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 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]

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.

Common elements

Self-reference, contradiction and infinite regress are core elements of many paradoxes.^[15] 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".^[16] Another example occurs in the barber paradox, which poses the question of whether a barber who shaves all and only those men 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.^[15] The liar paradox, "This statement is false," exhibits contradiction because the statement cannot be false and true at the same time.^[17] 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.^[15]

Vicious circularity, or infinite regress

Another core aspect of paradoxes is non-terminating recursion, in the form of circular reasoning or infinite regress.^[15] 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.^[15][18]

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 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 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 logicians and philosophers. "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.^[10]

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.^[19] 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:^[20][21]

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 who was born on a leap day would have had only five birthdays. The same is true for a twenty-three-year-old, unless either live through a century boundary, in which case they might have had only four birthdays. Likewise, Arrow's impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. 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 paradox. In 20th-century science, Hilbert's paradox of the Grand Hotel, Schrödinger's cat, Wigner's friend or the 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. Another example is the inductive form of the horse paradox, which falsely generalises from true specific statements. 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. 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, that no dialetheia exist, but they are sometimes accepted in Eastern traditions (e.g. in the Mohists,^[22] the Gongsun Longzi,^[23] and in Zen^[24]) and in some 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 Russell's paradox belonging to the former category, and the liar paradox and Grelling's paradoxes to the latter.^[25] 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.^[26]

In philosophy

A taste for paradox is central to the philosophies of Laozi, Zeno of Elea, Zhuangzi, Heraclitus, Bhartrhari, Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes in the Philosophical 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.^[27]

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.^[28]

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.