Paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation.[1][2] 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.[3][4] A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time.[5][6][7] They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".[8]
For other uses, see Paradox (disambiguation).
In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking,[9] while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed.[10][11] Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.[12]
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.[13] Paradoxes can also take the form of images or other media. For example, M.C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly.[14]
Informally, the term paradox is often used to describe a counterintuitive result.
W. V. O. Quine (1962) distinguished between three classes of paradoxes:[20][21]
According to Quine's classification of paradoxes:
A fourth kind, which may be alternatively interpreted as a special case of the third kind, has sometimes been described since Quine's work:
Ramsey's classification[edit]
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.[22] 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.[23]
In medicine[edit]
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.[25]
The actions of antibodies on antigens 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.