Paradox - a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time.
Some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. For example, inn George Orwell's “Animal Farm”, the words "All animals are equal, but some are more equal than others" are part of the cardinal rules. Clearly this statement does not make logical sense. However, the point of a paradox is to point out a truth, even if the statements contradict each other.
Some paradoxes reveal errors in definitions that are assumed to be rigorous, and have caused axioms of both mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed.
W. V. Quine (1962) distinguished between three classes of paradoxes:
A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless.
A falsidical paradox establishes a result that not only appears false but actually is false, due to a fallacy in the demonstration.
A paradox that is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning.
Temporal Paradox - an apparent or logical contradiction that is associated with the idea of time and time travel. In physics, temporal paradoxes fall into two broad groups: consistency paradoxes exemplified by the grandfather paradox; and causal loops. More broadly, a variation of the Fermi paradox also applies to time travel.
A causal loop is a paradox of time travel that occurs when a future event is the cause of a past event, which in turn is the cause of the future event. Both events then exist in spacetime, but their origin cannot be determined. A causal loop may involve an event, a person or object, or information.
The consistency paradox or grandfather paradox occurs when the past is changed in any way, thus creating a contradiction. A time traveler can do anything that did happen, but can't do anything that didn't happen. Doing something that didn't happen results in a contradiction. Consistency paradoxes occur whenever changing the past is possible.'
The Fermi paradox can be adapted for time travel, and phrased "if time travel were possible, where are all the visitors from the future?" Answers vary, from time travel not being possible, to the possibility that visitors from the future can not reach any arbitrary point in the past, or that they disguise themselves to avoid detection.
Here are a few classic Paradoxes that demonstrate the use of language and mathematics:
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 in order to avoid combat is deemed not insane for that very reason, and will therefore not be declared insane.
Barber paradox: A barber (who is a man) shaves all and only those men who do not shave themselves. Does he shave himself?
Liar's paradox: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question 'no'?", and "I'm lying."
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 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.
Buridan's bridge: If Plato says "If you make a false statement, I will throw you in the water", and Socrates responds, "You will throw me in the water", there is no way for Plato to keep his promise.
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)