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.

The statement is not 1+1 but rather 1+1=1.

What the duality principle says is that "if you exchange every symbol by its dual in a formula you get the dual result".

The difference between an expression and a statement is that statement is like an equation and expression is like variable.

"Duality of a statement is true" - What this means is that when you equate the dual of both the expressions in LHS and RHS of a statement(equation), the statement still makes sense.

Observe that in your example both 1+1 = 1 & 0.0 = 0 ,which are dual to each other, are true statements.