength limit, the total scattering cross section of an impenetrable sphere is twice its geometrical cross-sectional area (which is the value obtained in classical mechanics).
Hardy's paradox How can we make inferences about past events that we haven't observed while at the same time acknowledge that the act of observing it affects the reality we are inferring to?
Klein paradox When the potential of a potential barrier becomes similar to the mass of the impinging particle, it becomes transparent.
Mott problem Spherically symmetric wave functions, when observed, produce linear particle tracks.
French paradox The observation that the French suffer a relatively low incidence of coronary heart disease, despite having a diet relatively rich in saturated fats, which are assumed to be the leading dietary cause of such disease.
Glucose paradox The large amount of glycogen in the liver cannot be explained by its small glucose absorption.
Hispanic paradox The finding that Hispanics in the U.S. tend to have substantially better health than the average population in spite of what their aggregate socio-economic indicators predict.
Israeli paradox The observation that Israelis suffer a relatively high incidence of coronary heart disease, despite having a diet very low in saturated fats, which are assumed to be the leading dietary cause of such disease.
Meditation paradox The amplitude of heart rate oscillations during meditation was significantly greater than in the pre-meditation control state and also in three non-meditation control groups
Mexican paradox Mexican children tend to have higher birth weights than can be expected from their socio-economic status.
Obesity survival paradox Although the negative health consequences of obesity in the general population are well supported by the available evidence, health outcomes in certain subgroups seem to be improved at an increased BMI.
Peto's paradox Humans get cancer with high frequency, while larger mammals, like whales, do not. If cancer is essentially a negative outcome lottery at the cell level, and larger organisms have more cells, and thus more potentially cancerous cell divisions, one would expect larger organisms to be more predisposed to cancer.
Pulsus paradoxus A pulsus paradoxus is a paradoxical decrease in systolic blood pressure during inspiration. It can indicate certain medical conditions in which there is reduced venous return of blood to the heart, such as cardiac tamponade or constrictive pericarditis. Also known as the Pulse Paradox.
Bootstrap paradox Can a time traveler send himself information with no outside source?
Polchinski's paradox A billiard ball can be thrown into a wormhole in such a way that it would emerge in the past and knock its incoming past self away from the wormhole entrance, creating a variant of the grandfather paradox.
Predestination paradox A man travels back in time to discover the cause of a famous fire. While in the building where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would inspire him, years later, to travel back in time. The bootstrap paradox is closely tied to this, in which, as a result of time travel, information or objects appear to have no beginning.
Grandfather paradox You travel back in time and kill your grandfather before he conceives one of your parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your grandfather.
Hitler's murder paradox You travel back in time and kill a famous person in history before they become famous; but if the person had never been famous then he could not have been targeted as a famous person.
Tzimtzum In Kabbalah, how to reconcile self-awareness of finite Creation with Infinite Divine source, as an emanated causal chain would seemingly nullify existence. Luria's initial withdrawal of God in Hasidic panentheism involves simultaneous illusionism of Creation (Upper Unity) and self-aware existence (Lower Unity), God encompassing logical opposites.
Lucas paradox Capital is not flowing from developed countries to developing countries despite the fact that developing countries have lower levels of capital per worker, and therefore higher returns to capital.
Mayfield's paradox Keeping everyone out of an information system is impossible, but so is getting everybody in.
Metzler paradox The imposition of a tariff on imports may reduce the relative internal price of that good.
Paradox of prosperity Why do generations that significantly improve the economic climate seem to generally rear a successor generation that consumes rather than produces?
Paradox of competition A class of paradoxes where - under the condition of a competitive situation - individual measures to do good or gain advantage, in the end lead to nullification of advantage or even worsening for the totality of economic actors as well as for the individual (Circuit paradoxes, Classical paradoxes, Marx paradoxes).
Paradox of thrift If everyone saves more money during times of recession, then aggregate demand will fall and will in turn lower total savings in the population.
Paradox of toil If everyone tries to work during times of recession, lower wages will reduce prices, leading to more deflationary expectations, leading to further thrift, reducing demand and thereby reducing employment.
Paradox of value, also known as diamond-water paradox: Water is more useful than diamonds, yet is a lot cheaper.
Productive failure Providing less guidance and structure and thereby causing more failure is likely to promote better learning.
Productivity paradox (also known as Solow computer paradox): Worker productivity may go down, despite technological improvements.
Paradox of Plenty The Paradox of Plenty (resource curse) refers to the paradox that countries and regions with an abundance of natural resources, specifically point-source non-renewable resources like minerals and fuels, tend to have less economic growth and worse development outcomes than countries with fewer natural resources.
Tullock paradox Bribing politicians costs less than one would expect, considering how much profit it can yield.
Stability–instability paradox When two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases.
Gender paradox Women conform more closely than men to sociolinguistics norms that are overtly prescribed, but conform less than men when they are not.
Moral paradox A situation in which moral imperatives clash without clear resolution.
Outcomes paradox Schizophrenia patients in developing countries seem to fare better than their Western counterparts.
Status paradox Several paradoxes involve the concept of medical or social status.
The Paradox of Anti-Semitism A book arguing that the lack of external persecutions and antagonisms results in the dissolution of Jewish identity, a theory that resonates in works of Dershowitz and Sartre.
Region-beta paradox People can sometimes recover more quickly from more intense emotions or pain than from less distressing experiences.
Self-absorption paradox The contradictory association whereby higher levels of self-awareness are simultaneously associated with higher levels of psychological distress and with psychological well-being.
Stockdale paradox "You must never confuse faith that you will prevail in the end—which you can never afford to lose—with the discipline to confront the most brutal facts of your current reality, whatever they might be."
Ironic process theory Ironic processing is the psychological process whereby an individual's deliberate attempts to suppress or avoid certain thoughts (thought suppression) renders those thoughts more persistent.
Ant on a rubber rope An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.
Bonini's paradox Models or simulations that explain the workings of complex systems are seemingly impossible to construct. As a model of a complex system becomes more complete, it becomes less understandable, for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents.
^Herzberger, Hans and Radhika Herzberger (1981). "Bhartrhari's Paradox" Journal of Indian Philosophy 9: 1-17 (slightly revised version of "Bhartrhari's Paradox" in Studies in Indian Philosophy. A memorial volume in honour of pandit Sukhlalji Sanghvi. (L.D. Series 84.) Gen. ed. Dalsukh Malvania et al. Amedabad, 1981).
^Newton, Roger G. (2002). Scattering Theory of Waves and Particles, second edition. Dover Publications. p. 68. ISBN0-486-42535-5.
^Trapnell, P. D., & Campbell, J. D. (1999). "Private self-consciousness and the Five-Factor Model of Personality: Distinguishing rumination from reflection". Journal of Personality and Social Psychology, 76, 284-304.