All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + for ., 0 for 1 and vice versa.
However when I say 1+1, whose dual statement according to the above is 0.0, I get opposite results, that is:
1 + 1 = 1
0 . 0 = 0
How should I understand this duality principle ?
Best Answer
"$1 + 1 = 1$" is a statement (a boolean statement, in fact), and indeed, $1 + 1 = 1$ happens to be a true statement.
Likewise, the entire statement "$0 \cdot 0 = 0$" is a true statement, since $0 \cdot 0$ correctly evaluates to false: and this is exactly what "$0 \cdot 0 = 0$" asserts, so it is a correct (true) statement about the falsity of $0 \cdot 0$.
The duality principle ensures that "if we exchange every symbol by its dual in a formula, we get the dual result".
More examples:
(a)
0 . 1 = 0: is a true statement asserting that "false and true evaluates to false"(b)
1 + 0 = 1: is the dual of (a): it is a true statement asserting that "true or false evaluates true."(c)
1 . 1 = 1: it is a true statement asserting that "true and true evaluates to true".(d)
0 + 0 = 0: (d) is the dual of (c): it is a true statement asserting, correctly, that "false or false evaluates to false".