Paradox in the context of Ship of Theseus

A tragedy is a genre of drama based on human suffering and, mainly, the terrible or sorrowful events that befall a main character or cast of characters. Traditionally, the intention of tragedy is to invoke an accompanying catharsis, or a "pain [that] awakens pleasure," for the audience. While many cultures have developed forms that provoke this paradoxical response, the term tragedy often refers to a specific tradition of drama that has played a unique and important role historically in the self-definition of Western civilization. That tradition has been multiple and discontinuous, yet the term has often been used to invoke a powerful effect of cultural identity and historical continuity—"the Greeks and the Elizabethans, in one cultural form; Hellenes and Christians, in a common activity," as Raymond Williams puts it.

Originating in the theatre of ancient Greece 2500 years ago, where only a fraction of the works of Aeschylus, Sophocles and Euripides survive, as well as many fragments from other poets, and the later Roman tragedies of Seneca; through its singular articulations in the works of Shakespeare, Lope de Vega, Jean Racine, and Friedrich Schiller to the more recent naturalistic tragedy of Henrik Ibsen and August Strindberg; Nurul Momen's Nemesis' tragic vengeance & Samuel Beckett's modernist meditations on death, loss and suffering; Heiner Müller postmodernist reworkings of the tragic canon, tragedy has remained an important site of cultural experimentation, negotiation, struggle, and change. A long line of philosophers—which includes Plato, Aristotle, Saint Augustine, Voltaire, Hume, Diderot, Hegel, Schopenhauer, Kierkegaard, Nietzsche, Freud, Benjamin, Camus, Lacan, and Deleuze—have analysed, speculated upon, and criticised the genre.

Heraclitus (/ˌhɛrəˈklaɪtəs/; Ancient Greek: Ἡράκλειτος Hērákleitos; fl. c. 500 BC) was an ancient Greek pre-Socratic philosopher from the city of Ephesus, which was then part of the Persian Empire. He exerts a wide influence on Western philosophy, both ancient and modern, through the works of such authors as Plato, Aristotle, Georg Wilhelm Friedrich Hegel, Friedrich Nietzsche, and Martin Heidegger.

Little is known of Heraclitus's life. He wrote a single work, of which only fragments survive. Even in ancient times, his paradoxical philosophy, appreciation for wordplay, and cryptic, oracular epigrams earned him the epithets "the dark" and "the obscure". He was considered arrogant and depressed, a misanthrope who was subject to melancholia. Consequently, he became known as "the weeping philosopher" in contrast to the ancient atomist philosopher Democritus, who was known as "the laughing philosopher".

In aesthetics, the concept of taste has been the interest of philosophers such as Plato, Hume, and Kant. It is defined by the ability to make valid judgments about an object's aesthetic value. However, these judgments are deficient in objectivity, creating the 'paradox of taste'. The term 'taste' is used because these judgments are similarly made when one physically tastes food.

Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently-innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.

The paradox is named after the logician Haskell Curry, who wrote about it in 1942. It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem.

In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.

Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy, among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series.

This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.

In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection. The difficulty is handled in axiomatic set theory by declaring that this collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.

This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and 1897). Like a number of "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.

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, which means the liar just lied. In "this sentence is a lie", the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

Assume that "this sentence is false" is true, then we can trust its content, which states the opposite and thus causes a contradiction. Similarly, we get a contradiction when we assume the opposite.

Buridan's ass is an illustration of a paradox in philosophy in the conception of free will. It refers to a hypothetical situation wherein an ass (or donkey) that is equally hungry and thirsty is placed precisely midway between a stack of hay and a pail of water. Since the paradox assumes the ass will always go to whichever is closer, it dies of both hunger and thirst since it cannot make any rational decision between the hay and water. A common variant of the paradox substitutes the hay and water for two identical piles of hay; the ass, unable to choose between the two, dies of hunger.

The paradox is named after the 14th-century French philosopher Jean Buridan, whose philosophy of moral determinism it satirizes. Although the illustration is named after Buridan, philosophers have discussed the concept before him, notably Aristotle, who put forward the example of a man equally hungry and thirsty, and Al-Ghazali, who used a man faced with the choice of equally good dates.

Quantum entanglement is the phenomenon wherein the quantum state of each particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical physics and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. This behavior gives rise to seemingly paradoxical effects: any measurement of a particle's properties results in an apparent and irreversible wave function collapse of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole.

Cleanth Brooks (/ˈkliːænθ/ KLEE-anth; October 16, 1906 – May 10, 1994) was an American literary critic and professor. He is best known for his contributions to New Criticism in the mid-20th century and for revolutionizing the teaching of poetry in American higher education. His best-known works, The Well Wrought Urn: Studies in the Structure of Poetry (1947) and Modern Poetry and the Tradition (1939), argue for the centrality of ambiguity and paradox as a way of understanding poetry. With his writing, Brooks helped to formulate formalist criticism, emphasizing "the interior life of a poem" (Leitch 2001) and codifying the principles of close reading.

Brooks was also the preeminent critic of Southern literature, writing classic texts on William Faulkner, and co-founder of the influential journal The Southern Review (Leitch 2001) with Robert Penn Warren.

The black hole information paradox is an unsolved problem in physics and a paradox that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from which nothing—not even light—can escape. In the 1970s, Stephen Hawking applied the semiclassical approach of quantum field theory in curved spacetime to such systems and found that an isolated black hole would emit a form of radiation (now called Hawking radiation in his honor). He also argued that the detailed form of the radiation would be independent of the initial state of the black hole, and depend only on its mass, electric charge and angular momentum.

The information paradox appears when one considers a process in which a black hole is formed through a physical process and then evaporates away entirely through Hawking radiation. Hawking's calculation suggests that the final state of radiation would retain information only about the total mass, electric charge and angular momentum of the initial state. Since many different states can have the same mass, charge and angular momentum, this suggests that many initial physical states could evolve into the same final state. Therefore, information about the details of the initial state would be permanently lost; however, this violates a core precept of both classical and quantum physics: that, in principle only, the state of a system at one point in time should determine its state at any other time. Specifically, in quantum mechanics the state of the system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future. In 1993, Don Page argued that if a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. This is called the Page curve.

Fisherian runaway or runaway selection is a sexual selection mechanism proposed by the mathematical biologist Ronald Fisher in the early 20th century, to account for the evolution of ostentatious male ornamentation by persistent, directional female choice. An example is the colourful and elaborate peacock plumage compared to the relatively subdued peahen plumage; the costly ornaments, notably the bird's extremely long tail, appear to be incompatible with natural selection. Fisherian runaway can be postulated to include sexually dimorphic phenotypic traits such as behavior expressed by a particular sex.

Extreme and (seemingly) maladaptive sexual dimorphism represented a paradox for evolutionary biologists from Charles Darwin's time up to the modern synthesis. Darwin attempted to resolve the paradox by assuming heredity for both the preference and the ornament, and supposed an "aesthetic sense" in higher animals, leading to powerful selection of both characteristics in subsequent generations. Fisher developed the theory further by assuming genetic correlation between the preference and the ornament, that initially the ornament signalled greater potential fitness (the likelihood of leaving more descendants), so preference for the ornament had a selective advantage. Subsequently, if strong enough, female preference for exaggerated ornamentation in mate selection could be enough to undermine natural selection even when the ornament has become non-adaptive. Over subsequent generations this could lead to runaway selection by positive feedback, and the speed with which the trait and the preference increase could (until counter-selection interferes) increase exponentially.