A paradox was originally something that was contrary to received or common opinion.
The term paradox comes from the Greek para ("contrary to") and doxa ("opinion"). From that, the term came to be used for something that was contrary to, or contradicted, common sense. Today the term is usually used for something that is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition.
A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition
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. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a twenty-one-year-old would have had only five birthdays, if he was born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people.
- A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the demonstration. The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples, generally relying on a hidden division by zero. Another example is the inductive form of the Horse paradox, falsely generalizes from true specific statements.
- A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling–Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.
A fourth kind has sometimes been described since Quine's work.
- A paradox which is both true and false at the same time in the same sense is called a dialetheism. In Western logics it is often assumed, following Aristotle, that no dialetheia exist.
