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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. In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting
critical thinking Critical thinking is the analysis of facts to form a judgment. The subject is complex; several different Critical thinking#Definitions, definitions exist, which generally include the rational, skepticism, skeptical, and unbiased analysis or eval ...
, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
of mathematics and logic to be re-examined. One example is
Russell's paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
on the identification of sets with
properties Property (''latin: Res Privata'') in the abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether phys ...
or predicates were flawed. Others, such as
Curry's paradox Curry's paradox is a paradox A paradox, also known as an antinomy, 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, ...
, cannot be easily resolved by making foundational changes in a logical system. Examples outside logic include the
ship of Theseus In the metaphysics Metaphysics is the branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosoph ...
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
ironic Irony (), in its broadest sense, is a rhetorical device, literary technique, or event in which what on the surface appears to be the case or to be expected differs radically from what is actually the case. Irony can be categorized into differ ...

ironic
or unexpected, such as "the paradox that standing is more tiring than walking".


Introduction

Common themes in paradoxes include
self-reference Self-reference occurs in natural language, natural or formal languages when a Sentence (linguistics), sentence, idea or Well-formed formula, formula refers to itself. The reference may be expressed either directly—through some intermediate s ...
,
infinite regress An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified becaus ...

infinite regress
,
circular definition A circular definition is a definition A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with s ...
s, and confusion or equivocation between different levels of
abstraction Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...

abstraction
. 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 In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, ...

liar paradox
. The statement is referring to itself. Another example of self-reference is the question of whether the
barber A barber is a person whose occupation is mainly to cut, dress, groom, style and shave men's and boys' hair or beards. A barber's place of work is known as a "barbershop" or a "barber's". Barbershops are also places of social interaction and publi ...

barber
shaves himself in the
barber paradox The barber paradox is a puzzle A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logical way, in order to arrive at the correct o ...
. 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 his wish, he did not grant his wish, and if he refuses to grant his wish, then he did indeed grant his wish, therefore making it impossible either to grant or not grant his 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-truthA half-truth is a deceptive statement that includes some element of truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things ...
s ("''impossible'' is not in my vocabulary") or rely on a hasty assumption. (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 if the boy's mother is a surgeon.) Paradoxes that are not based on a hidden error generally occur at the fringes of context or
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...

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
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
ians and
philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ...

philosopher
s. "This sentence is false" is an example of the well-known
liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, ...

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 In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...

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-experiment A thought experiment is a hypothetical situation in which a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method The scientific me ...
s can also yield interesting paradoxes. The
grandfather paradox The grandfather paradox is a paradox 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 seemi ...
, for example, would arise if a
time-travel Time travel is the concept of movement between certain points in time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparently irreversible process, irreversible succession f ...

time-travel
er 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 In chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from ...

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 ''The Pirates of Penzance; or, The Slave of Duty'' is a comic opera in two acts, with music by Arthur Sullivan and libretto by W. S. Gilbert, W. S. Gilbert. The opera's official premiere was at the Fifth Avenue Theatre in New York Cit ...
'' establishes the surprising fact that a twenty-one-year-old would have had only five birthdays had he been born on a
leap day February 29, also known as leap day or leap year day, is a date added to leap year A leap year (also known as an intercalary year or wikt:bissextile, bissextile year) is a calendar year that contains an additional day (or, in the case of ...
. Likewise,
Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
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 In quantum mechanics, Schrödinger's cat is a thought experiment that illustrates a paradox of quantum superposition. In the thought experiment, a hypothetical cat may be considered simultaneously both alive and dead as a result of being linked to ...
,
Wigner's friend Wigner's friend is a thought experiment A thought experiment is a hypothetical situation in which a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the ...
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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. 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 Antinomy (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as o ...
, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the
Grelling–Nelson paradox The Grelling–Nelson paradox is an antinomy, or a semantic Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of meaning, reference, or truth. The term can be used to refer to subfields of several distinc ...
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 A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

Aristotle
, that no ''dialetheia'' exist, but they are sometimes accepted in Eastern traditions (e.g. in the
Mohists Mohism or Moism () was an ancient Chinese philosophy Chinese philosophy originates in the Spring and Autumn period () and Warring States period (), during a period known as the "Hundred Schools of Thought", which was character ...
,The
Logicians Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, account, re ...
(
Warring States period The Warring States period () was an era in ancient Chinese history characterized by warfare, as well as bureaucratic and military reforms and consolidation. It followed the Spring and Autumn period#REDIRECT Spring and Autumn period The Spri ...

