An example is "This statement is false", a form of the liar paradox. The statement is referring to itself. Another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be "Is the answer to this question 'No'?" In this case, replying "No" would be stating that the answer is not "No". If the reply is "Yes", it would be stating that it is "No", as the reply was "Yes". But because the question was answered with a "Yes", the answer is not "No". A negative response without saying the word "No", such as "It isn't", would, however, leave the question answered without bringing about a paradox. Another example is the affirmation 'Nothing is Impossible', meaning that it is impossible for something to be impossible, thus contradicting itself.
"This statement is false"; the statement cannot be false and true at the same time.
Vicious circularity, or infinite regress
"This statement is false"; if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the following group of statements:
"The following sentence is true."
"The previous sentence is false."
"What happens when Pinocchio says, 'My nose will grow now'?"
Other paradoxes involve false statements or half-truths and the resulting biased assumptions. This form is common in howlers.
For example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son."
The apparent paradox is caused by a hasty generalization, for if the surgeon is the boy's father, the statement cannot be true. The paradox is resolved if it is revealed that the surgeon is a woman — the boy's mother.
Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, and require extending the context or language in order to lose their paradoxical quality. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. "This sentence is false" is an example of the well-known liar paradox: it is a sentence which cannot be consistently interpreted as either true or false, because if it is known to be false, then it is known that it must be true, and if it is known to be true, then it is known that it must be false. Therefore, it can be concluded that it is unknowable. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory.
Thought experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a time traveller were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. This is a specific example of the more general observation that a time-traveller's interaction with the past — however slight — would entail making changes that would, in turn, change the future in which the time-travel was yet to occur, and would thus change the circumstances of the time-travel itself.
Quine's classification of paradoxes
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 had been born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. The Monty Hall paradox demonstrates that a decision which has an intuitive…