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Bertrand Russell: Logic
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Bertrand Russell is a slight, dark-haired man, with a prominent forehead, bright eyes, strong features except for a retreating chin, nervous hands and alert quick movements. In manner of dress and outward bearing he is most carefully trimmed, conventionally correct and punctiliously polite, and in speech he has an almost affectedly clear enunciation of words and preciseness of expression. In morals he is a puritan; in personal habits almost an ascetic, except that he lives for efficiency and therefore expects to be kept in the best physical condition. But intellectually he is audacious - an iconoclast, detesting religions or social conventions, suspecting sentiment, believing in the 'order of thought' and the order of things, in logic and in science. He is a delightful talker, especially in general conversation. He dislikes bores and hates any kind of self-seeking selfishness or coarse-grainedness.
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At Cambridge Russell became interested in the relatively new discipline of mathematical logic in which he was to be a pioneer. With Guiseppe Peano he was one of the few to recognize the genius of Gottlob Frege and his new system of logic. In 1902 he wrote to Frege, presenting what is now known as Russell's paradox, and asking how Frege's system would deal with it. (Unfortunately, as Frege acknowledged, the system could not accommodate it.) The paradox is one of the paradoxes of set theory and rests on the (then ill-defined) notion of a set. Some sets are members of themselves (the set of all sets is an example because it is itself a set; the set of cats is not an example, as it is not itself a cat). Consider the set of all sets that are not members of themselves: is it a member of itself? If it is, it is not and vice versa.
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Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox. Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves "R." If R is a member of itself, then by definition it must not be a member of itself.
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, Russell became aware of Cantor's proof that there was no greatest cardinal number, which Russell believed was mistaken. The Cantor Paradox in turn was shown (for example by Crossley) to be a special case of the Russell Paradox. This caused Russell to analyze classes, for it was known that given any number of elements, the number of classes they result in is greater than their number. This in turn led to the discovery of a very interesting class - namely, the class of all classes. It contains two kinds of classes: those classes that contain themselves, and those that do not. Consideration of this class led him to find a fatal flaw in the so-called principle of comprehension, which had been taken for granted by logicians of the time.
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Russell rejects substances and essences in the traditional sense. But he admits six sorts of beings or substances, or substance substitutes: (1) All entities, including both being and existence, have timeless being in 1903. (2) Universals' have being in 1912. (3) Being is general timelessness in 1914. (4) Being is logical atoms in 1918. (5) Being is object words in 1940.
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Russell frequently claimed that he was more convinced of his method of doing philosophy, the method of analysis, than of his philosophical conclusions. Science, of course, was one of the principal components of analysis, along with logic and mathematics. While Russell was a believer in the scientific method, knowledge derived from empirical research that is verified through repeated testing, he believed that science reaches only tentative answers, and that scientific progress is piecemeal, and attempts to find organic unities were largely futile.[43] Indeed, he believed the same was true of philosophy. Another founder of modern philosophy of science, Ernst Mach, placed less reliance on method, per se, for he believed that any method that produced predictable results was satisfactory and that the principal role of the scientist was to make successful predictions.[44] While Russell would doubtless agree with this as a practical matter, he believed that the ultimate objective of both science and philosophy was to understand reality, not simply to make predictions.
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