[supernewton]

The thing you're missing is that at no point is it assumed that there are exactly two elements in a boolean algebra. In fact you can have a boolean algebra with four elements (see https://en.wikipedia.org/wiki/Boolean_algebra_(structure)).

[coonjecture]

It seems the author is using the word logic, so logic boolean algebra suggests the classical case. Perhaps what is not trivial is that one can use that rule to deduce the other axioms. So that is not the theorem what is important but that one can prove any tautology using that simple axiom.