Duality principle in boolean algebra 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 ?
"$1 + 1 = 1$" 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".


Everywhere we see 1, change to 0.  
Everywhere we see 0, change to 1.  
Similarly, + to $\cdot$, and $\cdot$ to +.




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".
In my humble opinion, it is more helpful to describe the duality relationship in terms of variables. Example:
Expression: x + x = x
Dual of expression: x * x = x
x could be 1 or 0. In all iterations, the expressions are correct.
Good Luck!