Why is Binary $1+1$ not the same as Boolean $1+1$?

Question

Accepted Answer

Binary $1+1 = 10$ (with carry), while Boolean $1+1 = 1$

Answer

They are exactly the same

Answer

Binary $1+1 = 1$, while Boolean $1+1 = 10$

Answer

Both equal 2

Why is Binary $1+1$ not the same as Boolean $1+1$ ?