A paradox is a true statement or group of statements that leads to a contradiction In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “ or a situation which defies intuition Intuition is the apparent ability to acquire knowledge without inference or the use of reason. “The word ‘intuition’ comes from the Latin word 'intueri', which is often roughly translated as meaning ‘to look inside’ or ‘to contemplate’." Intuition provides us with beliefs that we cannot necessarily justify. For this reason, it. The term is also used for an apparent contradiction In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “ that actually expresses a non-dual Nondual can refer to a belief, condition, theory, practice, or quality. The academic disciplines that study Nondualism in its spiritual permutations and cultural evocations are transpersonal psychology and the anthropology of religion. Though nondualism proper has historically been glossed as "monism" or "qualified monism" with truth (cf. kōan Kōans originate in the sayings and events in the lives of sages and legendary figures, usually those authorized to teach in a lineage that regards Bodhidharma as its ancestor. Kōans reflect the enlightened or awakened state of such persons and sometimes confound the habit of discursive thought or shock the mind into awareness.[citation needed], Catuskoti Catuṣkoṭi is a logical argument(s) of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple applications and has been important in the Dharmic traditions of Indian logic and the Buddhadharma logico-epistemological traditions, particularly those of the Madhyamaka school). Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises In logic, an argument is a set of one or more declarative sentences known as the premises along with another declarative sentence (or "proposition") known as the conclusion. Aristotle held that any logical argument could be reduced to two premises and a conclusion. Premises are sometimes left unstated in which case they are called themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “. It is also used to describe situations that are ironic Irony is a situation, literary technique, or rhetorical device, in which there is an incongruity or discordance that goes strikingly beyond the most simple and evident meaning of words or actions. Verbal and situational irony is often intentionally used as emphasis in an assertion of a truth. The ironic form of simile, irony used in sarcasm, and.^[1]

But many paradoxes, such as Curry's paradox Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. It is named after the logician Haskell Curry, do not yet have universally accepted resolutions.

Sometimes the term paradox is used for situations that are merely surprising. The birthday paradox In probability theory, the birthday problem, or birthday paradox pertains to the probability that in a set of randomly chosen people some pair of them will have the same birthday. In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will have the same birthday. Such a result is counter-, for instance, is unexpected but perfectly logical. The logician Willard V. O. Quine Willard Van Orman Quine (known to intimates as "Van"), was an American analytic philosopher and logician. From 1930 until his death 70 years later, Quine was continuously affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of mathematics, and finally as an emeritus distinguishes falsidical paradoxes, which are seemingly valid, logical demonstrations of absurdities, from veridical paradoxes, such as the birthday paradox, which are seeming absurdities that are nevertheless true.^[2] Paradoxes in economics Economics is the social science that is concerned with the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)". Current tend to be the veridical type, typically counterintuitive outcomes of economic theory, such as Simpson's paradox. In literature Literature,, is the art of written works. Literally translated, the word means acquaintance with letters (as in the Arts and Letters"). In Western culture the most basic written literary types include fiction and nonfiction a paradox can be any contradictory In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of noncontradiction states that “ or obviously untrue statement, which resolves itself upon later inspection An inspection is, most generally, an organized examination or formal evaluation exercise. It involves the measurements, tests, and gauges applied to certain characteristics in regard to an object or activity. The results are usually compared to specified requirements and standards for determining whether the item or activity is in line with these.

Logical paradox

Common themes in paradoxes include self-reference Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly; through some intermediate sentence or formula; or by means of some encoding. In philosophy, it also refers to the ability of a subject to speak of or refer to himself, herself, or itself: to have the, the infinite regress An infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, and for any proposition in the series Pn, the truth of Pn requires the support of the truth of Pn+1. There would never be adequate support for P1, because the infinite sequence needed to provide such support could not be, circular definitions A circular definition is one that assumes a prior understanding of the term being defined. By using the term being defined as a part of the definition, a circular definition provides no new or useful information; either the audience already knows the meaning of the term(s), or the definition is deficient in including the term(s) to be defined in, and confusion between different levels of abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal, "real," or "concrete" concepts.

Patrick Hughes outlines three laws of the paradox:^[3]

• Self reference - An example is "This statement is false", a form of the Liar paradox In philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent classical binary truth. The statement is referring to itself. Another example of self reference is the question of whether the barber shaves himself in the Barber paradox. One more example would be "Is the answer to this question no?" In this case, if you replied no, you would be stating that the answer is not no. If you reply yes, you are stating that it is no, because you said yes.
• Contradiction - "This statement is false"—the statement cannot be false and true at the same time.
• 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 statement below is false."

"The statement above is true."

