This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. These paradoxes may be due to fallacious reasoning (falsidical), or an unintuitive solution (veridical). 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.
Logic
: 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. : 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. : 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. : In any pub, there is a customer such that if that customer is drinking, everybody in the pub is drinking. : Disjunction introduction poses a problem for modal inferences, permitting arbitrary modal statements to be inferred. : Inconsistent premises always make an argument valid. : 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. : (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black. : Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred. : If the temperature is 90 and the temperature is rising, that would seem to entail that 90 is rising. : The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The surprise examination and Bottle Imp paradox use similar logic.
Self-reference
These paradoxes, insolubilia (insolubles), 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.
: A male barber shaves all and only those men who do not shave themselves. Does he shave himself? (Russell's popularization of his set theoretic paradox.) Not to be confused with the Barbershop paradox. : The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable. : The phrase "the first number not nameable in under ten words" appears to name it in nine words. : 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? : 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. : "If this sentence is true, then Germany borders China." : A Cretan says: "All Cretans are liars". This paradox works in mainly the same way as the liar paradox. : Is the word "heterological", meaning "not applicable to itself", a heterological word? (A close relative of Russell's paradox.) : If there was a name for a natural number that is identical to a name of the successor of that number, there would be a natural number equal to its successor. : Purportedly said by Socrates. : By formulating an equivalent to Richard's paradox, untyped lambda calculus is shown to be inconsistent. : "This sentence is not known." : "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question 'no'?", and "I'm lying." : "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. : Two sentences that each say the other is not true. : What would happen if Pinocchio said "My nose grows now"?, an image of Pinocchio with a speech bubble "My nose will grow now!" has become a minor Internet phenomenon (Google search, Google image search). : "'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. : 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. : "It is opposite day today." Therefore, it is not opposite day, but if one says 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. : We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory. : Does the set of all those sets that do not contain themselves contain itself?
Vagueness
: It seems as if 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. (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?
Mathematics
: A fallacious argument by induction that appears to prove that all horses are the same color. : An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl. : The number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve. : Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement. : The first number that can be considered "dull" rather than "interesting" becomes interesting because of that fact. : If potatoes consisting of 99% water dry until they are 98% water, they lose 50% of their weight. : Does the set of all those sets that do not contain themselves contain itself?
Statistics
: Effect size may not be indicative of practical meaning. : Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy. : A complicating factor arising in statistical tests of proportions. : Describes a problem in model selection where predictor variables with no explanatory power can appear artificially important. : For almost everyone, their friends have more friends than they do. : (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. : 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 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. , or the : A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data. : 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.
Probability
: A paradox of conditional probability closely related to the Boy or Girl paradox. : Different common-sense definitions of randomness give quite different results. : In a random group of only 23 people, there is a better than 50/50 chance two of them have the same birthday. : Conditional probability density functions are not invariant under coordinate transformations. : A two-child family has at least one boy. What is the probability that it has a girl? : A test that is accurate the vast majority of the time could show that someone has a disease, but the probability that they actually have it could still be tiny. : Shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination. : One can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A. , also known as the Monty Hall paradox: An unintuitive consequence of conditional probability. : A wager between two people seems to favour them both. Very similar in essence to the two-envelope paradox. : The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin. : A probability problem that can be correctly answered as one half or one third depending on how the question is approached. , also known as the Three Prisoners paradox: A variation of the Monty Hall problem. : Two indistinguishable envelopes each contains a positive sum of money. One envelope contains twice as much as the other. One may pick one envelope and keep whatever amount it contains. After they picked an envelope at random but before they open it, they are given the chance to take the other envelope instead.
Infinity and infinitesimals
: If the ordinal numbers formed a set, it would be an ordinal number that is smaller than itself. : 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.
: Though most numbers are not squares, there are no more numbers than squares. (See also Cantor's diagonal argument) : If a hotel with infinitely many rooms is full, it can still take in more guests. : Countably infinite models of set theory contain sets that are uncountable in the sense of the model. : "One will never reach point B from point A as they must always get half-way there, and half of the half, and half of that half, and so on..." This is also a physical paradox. Supertasks may result in paradoxes such as : An infinite number of gods place barriers to stop a man advancing, but there can be no individual god responsible for preventing him. : An infinite number of assassins can create an explicit self-contradiction by scheduling their assassinations at certain times. : The sum of 1−1+1−1+1−1... can be either one, zero, or one-half. : After alternately adding and removing balls to a vase infinitely often, how many balls remain? : After flicking a lamp on and off infinitely often, is it on or off?