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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 seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, 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 identificatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 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. If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on. History The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter. According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let ''R'' be the set of all sets that ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grandfather Paradox
A temporal paradox, time paradox, or time travel paradox is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time and time travel. The notion of time travel to the future complies with current understanding of physics via relativistic time dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past and are often used to demonstrate its impossibility. In physics, temporal paradoxes fall into two broad groups: consistency paradoxes exemplified by the grandfather paradox; and causal loops. Other paradoxes associated with time travel are a variation of the Fermi paradox and paradoxes of free will that stem from causal loops such as Newcomb's paradox. Causal loop A causal loop is a paradox of time travel that occurs when a future event is the cause of a past event, which in turn is the cause of the future event. Both events then exist in spacetime, but their origin cannot be determined. A causal loop m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curry's Paradox
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''", requiring 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. It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem. In natural language Claims of the form "if A, then B" are called conditional claims. Curry's paradox uses a particular kind of self-referential conditional sentence, as demonstrated in this example: Even though Germany does not border China, the example sentence certainly is a natural-language sentence, and so the truth of that sentence can be analyzed. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time-travel
Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a widely recognized concept in philosophy and fiction, particularly science fiction. The idea of a time machine was popularized by H. G. Wells' 1895 novel '' The Time Machine''. It is uncertain if time travel to the past is physically possible, and such travel, if at all feasible, may give rise to questions of causality. Forward time travel, outside the usual sense of the perception of time, is an extensively observed phenomenon and well-understood within the framework of special relativity and general relativity. However, making one body advance or delay more than a few milliseconds compared to another body is not feasible with current technology. As for backward time travel, it is possible to find solutions in general relativity that al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barber Paradox
The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him.''The Philosophy of Logical Atomism'', reprinted in ''The Collected Papers of Bertrand Russell, 1914-19'', Vol 8., p. 228 The puzzle shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself, which implies that no barber exists. Paradox The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? Any answer to this question results in a contradiction: The barber cannot shave himself, as he only shaves those who do ''not'' shave themselves. Thus, if he shaves himself he ceases to be the barber specified. Conversely, if the barber does not shave himself, then he fits into the group o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-reference
Self-reference occurs in natural or formal languages when a sentence, idea 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 itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive. In logic, mathematics and computing In classical philosophy, paradoxes were created by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies 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, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thought-experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment. Johann Witt-Hansen established that Hans Christian Ørsted was the first to use the German term ' (lit. thought experiment) circa 1812. Ørsted was also the first to use the equivalent term ' in 1820. By 1883 Ernst Mach used the term ' in a different way, to denote exclusively the conduct of a experiment that would be subsequently performed as a by his students. Physical and mental experimentation could then be contrasted: Mach asked his students to provide him with explanations whenever the results from their subsequent, real, physical experiment differed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property (philosophy)
In logic and philosophy (especially metaphysics), a property is a characteristic of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property, however, differs from individual objects in that it may be instantiated, and often in more than one object. It differs from the logical/mathematical concept of class by not having any concept of extensionality, and from the philosophical concept of class in that a property is considered to be distinct from the objects which possess it. Understanding how different individual entities (or particulars) can in some sense have some of the same properties is the basis of the problem of universals. Terms and usage A property is any member of a class of entities that are capable of being attributed to objects. Terms similar to ''property'' include ''predicable'', ''attribute'', ''quality'', ''feature'', ''characteristic'', ''ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ship Of Theseus
The Ship of Theseus is a thought experiment about whether an object that has had all of its original components replaced remains the same object. According to legend, Theseus, the mythical Greek founder-king of Athens, had rescued the children of Athens from King Minos after slaying the minotaur and then escaped on a ship to Delos. Every year, the Athenians commemorated this legend by taking the ship on a pilgrimage to Delos to honor Apollo. The question was raised by ancient philosophers: After several centuries of maintenance, if every part of the Ship of Theseus had been replaced, one at a time, was it still the same ship? In contemporary philosophy, this thought experiment has applications to the philosophical study of identity over time, and has inspired a variety of proposed solutions in contemporary philosophy of mind concerned with the persistence of personal identity. History In its original formulation, the "Ship of Theseus" paradox concerns a debate over wheth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
