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Required fields*
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4If you had followed the link on wikipedia you could have found that the concept of a boolean algebra is closed related with that of a Galois field of two elements (en.wikipedia.org/wiki/GF%282%29). The symbols 0 and 1 are conventionally used to denote the additive and multiplicative identities, respectively, because the real numbers are also a field whose identities are the numbers 0 and 1.Giorgio– Giorgio2013-05-15 22:08:08 +00:00Commented May 15, 2013 at 22:08
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1@NeilG I think Giorgio is trying to say it's more than just a convention. 0 and 1 in boolean algebra are basically the same as 0 and 1 in GF(2), which behave almost the same as 0 and 1 in real numbers with regards to addition and multiplication.svick– svick2013-05-15 23:47:55 +00:00Commented May 15, 2013 at 23:47
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1@svick: No, because you can simply rename multiplication and saturating addition to be OR and AND and then flip the labels so that 0 is True and 1 is False. Giorgio is saying that it was a convention of Boolean logic, which was adopted as a convention of computer science.Neil G– Neil G2013-05-16 00:49:12 +00:00Commented May 16, 2013 at 0:49
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1@Neil G: No, you cannot flip + and * and 0 and 1 because a field requires distributivity of multiplication over addition (see en.wikipedia.org/wiki/Field_%28mathematics%29), but if you set + := AND and * := XOR, you get T XOR (T AND F) = T XOR F = T, whereas (T XOR T) AND (T XOR F) = F AND T = F. Therefore by flipping the operations and the identities you do not have a field any more. So IMO defining 0 and 1 as the identities of an appropriate field seems to capture false and true pretty faithfully.Giorgio– Giorgio2013-05-16 01:08:55 +00:00Commented May 16, 2013 at 1:08
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1@giorgio: I have edited the answer to make it obvious what is going on.Neil G– Neil G2013-05-17 05:55:15 +00:00Commented May 17, 2013 at 5:55
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