Duality principle in boolean algebra
"$1 + 1 = 1$" 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".
The statement is the full equation, including the = sign. 1+1 is neither true nor false: it takes the value 1, but it is not actually saying anything. Analogously, the expression "Tom has a cat" is neither true nor false (without specifying who Tom is) - it is an expression which could be true or false, depending on who we mean when we say "Tom".
On the other hand, the statement 1+1=0 is a false. Analogously, the statement "If Tom has a cat then Tom has no cats" is false, no matter who we mean when we say "Tom".
In this case, 1+1=1 is the true statement. Its dual is 0.0=0, which is also a true statement.