Self-reference,
contradiction and
infinite regress are core elements of many paradoxes. Other common elements include
circular definitions, and confusion or equivocation between different levels of
abstraction.
Self-reference Self-reference occurs when a
sentence, idea or
formula refers to itself. Although statements can be self referential without being paradoxical ("This statement is written in English" is a true and non-paradoxical self-referential statement), self-reference is a common element of paradoxes. One example occurs in the
liar paradox, which is commonly formulated as the self-referential statement "This statement is false". Another example occurs in the
barber paradox, which poses the question of whether a
barber who shaves all and only those who do not shave themselves will shave himself. In this paradox, the barber is a self-referential concept.
Contradiction Contradiction, along with self-reference, is a core feature of many paradoxes. The barber paradox is contradictory because it implies that the barber shaves himself if and only if the barber does not shave himself. As with self-reference, a statement can contain a contradiction without being a paradox. "This statement is written in French" is an example of a contradictory self-referential statement that is not a paradox and is instead false. The barber paradox also exemplifies vicious circularity: The barber shaves those who do not shave themselves, so if the barber does not shave himself, then he shaves himself, then he does not shave himself, and so on.
Other elements Other paradoxes involve false statements and
half-truths or rely on hasty assumptions (A father and his son are in a car crash; the father is killed and the boy is rushed to the hospital. The doctor says, "I can't operate on this boy. He's my son." There is no contradiction, the doctor is the boy's mother.). Paradoxes that 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 that cannot be consistently interpreted as either true or false, because if it is known to be false, then it can be inferred that it must be true, and if it is known to be true, then it can be inferred that it must be false.
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 traveler were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. This is a specific instance of the
butterfly effectin that any interaction a time traveler has with the past would alter conditions such that divergent events "propagate" through the world over time, ultimately altering the circumstances in which the time travel initially takes place. Often a seemingly paradoxical conclusion arises from an inconsistent or inherently contradictory definition of the initial premise. In the case of that apparent paradox of a time traveler killing his own grandfather, it is the inconsistency of defining the past to which he returns as being somehow different from the one that leads up to the future from which he begins his trip, but also insisting that he must have come to that past from the same future as the one that it leads up to. ==Quine's classification==