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seen Dec 31 '13 at 20:48

May
17
comment Why is 0 false?
@Giorgio: I corrected my construction per your comment in my last comment…
May
17
revised Why is 0 false?
added 1 characters in body
May
17
comment Why is 0 false?
Ah, sorry, I made a mistake. In the flipped field XNOR is identified with +. Good catch!
May
17
comment Why is 0 false?
@giorgio: I have edited the answer to make it obvious what is going on.
May
17
revised Why is 0 false?
added 379 characters in body
May
17
comment Why is 0 false?
You will find that both of these identifications are done over the same field and both are consistent with the rules of Boolean logic. Your note is unfortunately incorrect :)
May
17
comment Why is 0 false?
Flipped: Given a field with two elements and operations * and +, we identify True with 0 and False with 1. We identify OR with * and XOR with +.
May
17
comment Why is 0 false?
Original: Given the field with two elements and operations * and +, we identify True with 1 and False with 0. We identify AND with * and XOR with +.
May
16
comment Why is 0 false?
let us continue this discussion in chat
May
16
comment Why is 0 false?
@Girgio: That was also true in the original field. You do not identify OR and AND with addition multiplication. With the usual definition 0 corresponding to false and 1 to true, you identify XOR with addition and AND with multiplication. With the reversed definition you identify XOR with addition and OR with multiplication.
May
16
comment Why is 0 false?
@Giorgio: No, XOR is always addition, but OR becomes multiplication when 0 becomes true and 1 false.
May
16
comment Why is 0 false?
@Giorgio: In other words, fields need not have operations like conjunction and disjunction, but rather must have addition and multiplication. We identify the former with the latter.
May
16
comment Why is 0 false?
@Giorgio: A field has operations + and * with members 0 and 1. You then identify True and False with 1 and 0 (or 0 and 1) and + and * with XOR and AND (or XOR and OR). If you like you can define saturating addition for the Boolean operator OR (or AND). Parentheses indicate the reversed case.
May
16
comment Why is 0 false?
@Giorgio: I set saturating addition=And, and *=Or (not XOR). Xor is just addition. Then you will find that your relations work and all the regular properties are preserved. You can see the flipping as an application of De Morgan's identity if that is clearer for you.
May
16
comment Why is 0 false?
@shambulator: In your example, you would have had the same problem with the regular correspondence; does True or True = False? The regular definition is 1: True, 0: False, Or: Saturating addition, And: Multiplication. In the reversed definition, we flip both pairs of labels.
May
16
comment Why is 0 false?
@artem: Why not associate 1 with the empty set, and with False?
May
16
comment Why is 0 false?
@artem: "does not seem natural" …because of convention. :)
May
16
comment Why is 0 false?
@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.
May
15
comment Why is 0 false?
@Giorgio: Great, add an answer indicating that it's historical convention and I'll upvote you.
May
15
answered Why is 0 false?