List of statements that appear to contradict themselves
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. Although considered paradoxes, some of these are simply based on fallacious reasoning (falsidical), or an unintuitive solution (veridical). Informally, the term paradox is often used to describe a counter-intuitive result.
However, some of these paradoxes qualify to fit into the mainstream viewpoint of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called antinomy, point out genuine problems in our understanding of the ideas of truth and description.
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: If a presumption needs to be made that a specific result can be deduced from premises, then the result can never be deduced. An inference rule, which is valid (or not), cannot be a premise, which is true (or false), otherwise one has an infinite regress. Also known as Carroll's paradox and is not to be confused with the "Achilles and the tortoise" paradox by Zeno of Elea.
Catch-22: A situation in which someone is in need of something that can only be had by not being in need of it. A soldier who wants to be declared insane to avoid combat is deemed not insane for that very reason and will therefore not be declared insane.
Drinker paradox: In any pub there is a customer such that if that customer is drinking, everybody in the pub is drinking.
Paradox of free choice: Disjunction introduction poses a problem for modal inferences, permitting arbitrary modal statements to be inferred.
Lottery paradox: If there is one winning ticket in a large lottery, it is reasonable to believe of any particular lottery ticket that it is not the winning ticket, 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' paradox: Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred.
Temperature paradox: If the temperature is 90 and the temperature is rising, that would seem to entail that 90 is rising.
These paradoxes have in common a contradiction arising from either self-reference or circular reference, in which several statements refer to each other in a way that following some of the references leads back to the starting point.
Bhartrhari's paradox: The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.
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 guesses that the child will not be returned?
Paradox of the Court: A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment.
Liar paradox: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question 'no'?", and "I'm lying."
Card paradox: "The next statement is true. The previous statement is false." A variant of the liar paradox in which neither of the sentences employs (direct) self-reference, instead this is a case of circular reference.
No-no paradox: Two sentences that each say the other is not true.
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. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not.
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, which contradicts the fact that it has previously been stated that it is an opposite day.
Ship of Theseus: It seems like one can replace any component of a ship, and it is still the same ship. So they can replace them all, one at a time, and it is still the same ship. However, they can then take all the original pieces, and assemble them into a ship. That, too, is the same ship they began with.
Sorites paradox (also known as the paradox of the heap): If one removes a single grain of sand from a heap, they still have a heap. If they keep removing single grains, the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?
Inspection paradox: (Bus waiting time paradox) For a given random distribution of bus arrivals, the average rider at a bus stop observes more delays than the average operator of the buses.
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.
Cantor's paradox: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor's theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.
Benardete's paradox: Apparently, a man can be "forced to stay where he is by the mere unfulfilled intentions of the gods".
Grim Reaper paradox: An infinite number of assassins can create an explicit self-contradiction by scheduling their assassinations at certain times.
Grandi's series: The sum of 1−1+1−1+1−1... can be either one, zero, or one-half.
Ross–Littlewood paradox: After alternately adding and removing balls to a vase infinitely often, how many balls remain?
Thomson's lamp: After flicking a lamp on and off infinitely often, is it on or off?
Geometry and topology
Banach–Tarski paradox: A ball can be cut into a finite number of pieces and re-assembling the pieces will get two balls, each of equal size to the first. The von Neumann paradox is a two-dimensional version.
Paradoxical set: A set that can be partitioned into two sets, each of which is equivalent to the original.
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).
Hausdorff paradox: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.
Hooper's paradox: An image with many pieces whose size is 32 m², but drops down to 30 m² when its pieces are rearranged
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.
Paradox of place: If everything that exists has a place, that place must have a place, and so on ad infinitum.
Paradox of the grain of millet: When a grain of millet falls it makes no sound, but when a thousand grains fall they do, thus many of nothing become something.
The moving rows: Suppose two rows are moving past a stationary row in opposite directions. If a member of a moving row moves past a member of the stationary row in an indivisible instant of time, they move past two members of the row that is moving in the other direction in this instant of time.
Algol paradox: In some binary star systems 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.
Olbers' paradox: Why is the night sky dark if there is an infinity of stars, covering every part of the celestial sphere?
Achilles and the tortoise: If the tortoise is ahead of Achilles, by the time Achilles reaches the tortoise's current position, the tortoise will have moved a bit further ahead, which goes on indefinitely.
Archer's paradox: An archer must, in order to hit his target, not aim directly at it, but slightly to the side. Not to be confused with the arrow paradox.
Arrow paradox : If we divide time into discrete 0-duration slices, no motion is happening in each of them, so taking them all as a whole, motion is impossible.
Knudsen paradox: Based on the Navier–Stokes equations, one would expect the mass flux in a channel to decrease with increasing Knudsen number, but there is a distinct minimum around Knudsen number 0.8.
Dichotomy paradox: To reach its target, an airborne arrow must first reach an infinite number of midpoints between its current position and the target.
Elevator paradox: Even though hydrometers are used to measure fluid density, a hydrometer will not indicate changes of fluid density caused by changing atmospheric pressure.
Feynman sprinkler: Which way does a sprinkler rotate when submerged in a tank and made to suck in the surrounding fluid?
Norton's dome: Are there non-deterministic systems in Newtonian mechanics?
Painlevé paradox: Rigid-body dynamics with contact and friction is inconsistent.
Tea leaf paradox: When a cup of tea is stirred, the leaves assemble in the center, even though centrifugal force pushes them outward.
Upstream contamination: When a fluid is poured from a higher container onto a lower one, particles can climb up the falling water.
