A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In
, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, 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
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. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system. 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. 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. In common usage, the word "paradox" often refers to statements that are
or unexpected, such as "the paradox that standing is more tiring than walking".


Common themes in paradoxes include self-reference, infinite regress, circular definitions, and confusion or equivocation between different levels of
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. T ...

. Patrick Hughes outlines three laws of the paradox: ;Self-reference:An example is the statement "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. Yet another example involves the question "Is the answer to this question 'No'?" ;Contradiction:"This statement is false"; the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldn't come true. This contradicts itself because if the genie grants their wish, they did not grant their wish, and if the genie refuses to grant their wish, then he did indeed grant their wish, therefore making it impossible either to grant or not grant their wish without leading to a contradiction. ;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." Other paradoxes involve false statements and half-truths ("'impossible' is not in my vocabulary") 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 paradox, 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 Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...

, 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
ians and
philosopher A philosopher is a person who practices or investigates philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Suc ...

s. "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
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 example of the more general observation of the butterfly effect, or 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. 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: According to Quine's classification of paradoxes: * A veridical paradox produces a result that appears absurd, but is demonstrated to be true nonetheless. 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 had he been born on a leap day. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. Monty Hall paradox (or equivalently three prisoners problem) demonstrates that a decision that has an intuitive fifty–fifty chance is in fact heavily biased towards making a decision that, given the intuitive conclusion, the player would be unlikely to make. In 20th-century science, Hilbert's paradox of the Grand Hotel, Schrödinger's cat,
Wigner's friend Wigner's friend is a thought experiment in theoretical Quantum mechanics, quantum physics, first conceived by the physicist Eugene Wigner in 1961, Reprinted in and further developed by David Deutsch in 1985. The scenario involves an indirect obser ...
or Ugly duckling theorem are famously vivid examples of a theory being taken to a logical but paradoxical end. * A falsidical paradox establishes a result that not only ''appears'' false but actually ''is'' false, due to a
fallacy A fallacy is the use of Validity (logic), invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual ...
in the demonstration. The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples of this, often relying on a hidden
division by zero In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
. Another example is the inductive form of the horse paradox, which falsely generalises from true specific statements.
Zeno's paradoxes Zeno's paradoxes are a set of philosophy, philosophical problems generally thought to have been devised by Magna Graecia, Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's sens ...
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. Therefore, falsidical paradoxes can be classified as fallacious arguments. * A paradox that is in neither class may be an antinomy, 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. A fourth kind, which may be alternatively interpreted as a special case of the third kind, has sometimes been described since Quine's work: * A paradox that is both true and false at the same time and in the same sense is called a '' dialetheia''. In Western logics, it is often assumed, following
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
, that no ''dialetheia'' exist, but they are sometimes accepted in Eastern traditions (e.g. in the Mohists,The Logicians (
Warring States period The Warring States period () was an era in History of China#Ancient China, ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period and concluded ...

"Miscellaneous paradoxes"
''Stanford Encyclopedia of Philosophy''
the Gongsun Longzi,Graham, Angus Charles. (1990). and in Zen) and in paraconsistent logics. It would be mere equivocation or a matter of degree, for example, to both affirm and deny that "John is here" when John is halfway through the door, but it is self-contradictory simultaneously to affirm and deny the event.

Ramsey's classification

Frank Ramsey drew a distinction between logical paradoxes and semantic paradoxes, with 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 Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epis ...

In philosophy

A taste for paradox is central to the philosophies of
Laozi Laozi (), also known by #Names, numerous other names, was a Chinese legend, semilegendary ancient China, ancient Chinese Taoism, Taoist Chinese philosophy, philosopher. Laozi ( zh, ) is a Chinese honorific, generally translated as "the Old Ma ...
Zeno of Elea Zeno of Elea (; grc, wikt:Ζήνων, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He i ...
, Zhuangzi,
Heraclitus Heraclitus of Ephesus (; grc-gre, wikt:Ἡράκλειτος, Ἡράκλειτος , "Glory of Hera"; ) was an Ancient Greece, ancient Greek Pre-Socratic philosophy, pre-Socratic philosopher from the city of Ephesus, which was then part of th ...
, Bhartrhari,
Meister Eckhart Eckhart von Hochheim ( – ), commonly known as Meister Eckhart, Master EckhartPhilosophical Fragments'' that:
But one must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to will the collision, although in one way or another the collision must become its downfall. This, then, is the ultimate paradox of thought: to want to discover something that thought itself cannot think.

In medicine

paradoxical reaction A paradoxical reaction (or paradoxical effect) is an effect of a chemical substance, such as a medical drug, that is opposite to what would usually be expected. An example of a paradoxical reaction is pain caused by a pain relief medication. Parado ...
to a
drug A drug is any chemical substance that causes a change in an organism's physiology or psychology when consumed. Drugs are typically distinguished from food and substances that provide nutritional support. Consumption of drugs can be via inh ...
is the opposite of what one would expect, such as becoming agitated by a
sedative A sedative or tranquilliser is a substance that induces sedation by reducing irritability or Psychomotor agitation, excitement. They are Central nervous system, CNS depressants and interact with brain activity causing its deceleration. Various k ...
or sedated by a
stimulant Stimulants (also often referred to as psychostimulants or colloquially as uppers) is an overarching term that covers many drugs including those that increase activity of the central nervous system and the body, drugs that are pleasurable and inv ...
. 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 Attention deficit hyperactivity disorder (ADHD) is a neurodevelopmental disorder characterised by excessive amounts of inattention, hyperactivity, and impulsivity that are pervasive, impairing in multiple contexts, and otherwise Development ...
(also known as ADHD), while others are rare and can be dangerous as they are not expected, such as severe agitation from a
benzodiazepine Benzodiazepines (BZD, BDZ, BZs), sometimes called "benzos", are a class of depressant, depressant drugs whose core chemical structure is the fusion of a benzene ring and a diazepine ring. They are prescribed to treat conditions such as anxiety d ...
. The actions of
antibodies An antibody (Ab), also known as an immunoglobulin (Ig), is a large, Y-shaped protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residue (biochemistry), residues. Proteins perform ...
antigen In immunology, an antigen (Ag) is a molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. ...
s 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




* Frode Alfson Bjørdal, ''Librationist Closures of the Paradoxes'', Logic and Logical Philosophy, Vol. 21 No. 4 (2012), pp. 323–361. * Mark Sainsbury, 1988, Paradoxes, Cambridge: Cambridge University Press * William Poundstone, 1989, Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge, Anchor * Roy Sorensen, 2005, A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind, Oxford University Press * Patrick Hughes, 2011, Paradoxymoron: Foolish Wisdom in Words and Pictures, Reverspective

External links

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