We will not need all details of Quantum Logic. Note that quantum logic is a sublogic of classical logic. Traditional semantics of weak logic are algebraic. A proposition is interpreted by a subset. TRUE by the whole set.
Intuitionistic logic admits a semantics given by topological spaces.
Propositions are interpreted by open sets. The complement of an open set is not necessarily open so that p v -p is not necessarily interpreted by the whole space. So p v -p is not intuitionistically valid. The negation is interpreted by the orthogonal subspaces. In quantum logic we lose the distibutivity axioms.