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 an ostensible observer has.
Fermi paradox: If there are, as various arguments suggest, many other sentient species in the universe, then where are they? Should their presence not be obvious?
Heat death paradox: If the universe were infinitely old, it would be in thermodynamic 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?
Faraday paradox: An apparent violation of Faraday's law of electromagnetic induction.
Aharonov–Bohm effect: A charged particle is affected by an electromagnetic field even though it has no local contact with that field.
Bell's theorem: Why do measured quantum particles not satisfy mathematical probability theory?
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.
Schrödinger's cat paradox: According to the Copenhagen interpretation of quantum mechanics, a cat could be simultaneously alive and dead, as long as it remains unobserved.
Uncertainty principle: There is a fundamental limit to the precision with which certain pairs of physical properties of a particle, known as complementary variables, such as position and momentum can be known. This is often confused with a similar effect in physics called the observer effect.
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 United States 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.
Mexican paradox: Mexican children tend to have higher birth weights than can be expected from their socio-economic status.
Obesity paradox: In some medical conditions, obesity is associated with increased survival, although there is a strong association with shortened lifespan in the general population.
Peto's paradox: Humans and other small-to-medium-sized mammals 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.
Second wind: The "second wind" is a sudden period of increased wakefulness in individuals deprived of sleep that tends to coincide with the individual's circadian rhythm. Although the individual is more wakeful and aware of their surroundings, they are continuing to accrue sleep debt and thus, are actually exacerbating their sleep deprivation.
Levinthal paradox: The length of time that it takes for a protein chain to find its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations.
SAR paradox: Exceptions to the principle that a small change in a molecule causes a small change in its chemical behavior are frequently profound.
Bootstrap paradox (also ontological paradox): You send information/an object to your past self, but you only have that information/object because in the past, you received it from your future self. This means the information/object was never created, yet still exists.
Predestination paradox: Someone travels back in time to discover the cause of a famous fire. While in the building where the fire started, they accidentally knock over a kerosene lantern and causes a fire, the same fire that would inspire them, 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: If one travels back in time and kills their grandfather before he conceives one of their parents, which precludes their own conception and, therefore, they couldn't go back in time and kill their grandfather.
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.
Buridan's bridge: Plato says: "If your next statement is true, I will allow you to cross, but if it is false, I will throw you in the water." Socrates responds: "You will throw me in the water." Whatever Plato does, he will seemingly break his promise. Similar to the crocodile dilemma.
Paradox of fiction: How can people experience strong emotions from purely fictional things?
Fitch's paradox: If all truths are knowable, then all truths must in fact be known.
Paradox of free will: If God knows in advance what a person will decide, how can there be free will?
Goodman's paradox: Why can induction be used to confirm that things are "green", but not to confirm that things are "grue"?
Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something else, one achieves happiness.
Polanyi's paradox: "We know more than we can tell", Polanyi's paradox brings to attention the cognitive phenomenon that there exist tasks which human beings understand intuitively how to perform but cannot verbalise the rules behind.
Preface paradox: The author of a book may be justified in believing that all their statements in the book are correct, at the same time believing that at least one of them is incorrect.
Problem of evil: (Epicurean paradox) The existence of evil seems to be incompatible with the existence of an omnipotent, omniscient, and morally perfect God.
Rule-following paradox: Even though rules are intended to determine actions, "no course of action could be determined by a rule, because any course of action can be made out to accord with the rule".
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..." (This is also a paradox of the infinite)
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 Hasidicpanentheism involves simultaneous illusionism of Creation (Upper Unity) and self-aware existence (Lower Unity), God encompassing logical opposites.
One class of paradoxes in economics are the paradoxes of competition, in which behavior that benefits a lone actor would leave everyone worse off if everyone did the same. These paradoxes are classified into circuit, classical and Marx paradoxes.
Allais paradox: A change in a possible outcome that is shared by different alternatives affects people's choices among those alternatives, in contradiction with expected utility theory.
The Antitrust Paradox: A book arguing that antitrust enforcement artificially raised prices by protecting inefficient competitors from competition.
Lerner paradox: The imposition of a tariff on imports may raise the relative world price of that good.
Louboutin paradox: Paradox of luxury goods. The more expensive some commodity is, the less it is used after acquiring.
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 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.
Productivity paradox: (also known as Solow computer paradox): Worker productivity may go down, despite technological improvements.
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.
Wollheim's paradox: A person can simultaneously advocate two conflicting policy options, A and B, provided that the person believes that democratic decisions should be followed.
Psychology and sociology
Gender paradox: Women conform more closely than men to sociolinguistics norms that are overtly prescribed, but conform less than men when they are not.
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.
Meat paradox: People care about animals, but embrace diets that involve harming them.
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.
Paradox of suspense: Sometimes, retelling of familiar stories appears to still induce suspense, despite the fact that the audience already knows how the story will unfold.
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.
Status paradox: Several paradoxes involve the concept of medical or social status.
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."
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.
The Paradox of Choice: A book arguing that eliminating consumer choices can greatly reduce anxiety for shoppers.
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.
^Eldridge-Smith, Peter; Eldridge-Smith, Veronique (13 January 2010). "The Pinocchio paradox". Analysis. 70 (2): 212–215. doi:10.1093/analys/anp173. ISSN1467-8284. As of 2010[update], an image of Pinocchio with a speech bubble "My nose will grow now!" has become a minor Internet phenomenon (Google search, Google image search). It seems likely that this paradox has been independently conceived multiple times.
^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.