A paradox is a statement or sentiment that is seemingly contradictory or opposed to common sense and yet is perhaps true in fact. Conversely, a paradox may be a statement that is actually self-contradictory (and therefore false) even though it appears true. Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together.
The recognition of ambiguities, equivocations and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics. But many paradoxes (e.g. Curry's Paradox) do not yet have universally accepted resolutions.
It can be argued that there are four classes of paradoxes:
Veridical Paradoxes: which produce a result that appears absurd but can be demonstrated to be nevertheless true.
Falsidical Paradoxes: which produce a result that not only appears false but actually is false.
Antinomies: which are neither veridical nor falsidical, but produce a self-contradictory result by properly applying accepted ways of reasoning.
Dialetheias: which produce a result which is both true and false at the same time and in the same sense.
Paradoxes often result from self-reference (where a sentence or formula refers to itself directly), infinity (an argument which generates an infinite regress, or infinite series of supporting references), circular definitions (in which a proposition to be proved is assumed implicitly or explicitly in one of the premises), vagueness (where there is no clear fact of the matter whether a concept applies or not), false or misleading statements (assertions that are either willfully or unknowingly untrue or misleading), and half-truths (deceptive statements that include some element of truth).
Some famous paradoxes include:
Epimenides' Liar Paradox: Epimenides was a Cretan who said "All Cretans are liars." Should we believe him?
Liar Paradox (2): "This sentence is false."
Liar Paradox (3): "The next sentence is false. The previous sentence is true."
Curry's Paradox: "If this sentence is true, then Santa Claus exists."
Quine's Paradox: "yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
Russell's Barber Paradox: If a barber shaves all and only those men in the village who do not shave themselves, does he shave himself?
Grandfather Paradox: Suppose a time traveler goes back in time and kills his grandfather when the latter was only a child. If his grandfather dies in childhood, then the time traveler cannot be born. But if the time traveler is never born, how can he have traveled back in time in the first place?
Zeno's Dichotomy Paradox: Before a moving object can travel a certain distance (e.g. a person crossing a room), it must get halfway there. Before it can get halfway there, it must get a quarter of the way there. Before traveling a quarter, it must travel one-eighth; before an eighth, one-sixteenth; and so on. As this sequence goes on forever, an infinite number of points must be crossed, which is logically impossible in a finite period of time, so the distance will never be covered (the room crossed, etc).
Zeno's Paradox of Achilles and the Tortoise: If Achilles allows the tortoise a head start in a race, then by the time Achilles has arrived at the tortoise's starting point, the tortoise has already run on a shorter distance. By the time Achilles reaches that second point, the tortoise has moved on again, etc, etc. So Achilles can never catch the tortoise.
Zeno's Arrow Paradox: If an arrow is fired from a bow, then at any moment in time, the arrow either is where it is, or it is where it is not. If it moves where it is, then it must be standing still, and if it moves where it is not, then it can't be there. Thus, it cannot move at all.
Theseus' Ship Paradox: After Theseus died, his ship was put up for public display. Over time, all of the planks had rotted at one time or another, and had been replaced with new matching planks. If nothing remained of the actual "original" ship, was this still Theseus' ship?
Sorites (Heap of Sand) Paradox: If you take away one grain of sand from a heap, it is still a heap. If grains are individually removed, is it still a heap when only one grain remains? If not, when did it change from a heap to a non-heap?
Hempel's Raven Paradox: If all ravens are black, then in strict terms of logical equivalence, everything that is not black is not a raven. So every sighting of a blue sweater or a red cup confirms the hypothesis that all ravens are black.
Petronius' Paradox" "Moderation in all things, including moderation."
Paradoxical Notice: "Please ignore this notice."
Dull Numbers Paradox: If there is such a thing as a dull number, then we can divide all numbers into two sets - interesting and dull. In the set of dull numbers there will be only one number that is the smallest. Since it is the smallest dull number it becomes, ipso facto, an interesting number. We must therefore remove it from the dull set and place it in the other. But now there will be another smallest uninteresting number. Repeating this process will make any dull number interesting.
Protagoras' Pupil Paradox: A lawyer made an arrangement with one of his pupils whereby the pupil was to pay for his instruction after he had won his first case. After a while, the lawyer grew impatient with the pupil's lack of clients, and decided to sue him for the amount owed. The lawyer's logic was that if he, the lawyer, won, the pupil would pay him according to the judgment of the court; if the pupil won, then he would have to honor the agreement and pay anyway. The pupil, however, argued that if he, the pupil, won, then by the judgment of the court he need not pay the lawyer; and if the lawyer won, then the agreement did not come into force and the pupil need not pay the lawyer.
Moore's paradox: "It will rain but I don't believe that it will."
Schrödinger's Cat: There is a cat in a sealed box, and the cat's life or death is dependent on the state of a particular subatomic particle. According to quantum mechanics, the particle only has a definite state at the exact moment of quantum measurement, so that the cat remains both alive and dead until the moment the box is opened.
"Turtles all the way down": A story about an infinite regress, often attributed to Bertrand Russell but probably dating from centuries earlier, based on an old (possibly Indian) cosmological myth that the earth is a flat disk supported by a giant elephant that is in turn supported by a giant turtle. In the story, when asked what then supported the turtle, the response was "it's turtles all the way down".