
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 ...

, 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 ...

, 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 ...

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 ...

,
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 ...

.
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, ...

. 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 ...

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 ...

, 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 ...

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 ...

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, ...

: 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 ...

, 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 ...

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 ...

, 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
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...
{{Redirect category shell, 1=
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...
demonstrates difficulties in mapping voting results to the will of the people.