Other paradoxes involve false statements A false statement is a statement that is either willfully or unknowingly untrue. Though "fallacy" is often used as a synonym for "false statement", this is not what is meant by "fallacy" in logic or most formal contexts or half-truths A half-truth is a deceptive statement that includes some element of truth. The statement might be partly true, the statement may be totally true but only part of the whole truth, or it may utilize some deceptive element, such as improper punctuation, or double meaning, especially if the intent is to deceive, evade, blame or misrepresent the truth and the resulting biased A cognitive bias is the human tendency to draw incorrect conclusions in certain circumstances based on cognitive factors rather than evidence. Such biases are thought to be a form of "cognitive shortcut", often based upon rules of thumb, and include errors in statistical judgment, social attribution, and memory. Cognitive biases are a assumptions.

For example, consider a situation in which a father and his son are driving down the road. The car collides with a tree and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery Surgery is a medical specialty that uses operative manual and instrumental techniques on a patient to investigate and/or treat a pathological condition such as disease or injury, to help improve bodily function or appearance, and sometimes for religious reasons. An act of performing surgery may be called a surgical procedure, operation, or simply. On entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son."

The apparent paradox is caused by a hasty generalization Hasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence. It commonly involves basing a broad conclusion upon the statistics of a survey of a small group that fails to sufficiently represent the whole population. Its opposite fallacy is called slothful induction, or; if the surgeon is the boy's father, the statement cannot be true. The paradox is resolved if it is revealed that the surgeon is a woman, the boy's mother.

Paradoxes which are not based on a hidden error generally happen at the fringes of context or language Language is a term most commonly used to refer to so called "natural languages" — the forms of communication considered peculiar to humankind. By extension the term also refers to the type of human thought process which creates and uses language. Essential to both meanings is the systematic creation, maintenance and use of systems of, and require extending the context or language to lose their paradoxical quality. Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians Logic is the study of reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic and philosophers Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the. This sentence is false is an example of the famous liar paradox In philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent classical binary truth: it is a sentence which cannot be consistently interpreted as true or false, because if it is known to be false then it is known that it must be true, and if it is known to be true then it is known that it must be false. Therefore, it can be concluded that it is unknowable. Russell's paradox In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other, which shows that the notion of the set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

Thought experiments A thought experiment is a proposal for an experiment that would test or illuminate a hypothesis, theory, or principle can also yield interesting paradoxes. The grandfather paradox The grandfather paradox is a proposed paradox of time travel first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent (The Imprudent Traveller). Nevertheless, similar (and even more mind-boggling) paradoxes had already been described, for instance by Robert A. Heinlein in "By His Bootstraps", for example, would arise if a time traveler Time travel is the concept of moving between different points in time in a manner analogous to moving between different points in space, either sending objects backwards in time to some moment before the present, or sending objects forward from the present to the future without the need to experience the intervening period (at least not at the were to kill his own grandfather before his mother or father was conceived, thereby preventing his own birth. This paradox can be resolved by postulating that time travel leads to parallel or bifurcating universes, or that only contradiction-free timelines are stable The Novikov self-consistency principle, also known as the Novikov self-consistency conjecture, is a principle developed by Dr. Igor Novikov in the mid-1980s to solve the problem of paradoxes in time travel, which is theoretically permitted in certain solutions of general relativity . Stated simply, the Novikov consistency principle asserts that if.

W. V. Quine Willard Van Orman Quine (known to intimates as "Van") was an American philosopher and logician in the analytic tradition. From 1930 until his death 70 years later, Quine was continuously affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of mathematics, and (1962) distinguished between three classes of paradoxes:

• A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, 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. The opera's official premiere was at the Fifth Avenue Theatre in New York City on 31 December 1879, where the show was well-received by both audiences and critics. Its London debut was on 3 April 1880, at the establishes the surprising fact that a twenty-one-year-old would have had only five birthdays, if he was born on a leap day February 29 in the Gregorian calendar, the most widely used today, is a date that occurs only once every four years, in years evenly divisible by 4, such as 1996, 2000, 2004, 2008, 2012 or 2016 . These are called leap years. February 29 is the 60th day of the Gregorian calendar in such a year, with 306 days remaining until the end of that year. It. Likewise, Arrow's impossibility theorem In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more discrete options, no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria. These criteria are called demonstrates difficulties in mapping voting results to the will of the people.
• 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, generally relying on a hidden division by zero In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend. Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary arithmetic, the expression has no meaning. Another example is the inductive form of the Horse paradox, falsely generalizes from true specific statements.
• A paradox which is in neither class may be an antinomy Antinomy literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology, 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 has sometimes been described since Quine's work.

