A paradox is a proposition which is, or seems, absurd or impossible but which nonetheless seems to be, or is, true because it is the conclusion of what seems to be, or is, a sound argument.
Thus paradox always involves a conflict between two propositions, one of them a ‘common sense’ proposition, a proposition which we all intuitively ‘know’ to be true, and the other, the paradoxical proposition, a proposition which is supported by ‘theoretical’ considerations, e.g., is the conclusion of a philosophical argument. The word paradox comes from the Greek παράδοξος (paradoxos) ‘contrary to expectation’, an adjective formed from the prepositional phrase παρὰ δόξαν (para doxan), ‘contrary to expectation, opinion, or belief’; and this etymology is reflected in our use of the word: confronted with a paradox, we are surprised because what we have always unthinkingly assumed to be true is brought into question.
A clear example of a paradox is that of Achilles and the Tortoise, one of several paradoxes devised by the Greek philosopher Zeno of Elea (c490-c430 BCE). The purpose of the paradox is to show that ‘the slowest runner will never be caught by the fastest runner’ (Aristotle, Physics Z 9, 239b14). Zeno holds that if, in a race between Achilles and a tortoise the latter is given a start, it is impossible for Achilles to draw level with it, let alone overtake it and win the race. His argument is that if Achilles is to draw level with the tortoise, he must first reach the point at which the tortoise started, but in the time it takes him to do that, the tortoise will have moved on a little, and so he will then have to reach that point, but in the time he needs to reach that point the tortoise will again have moved on, if only a little, and in the time it takes him to reach that point, the tortoise will again have moved on, ,,,, and so on ad infinitum. If Achilles is to draw level with the tortoise he must complete an infinite series of tasks, which is impossible. He may draw closer and closer to the tortoise but can never reach it, let alone overtake it (Aristotle, Physics Z 9, 239b14-29).
Clearly our response to this paradox will be to insist that our ‘commonsense’ belief that Achilles can draw level with the tortoise and win the race is true, and that Zeno’s argument which claims to prove otherwise must be flawed, difficult though it may be to identify the flaw. (Incidentally, philosophers still dispute what the flaw is.) Our response to other paradoxes, however, may be different. Consider, for example, the Liar Paradox.
There are several different forms of the Liar Paradox, which was devised by the Greek philosopher Eubulides (4th century BCE). The paradox challenges our ‘commonsense’ assumptions that every statement must be either true or false and that no statement can be both true and false. In its simplest form it asks us to consider the statement ‘What I am saying now is false’ (let us call this statement S). If S is true, then ‘What I am saying now is false" is true. So S must be false. In other words the assumption that S is true leads to the contradictory conclusion that S is false. On the other hand if S is false, then "What I am saying now is false" is false. Therefore, S must be true. In other words the assumption that S is false also leads to a contradictory conclusion. Either way, S is, it seems, both true and false, which is a paradox.
A possible response to this paradox is to deny our ‘commonsense’ assumption that every statement must be either true or false: perhaps some statements, e.g., certain statements which refer to themselves, lack a truth value, i.e., are neither true nor false. This would deprive the argument which generates the paradox of one of its basic assumptions.
Other paradoxes call for yet another type of response. Consider ‘The Child is father of the Man’ (a line from the poem My Heart Leaps Up, by William Wordsworth (1770-1850)). Since children cannot be fathers, let alone fathers of adults, the line states, or appears to state, a paradox. However, Wordsworth clearly intends ‘father’ to be understood not literally, but metaphorically: what he means is that our experiences as children form our characters as adults. Hence the paradox can be removed by paraphrase or 'translation'. Indeed it may be argued that while Wordsworth’s line is undeniably paradoxical, it does not actually state a paradox, but merely appears to state one.
- Historical note: In earlier centuries the word paradox was often used of an idea that was (merely) hard to believe or went against the orthodox view. In 1616, e.g., a paradox was defined as: "an opinion maintained contrary to the common allowed opinion, as if one [were to] affirm that the earth doth move round, and the heavens stand still." (William Bullokar, cited in OED; spelling modernised for AWE.)
- On the difference between paradox and oxymoron see oxymoron.