"Miscellaneous paradoxes"
''Stanford Encyclopedia of Philosophy''
the
Gongsun Longzi Gongsun Long (, BCLiu 2004, p. 336), courtesy name Zibing (子秉), was a Chinese philosopher and writer who was a member of the School of Names (Logicians) of ancient Chinese philosophy Chinese philosophy originates in the S ...
,Graham, Angus Charles. (1990). and in
Zen Zen ( zh, t=禪, p=Chán; ja, text= 禅, translit=zen; ko, text=선, translit=Seon; vi, text=Thiền) is a school A school is an educational institution An educational institution is a place where people of different ages gai ...

Zen
) and in
paraconsistent logic A paraconsistent logic is an attempt at a logical system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical cal ...
s. 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
Russell’s paradox In the foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theori ...
belonging to the former category, and the
liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, ...

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 (; ) is the Outline of philosophy, branch of philosophy concerned with knowledge. Epistemologists study the nature, origin, and scope of knowledge, epistemic Justification (epistemology), justification, the Reason, rationality o ...

epistemology
.


In philosophy

A taste for paradox is central to the philosophies of
Laozi Lao Tzu (),"Lao Zi"
''
,
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Pre-Socratic philosophy is ancient Greek philosophy Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were st ...

Zeno of Elea
,
ZhuangziZhuangzi may refer to: *Zhuangzi (book), ''Zhuangzi'' (book) (莊子), an ancient Chinese collection of anecdotes and fables, one of the foundational texts of Daoism **Zhuang Zhou (莊周), the historical figure known as "Master Zhuang" ("Zhuangzi") ...
,
Heraclitus Heraclitus of Ephesus (; grc-gre, Ἡράκλειτος ; , ) was an Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), ...

Heraclitus
, Bhartrhari,
Meister Eckhart Eckhart von Hochheim ( – ), commonly known as Meister Eckhart or Eckehart, was a German Catholic theology, theologian, philosopher and German mysticism, mystic, born near Gotha (town), Gotha in the Thuringia, Landgraviate of Thuringia (now ce ...
,
Hegel Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citi ...
, Kierkegaard,
Nietzsche Friedrich Wilhelm Nietzsche (; or ; 15 October 1844 – 25 August 1900) was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as thos ...

Nietzsche
, and
G.K. Chesterton Gilbert Keith Chesterton (29 May 1874 – 14 June 1936) was an English writer, philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, mean ...
, among many others. Søren Kierkegaard, for example, writes in the ''
Philosophical Fragments ''Philosophical Fragments'' (Danish language, Danish title: ) is a Christian philosophical work written by Denmark, Danish philosopher Søren Kierkegaard in 1844. It was the second of three works written under the pseudonym ''Johannes Climacus''; ...
'' 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

A
paradoxical reactionA paradoxical reaction or paradoxical effect is an effect of a chemical substance, typically a medical drug, that is opposite to what would usually be expected. An example of a paradoxical reaction is pain Pain is a distressing feeling often cause ...
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 insuffl ...

drug
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 ki ...
or sedated by a
stimulant Stimulants (also often referred to as psychostimulants or colloquially as uppers) is an overarching term that covers many drug Uncoated tablets, consisting of about 90% acetylsalicylic acid, along with a minor amount of inert fillers and b ...
. Some are common and are used regularly in medicine, such as the use of stimulants such as
Adderall Adderall and Mydayis are trade names for a combination drug containing four salts of amphetamine Amphetamine (contracted from alpha- methylphenethylamine) is a central nervous system (CNS) stimulant Stimulants (also often referred t ...

Adderall
and
Ritalin Methylphenidate, abbreviated MP or MPH, sold under the trade name Ritalin, among others, is a stimulant drug used to treat attention-deficit/hyperactivity disorder (ADHD) and narcolepsy. It is a first line medication for ADHD. It may be taken ...

Ritalin
in the treatment of
attention deficit hyperactivity disorder Attention deficit hyperactivity disorder (ADHD) is a neurodevelopmental disorder Neurodevelopmental disorders are a group of disorders that affect the development of the nervous system, leading to abnormal brain function which may affect ...
(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 psychoactive drugs whose core chemical structure is the fusion of a benzene ring and a diazepine ring. The first such drug, chlordiazepoxide (Librium), was role of chance ...

benzodiazepine
. 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


References


Notes


Bibliography

* , ''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

* * * * * * {{Authority control Concepts in epistemology Concepts in logic Concepts in metaphysics Critical thinking Philosophical logic Thought