• A paradox which is both true and false at the same time in the same sense is called a dialetheism Dialetheism is the view that there are true contradictions, or dialetheia. More specifically, dialetheists believe that for some sentence or proposition P, both P and its negation, not-P , are true. Dialetheism is not itself a formal logic, but to endorse dialetheism without accepting some version of paraconsistent logic is to accept everything at. In Western logics it is often assumed, following Aristotle Aristotle (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most, that no dialetheia exist, but they are sometimes accepted in Eastern traditions^[which?] and in paraconsistent logics A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic. An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.

Paradox in literature

The paradox as a literary device A literary technique, literary device, or literary motif is an identifiable rule of thumb, convention or structure that is employed in literature and storytelling has been defined as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition Exposition is one of four rhetorical modes of discourse, along with argumentation, description, and narration. The purpose of exposition is to provide some background and inform the readers about the plot, character, setting, and theme of the essay/story or motion picture or unorthodox insight. It functions as a method of literary analysis which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.^[4]

Literary or rhetorical paradoxes abound in the works of Oscar Wilde and G. K. Chesterton; other literature deals with paradox of situation. Rabelais, Cervantes, Sterne, Borges, and Chesterton are all concerned with episodes and narratives designed around paradoxes. Statements such as Wilde's "I can resist anything except temptation" and Chesterton's "spies do not look like spies" are examples of rhetorical paradox. Further back, Polonius' observation in Hamlet that "though this be madness, yet there is method in't" is a memorable third.^[4]

A taste for paradox is central to the philosophies of Laozi, Heraclitus, Meister Eckhart, Kierkegaard, Nietzsche, and Tom Robbins, among many others.

Moral paradox

In moral philosophy, paradox in a loose sense plays a role in ethics debates. For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to armed neighbors actively intending murder. If the hostile neighbors succeed, it is impossible to follow the dictum. On the other hand, to attack, fight back, or restrain them is also not usually considered loving. This might be better termed an ethical dilemma rather than a paradox in the strict sense. However, for this to be a true example of a moral paradox, it must be assumed that "loving" and restraint cannot co-exist. In reality, this situation occurs often, notably when parents punish children out of love^{[citation needed]}.

Another example is the conflict between a moral injunction and a duty that cannot be fulfilled without violating that injunction. For example, take the situation of a parent with children who must be fed (the duty), but cannot afford to do so without stealing, which would be wrong (the injunction). Such a conflict between two maxims is normally resolved through weakening one or the other of them: the need for survival is greater than the need to abide by the law. However, as maxims are added for consideration, the questions of which to weaken in the general case and by how much pose issues related to Arrow's impossibility theorem; it may not be possible to formulate a consistent system of ethics rules with a definite order of preference in the general case, a so-called "ethical calculus".

Paradoxes in a more strict sense have been relatively neglected in philosophical discussion within ethics, as compared to their role in other philosophical fields such as logic, epistemology, metaphysics, or even the philosophy of science. Important book-length discussions appear in Derek Parfit's Reasons and Persons and in Saul Smilansky's 10 Moral Paradoxes.

See also

• Dilemma
• Ethical dilemma
• Formal fallacy
• Four valued logic
• Impossible object
• List of paradoxes
• Mu (negative)
• Self-refuting ideas

Footnotes

1. ^ http://dictionary.reference.com/browse/irony
2. ^ Van Orman Quine, Willard (1966). "The Ways of Paradox". The Ways of Paradox and Other Essays. Random House. p. 5.
3. ^ Hughes, Patrick; Brecht, George (1975). Vicious Circles and Infinity - A Panoply of Paradoxes. Garden City, New York: Doubleday. Library of Congress Catalog Card Number 74-17611. ISBN 0-385-09917-7. . p. 1-8.
4. ^ ^{a b} Rescher, Nicholas (2001). Paradoxes: Their Roots, Range, and Resolution. Chicago: Open Court. ISBN 0812694368. .

External links

• Stanford Encyclopedia of Philosophy:
  • "Paradoxes and Contemporary Logic" -- by Andrea Cantini.
  • "Insolubles" -- by Paul Vincent Spade.
• Paradoxes at the Open Directory Project
• "MathPages - Zeno and the Paradox of Motion"

"Paradox". Encyclopædia Britannica (11th ed.). 1911.

Categories: Philosophical logic | Paradoxes | Concepts in logic

See Also:

What Links Here:

Recently Viewed Pages:

• Home
• Opinionated: Blogs
• Functional Contextualism: Blogs
• A Tale of Two Cities: Blogs
• Gustav Mahler: Answers
• Kupczyński's Wager: Shopping
• Debate: Shopping

Most Popular Pages:

• Functional Contextualism: Blogs
• A Tale of Two Cities: Blogs
• Opinionated: Blogs
• Kupczyński's Wager: Shopping
• Gustav Mahler: Answers
• Debate: Shopping
• Paradox: Encyclopedia

Related 'Paradox' Searches: