Birkhoff-von Neumann quantum logic

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Birkhoff-von Neumann quantum logic, Not really familiar with it can anyone tell me anything about it?
Birkhoff-von Neumann quantum logic, the basic propositions are identified with the closed subspaces of a Hilbert space. There are four basic propositions in this system, each corresponding to a closed subspace of the Hilbert space:

1. A proposition that corresponds to the whole Hilbert space. This proposition is always true.
2. A proposition that corresponds to the empty subspace. This proposition is always false.
3. A proposition that corresponds to a non-empty, proper subspace of the Hilbert space. These propositions are neither true nor false in general.
4. A proposition that corresponds to the orthogonal complement of a non-empty, proper subspace of the Hilbert space. These propositions are also neither true nor false in general.

This logic differs from classical logic because, the truth values of the proposition aren't fixed, they can change based on the state of the system. This idea is known as superposition principle, in which a system can exist in multiple states simultaneously, and the measurement collapses this superposition state into one value.