A paradox is a self-contradictory statement or a statement that runs contrary to one's expectation.["By “paradox” one usually means a statement claiming something that goes beyond (or even against) ‘common opinion’ (what is usually believed or held)." ] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.
In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking,
Examples outside logic include the ship of Theseus from philosophy, a paradox that questions whether a ship repaired over time by replacing each and all of its wooden parts one at a time would remain the same ship.
Informally, the term paradox is often used to describe a counterintuitive result.
Common elements
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".
[ Extract of page 32] 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 liar paradox, "This statement is false," exhibits contradiction because the statement cannot be false and true at the same time.
[ Extract of page 376] 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.
Vicious circularity, or infinite regress
Another core aspect of paradoxes is non-terminating
recursion, in the form of circular reasoning or
infinite regress.
When this recursion creates a metaphysical impossibility through contradiction, the regress or circularity is vicious. Again, the liar paradox is an instructive example: "This statement is false"—if the statement is true, then the statement is false, thereby making the statement true, thereby making the statement false, and so on.
[ Extract of page 268]
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
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 and . "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 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 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
W. V. O. Quine (1962) distinguished between three classes of paradoxes:
Veridical paradox
A
veridical paradox produces a result that appears counter to
intuition, but is demonstrated to be true nonetheless:
-
That the Earth is an Spherical Earth that is heliocentrism, rather than the apparently obvious and common-sensical appearance of the Earth as a stationary flat Earth illuminated by a Sun that geocentrism.
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Condorcet's paradox demonstrates the surprising result that majority rule can be self-contradictory, i.e. it is possible for a majority of voters to support some outcome other than the one chosen (regardless of the outcome itself).
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The Monty Hall paradox (or equivalently three prisoners problem) demonstrates that a decision that has an intuitive fifty–fifty chance can instead have a provably different probable outcome. Another veridical paradox with a concise mathematical proof is the Birthday problem.
-
In 20th-century science, Hilbert's paradox of the Grand Hotel or the Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end.
-
The divergence of the harmonic series:
Falsidical paradox
A
falsidical paradox establishes a result that appears false and actually is false, due to a
fallacy in the demonstration. Therefore, falsidical paradoxes can be classified as
Fallacy:
-
The various invalid proof are classic examples of this, like the ones that attempt to prove that , which often rely on an inconspicuous division by zero.
-
The horse paradox, which falsely generalises from true specific statements
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Zeno's paradoxes are 'falsidical', concluding, for example, that a flying arrow never reaches its target or that a speedy runner cannot catch up to a tortoise with a small head-start.
Antinomy
An
antinomy is a paradox 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.
Sometimes described since Quine's work, a dialetheia is a paradox that is both true and false at the same time. It may be regarded as a fourth kind, or alternatively as a special case of antinomy. In logic, it is often assumed, following Aristotle, that no dialetheia exist, but they are allowed in some paraconsistent logics.
Ramsey's classification
Frank Ramsey drew a distinction between logical paradoxes and semantic paradoxes, with Russell's paradox belonging to the former category, and the
liar paradox and Grelling's paradoxes to the latter.
Ramsey introduced the by-now standard distinction between logical and semantical contradictions. Logical contradictions involve mathematical or logical terms like
class and
number, and hence show that our logic or mathematics is problematic. Semantical contradictions involve, besides purely logical terms, notions like
thought,
language, and
symbolism, which, according to Ramsey, are empirical (not formal) terms. Hence these contradictions are due to faulty ideas about thought or language, and they properly belong to
epistemology.
In medicine
A paradoxical reaction to a
drug is the opposite of what one would expect, such as becoming agitated by a
sedative or sedated by a
stimulant. Some are common and are used regularly in medicine, such as the use of stimulants such as
Adderall and
Ritalin in the treatment of attention deficit hyperactivity disorder (also known as ADHD), while others are rare and can be dangerous as they are not expected, such as severe agitation from a
benzodiazepine.
The actions of antibody on can rarely take paradoxical turns in certain ways. One example is antibody-dependent enhancement (immune enhancement) of a disease's virulence; another is the hook effect (prozone effect), of which there are several types. However, neither of these problems is common, and overall, antibodies are crucial to health, as most of the time they do their protective job quite well.
In the smoker's paradox, cigarette smoking, despite its proven harms, has a surprising inverse correlation with the epidemiological incidence of certain diseases.
See also
Notes
Bibliography
External links
