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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 or apparently 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Russell's Paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published 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. 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 are not members of themselves. (This set is sometimes called "the Russell set".) If ''R'' is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols: : Let R = \. Then R \in R \iff R \not \in R. Russell also showed that a version of the paradox co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. Assume that "this sentence is false" is true, then we can trust its content, which states the opposite and thus causes a contradition. Similarly, we get a contradiction when we assume the opposite. History The Epimenides paradox (c. 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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''". 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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.Russell, Bertrand (1919). "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 such 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
